Thesis Reference

Thesis

On the device-independent approach to quantum physics : advances in quantum nonlocality and multipartite entanglement detection

BANCAL, Jean-Daniel

Abstract

La physique quantique a participé au développement de nombreux domaines, qu'il s'agisse de l'informatique en permettant de traiter l'information électroniquement à l'aide de transistors, des communications, rendues possibles à grande échelle par la lumière laser guidée dans des fibre optique, ou bien de la médecine par les méthodes d'imagerie par résonnance magnétique nucléaire. Qui eut cru que l'hypothèse quantique formulée par Max Planck à l'aube du 20ème siècle aurait, de fil en aiguille, de telles répercussions? Malgré cela, la physique quantique reste encore passablement mystérieuse. L'un de ses aspects les plus intriguants étant sans doute son charactère nonlocal, c'est-à-dire sa capacité à violer des inégalités de Bell à l'aide de systèmes isolés les uns des autres.

Reference

BANCAL, Jean-Daniel. On the device-independent approach to quantum physics : advances in quantum nonlocality and multipartite entanglement detection. Thèse de doctorat : Univ. Genève, 2012, no. Sc. 4419

URN : urn:nbn:ch:unige-217102 DOI : 10.13097/archive-ouverte/unige:21710

Available at: http://archive-ouverte.unige.ch/unige:21710

Disclaimer: layout of this document may differ from the published version.

1 / 1 UNIVERSITE´ DE GENEVE` FACULTE´ DES SCIENCES Groupe de Physique Appliqu´ee - Optique Prof. N. Gisin

ON THE DEVICE-INDEPENDENT APPROACH TO QUANTUM PHYSICS

ADVANCES IN QUANTUM NONLOCALITY AND MULTIPARTITE ENTANGLEMENT DETECTION

THESE`

présentéeàla Facultédes Sciences de l’Universitéde Genève pour obtenir le grade de Docteur èsSciences, mention Physique

par

Jean-Daniel Bancal de Meyrin (GE)

Th`eseN◦ 4419

GENEVE` Atelier d’impression ReproMail 2012

R´esum´e

La physique quantique a participéau développement de nombreux domaines, qu’il s’agisse de l’informatique en permettant de traiter l’information électroniquement àl’aide de transistors, des communications, rendues possibles àgrande échelle par la lumière laser guidéedans des fibre optique, ou bien de la médecine par les méth- odes d’imagerie par résonnance magnétique nucléaire. Qui eut cru que l’hypothèse quantique formuléepar Max Planck àl’aube du 20ème siècle aurait, de fil en aiguille, de telles répercussions?

Malgrécela, la physique quantique reste encore passablement mystérieuse. L’un de ses aspects les plus intriguants étant sans doute son charactère nonlocal, c’est- à-dire sa capacitéàvioler des inégalitésde Bell àl’aide de systèmes isolésles uns des autres. Une telle violation suggère en effet que ces systèmes sont causalement reliés,ce qui semble contredire le fait qu’ils soient mutuellement séparés.

La manière directe avec laquelle la nonlocalitéquantique apparaˆıt dans des résultats expérimentaux lui permet d’être testéeen faisant appel àun minimum d’hypothèses. En particulier, aucune erreur de calibration sur des appareils de mesure individuels ne peut remettre en cause le résultat d’une telle expérience. Cette robustesse face aux erreurs d’implémentation, qui sont inhérentes àtoute manipulation expérimentale, ouvre la voie vers de nouvelles approches expérimen- tales. En effet elle montre qu’il est possible de répondre àcertaines questions en faisant appel àvirtuellement aucune hypothèse du moment que les systèmes mesurés sont suffisamment séparésles uns des autres.

A quelles questions peut-on répondre de cette fa¸con-là? A quoi peut servir la violation d’une inégalitéde Bell en général? Mais aussi, comment la nature s’y prend-elle pour violer une inégalitéde Bell? Et quelles sont les limites de la nonlocalitéquantique? Voici quelques-unes des questions abordées par cette thèse.

iii

Contents

Introduction 1

1 Bell tests in bipartite scenarios 5 1.1 No-signalling and local causality ...... 5 1.1.1 Local correlations ...... 5 1.1.2 No-signalling correlations ...... 7 1.1.3 Geometrical representation ...... 7 1.1.4 Experimental loopholes ...... 8 1.2 Bell test between an atom and an optical mode ...... 9 1.2.1 Creating atom-photon entanglement ...... 9 1.2.2 CHSH violation ...... 10 1.2.3 Space-like separation ...... 11 1.2.4 Conclusion ...... 11 1.3 Bell test with multiple pairs ...... 12 1.3.1 Two sources ...... 12 1.3.2 Noise model ...... 13 1.3.3 Bell violation ...... 14 1.4 Experimental violation of Bell inequalities with a commercial source of entanglement ...... 14 1.4.1 Experimental setup ...... 14 1.4.2 Test of several Bell inequalities ...... 15 1.4.3 Chained Bell inequality ...... 15 1.4.4 Conclusion ...... 17

2 Nonlocality with three and more parties 19 2.1 Deﬁning genuine multipartite nonlocality ...... 19 2.2 Multipartite Bell-like inequalities ...... 21 2.2.1 A general structure for (n, m, k) scenarios ...... 21 2.2.2 Recursion relation ...... 22 2.3 Nonlocality from local marginals ...... 24 2.3.1 An inequality ...... 25 2.3.2 Conclusion ...... 25 2.4 Tripartite nonlocal boxes ...... 26 2.4.1 The tripartite nosignalling polytope ...... 26 2.4.2 Conclusion ...... 26 2.5 A tight limit on quantum nonlocality ...... 26 2.5.1 Can you guess your neighbour’s input (GYNI)? ...... 26 2.5.2 Outlook ...... 27 2.6 Simulating projective measurements on the GHZ state ...... 28

v CONTENTS

2.6.1 Nonlocal resources ...... 28 2.6.2 Simulation ...... 28 2.6.3 Conclusion ...... 29

3 Device-independent entanglement detection 31 3.1 Imperfect measurements ...... 31 3.1.1 Eﬀects of systematic errors on tomography ...... 32 3.1.2 Eﬀects of systematic errors on entanglement witnesses ...... 32 3.2 Witnesses insensitive to systematic errors? ...... 33 3.2.1 Device-independent witnesses for genuine tripartite entanglement . 34 3.2.2 A witness for genuine multipartite entanglement ...... 34 3.3 Experimental demonstration ...... 35 3.3.1 Experimental setup and procedure ...... 35 3.3.2 Addressing errors ...... 36 3.3.3 Experimental results ...... 37 3.4 Conclusion ...... 37

4 Quantum information put into practice 39 4.1 Memoryless attack on the 6-state QKD protocol ...... 39 4.1.1 The 6-state protocol ...... 39 4.1.2 Secret key rate ...... 40 4.1.3 Discussion ...... 41 4.2 Private database queries ...... 42 4.2.1 Sketch of the protocol ...... 42 4.2.2 Discussion ...... 43

5 Finite-speed hidden influences 45 5.1 Finite-speed propagation and v-causal theories ...... 45 5.1.1 v-causal models and experimental limitations ...... 46 5.1.2 Influences without communication? ...... 47 5.2 The hidden influence polytope ...... 47 5.2.1 Quantum violation and faster-than-light communication ...... 49 5.3 Experimental perspectives ...... 50 5.4 Conclusion ...... 51

Conclusion and outlook 53

Acknowledgements 55

Bibliography 57

A Polytopes 63 A.1 Deﬁnition and terminology ...... 63 A.2 Some operations on polytopes ...... 64 A.2.1 Projection ...... 64 A.2.2 Slice ...... 66 A.2.3 Another tasks : ﬁnding facets lying under an inequality ...... 66

B Memoryless attack on the 6-state protocol – proof 67

Papers 71

vi Introduction

From its beginning in the 1920’s quantum physics has challenged our understanding of the world. Particles that could be conceived previously as points turned out to be provided with a wave evolving in time according to a law of motion. This conceptual change allowed for previously unsuspected phenomenons to be observed, like for instance the interference of a molecule with itself demonstrated several times experimentally (e.g. with C60 molecules in [1]). If the quantum theory is recognized for its extraordinary predictive power, the picture of the world that it suggests is not the subject of a common agreement. For instance, the question of whether the wavefunction ψ , a fundamental ingredient of the theory, should | i be understood as a proper physical object, i.e. a physical property of every quantum system, or rather as a tool from the theory which is only useful to predict the evolution of physically relevant objects, is still an active subject of research [2, 3, 4, 5]. One could argue that questions about the possible interpretation of the elements of the quantum theory are of secondary importance, provided that predictions match experimental results. But that would be putting aside the possibility for such considerations to reveal fundamental properties of nature. For instance, the quantum measurement process is commonly understood as an instantaneous change of the wavefunction throughout all space. If this process is indeed instantaneous, and if the wavefunction is a physical object, then measurement of a quantum system is a strongly nonlocal phenomenon, and one should expect physical quantities to be the subject of such instantaneous change at a distance. On the other hand, if the wavefunction can be understood as a tool of the theory, without a concrete physical counterpart, then the nonlocal character of this process might just be an artifact of the theory, without direct incidence on physically relevant quantities. Since a proper understanding of the elements of the quantum theory seems diﬃcult to reach without invoking arguable choices of additional assumptions, and since only properties of nature that can have a measurable impact are worth discussing anyway, we ask what properties of quantum physics can be detected directly from experimental data, without relying on more assumptions than the ones needed in order to make sense out of these data. In this way, we hope to be able to explore properties of quantum physics more directly. Moreover, we can expect to be able to check these properties on nature directly, because we follow an approach which fundamentally relies on experimental results. Following the seminal work of John Bell [6], we consider experimental setups characterized by a number n of identiﬁable systems, which can be measured in m possible ways, yielding each time one out of k possible values. The results of such a Bell-type experiment can be characterized by conditional probability distribution of the form P (ab xy) (here for | a scenario with n = 2 parties), which we refer for short as correlations. These correlations describe how often the results a and b are observed on two separated systems whenever measurement x and y are performed on them, respectively.

1 Introduction

An important property of correlations is that they are always accessible in principle: by sufficiently separating the systems under study, and performing enough measurements, the raw data produced during an experiment allows one to evaluate P (ab xy) directly. | Namely, if the measurements x and y are performed by the two parties Alice and Bob N times, leading to N(ab xy) N observations of the outcomes a and b, then xy | ≤ xy N(ab xy) P (ab xy) = lim | . (0.0.1) | Nxy N →∞ xy Statistical analysis can be used to infer the value of P (ab xy) with high probability when | the number of measurements performed is finite (N < ). xy ∞ Moreover the evaluation of the correlations P (ab xy) requires no knowledge about the | process creating the experimental results. It thus does not rely on any interpretation of the elements of a theory susceptible of describing these processes. Rather, all which is needed in order to make sense out of correlations is well-defined systems and indices to identify the inputs and outputs of the experiment in a reliable fashion. Since this description of an experiment requires no precise description of the working of the measurement devices we refer to it as device-independent. This makes correlations well adapted for our purpose. They are thus the central object of interest in this thesis. Note that apart from allowing to study nature with a minimum number of assumption, the device-independent hypotheses are also naturally adapted to the study of problems involving untrusted devices, such as quantum key distribution (QKD) [7], or to derive conclusions that are particularly robust with respect to practical uncertainties. While we present a possible application of the second kind in chapter 3, a significant part of this thesis is devoted to the study of correlations in multipartite scenarios.

Outline The content of this thesis is organized as follows. First, we recall Bell’s notion of local causality. This is useful for the rest of the thesis since most of it relates in some way to Bell inequalities. This leads us to discuss several studies on nonlocality in bipartite scenarios, including a proposal for a loophole-free Bell experiment combining measurement on an atom and a photon, and the analysis of Bell tests in presence of multipairs. We conclude this section with an experimental demonstration of nonlocal correlations conducted with a commercially-available entanglement source. The second part of this thesis discusses the notion of nonlocality in scenarios involving three or more parties. We discuss the definition of genuine multipartite nonlocality, and present a family of inequalities that can detect multipartite nonlocal correlations. In this section we also study more specifically the structure of multipartite correlations by analysing the set of tripartite no-signalling correlations and questioning the constraints that relate different marginals of a single multipartite system. Finally, we provide a bound on the nonlocality of quantum correlations, and a model that simulates measurements on a GHZ state with the help of bipartite nonlocal boxes. The next section is devoted to the detection of genuine multipartite entanglement in a device-independent manner. We examine in which case genuine multipartite entanglement can be witnessed based solely on the observation of some correlations. This allows one to witness multipartite entanglement, a property of quantum physics, in a way that is particularly resistant to practical imperfections. These results are illustrated experimentally.

2 The fourth section reminds us that quantum physics allows one to perform some tasks which would be impossible or harder otherwise. It contains the analysis of a specific attack on the 6-state QKD protocol, as well as a proposal for practical secure database queries. Finally, we close this thesis by considering the possibility of relaxing Bell’s condition of local causality to recover a causal explanation of quantum nonlocal correlations with a sense of proportion. We show that this is not possible without allowing for faster-than- light communication. Apart from the beginning of the first chapter, which contains several definitions used in the rest of the text, the different parts of this thesis can be read independently of each other. While the main text is meant to be concise, all complementary information should be found in the appendices and attached papers.

3 Introduction

4 Chapter 1

Bell tests in bipartite scenarios

1.1 No-signalling and local causality

From our everyday experience, we know that any transmission of information (i.e. communication) must be carried by a physical support: in order to let someone know about something we can say it to him, write him an SMS or a letter about it, etc, i.e. use either acoustic waves, electrons, or paper to carry this information to our friend. This idea can be expressed in the following principle: No-signalling principle. Any transmission of information must be carried by a physical support leaving the emitter after the message is chosen. This principle is satisfied by several if not all physical theories, including classical and quantum physics [8, 9]. In fact, the no-signalling principle is tightly related to quantum physics, since it can be seen restricts both the possibility of cloning quantum systems and the possibility of discriminating between quantum states, two peculiarities of quantum physics [10, 11]. In practice, many physical supports are available for communication in nature, like the ones mentioned above, but since the advent of special relativity, it is generally admitted that none of them can carry information faster than light in vacuum. The existence of an upperbound on the speed of all communication led Bell to enunciate the principle of local causality: “The direct causes (and effects) of events are near by, and even the indirect causes (and effects) are no further away than permitted by the velocity of light.” J. S. Bell [12] m In other words, the speed of light c = 29907920458 s , taken as an upper bound on any communication speed, naturally defines the limit between events in space-time which can have a direct causal relation with each other, and events which cannot. More precisely, for every event K the set of events that can be influenced by a decision taken at K is defined by its future light-cone. Similarly, all events that can influence a process taking place at K are contained in its past light-cone (c.f. Figure 1.1a).

1.1.1 Local correlations It is of particular interest to ask which correlations P (ab xy) can be observed, according | to the principle of local causality, in the situation of Figure 1.1b in which two measurements are performed in a space-like manner. Indeed, in this case, no information can be

5 Bell tests in bipartite scenarios

a) time future light-cone b) time

K AB

Λ1 Λ2 Λ3 past light-cone space space

Figure 1.1: a) Space-time diagram showing the regions containing events that can inﬂu- ence, or that can be inﬂuenced by K according to Bell’s principle of local causality. b) Bipartite Bell experiment in which events creating the outcomes a and b are space-like separated. Apart from the inputs x and y which are chosen at A and B, the outcome a can depend only on the regions Λ1 and Λ2, and b only on Λ2 and Λ3.

exchanged directly between the two measurement events. Letting the (free) choices of measurement setting x and y be made locally at the time of measurement in A and B, we see that the outcome a is produced by Alice’s measurement device before any information about Bob’s choice of measurement y can reach it. Thus, a cannot depend on y. Similarly the production of b by Bob’s measurement device cannot depend on x. Still, the measurement processes at A and B can depend on more variables than only x and y. In particular, a is allowed to depend on the whole content of its past light-cone, including Λ2, a region of space-time which can also influence the process creating b, and Λ1 which cannot influence b. Let us thus denote by λ1, λ2, λ3 all variables which belong to the corresponding regions Λ1, Λ2, Λ3 and which are relevant to make predictions about a and b. The most precise prediction of a that can be given prior to measurement and in agreement with the principle of local causality can then be described by a probability distribution of the form P (a x, λ , λ ). Allowing this distribution to also depend on A | 1 2 λ , resulting in a prediction of the form P (a x, λ , λ , λ ), can only gives a more precise 3 A | 1 2 3 description of a than a locally causal theory1. Similarly, the distribution P (b y, λ , λ , λ ) B | 1 2 3 describes predictions about b at least as well as any locally causal theory. Since the processes happening at A and B cannot influence each other, they are independent. The average bipartite correlations produced in this situation must thus be of the form P (ab xy) = p(λ)P (a x, λ)P (b y, λ) (1.1.1) | A | B | Xλ where λ = (λ , λ , λ ) and p(λ) is a probability distribution, i.e. p(λ) 0, p(λ) = 1. 1 2 3 ≥ λ We refer to this decomposition as the locality condition. Any correlations which can be P decomposed in this way are called local, and conversely any correlations which admit no such decomposition are referred to as being nonlocal. Note that here the regions Λ1, Λ2, Λ3 in Figure 1.1b extend up to immemorial times and depend on the precise space-time positions of A and B. Bell showed that different regions Λ0 with nicer properties can be chosen in order to reach the same decomposition

1 Remember that x and y are only chosen at A and B, in a way that is independent of λ1, λ2 and λ3. Allowing a prediction of a to depend on λ3 thus still doesn’t allow it to depend on y.

6 1.1 No-signalling and local causality

(1.1.1). Namely, any region Λ0 that screens oﬀ the regions Λi that we considered here, i.e. that already contains the information from Λ which is relevant to make predictions about a and b [13], is good enough to reach equation (1.1.1).

1.1.2 No-signalling correlations If no decomposition of the form (1.1.1) exists for some correlations P (ab xy), some kind of | influence must have taken place between the two measurement events. Yet, this influence might not be available through the correlations to transmit a message. Indeed, users having only access to the variables a, b, x, y can only encode a message to be carried from A to B by the influences in the choice of their inputs x and y. And this message can only be decoded from the observation of the outcomes a and b. Thus, in order to be able to use some correlations to communicate, the statistics of one party’s outcome must depend on the other party’s choice of measurement. In other words they must violate one of the no-signalling conditions:

P (a xy) = P (ab xy) = P (a x) y | | | ∀ Xb (1.1.2) P (b xy) = P (ab xy) = P (b y) x. | | | ∀ a X Note that these constraints are also suﬃcient: if P (b xy) = P (b x y) for some x, x , y, b, | 6 | 0 0 then Alice can always send a message to Bob by choosing between x and x0, and repeating the experiment enough times to allow for Bob to discriminate between these two probabilities. Correlations satisfying the conditions (1.1.2) are called no-signalling. Violation of the no-signalling conditions allows for communication, which is very common in nature. However, violation of these constraints between space-like separated events would allow for faster-than-light communication. Assuming that no physical support can carry information faster than light, this would directly contradict the principle of no-signalling.

1.1.3 Geometrical representation When talking about correlations, it is often useful to represent these probabilities in the vector space obtained by concatenating all components of P (ab xy). Let us brieﬂy describe | a few sets of correlations in this space. For concreteness, we consider here the scenario where a, b, x, y = 0, 1 can only take binary values. Every conditional probability distribution P (ab xy) can then be represented | as the vector

16 ~p = (P (00 00),P (10 00),P (00 10),P (10 10),...,P (11 11)) R (1.1.3) | | | | | ∈ which belongs to a 16-dimensional vector space. Since probabilities satisfy the normalization condition P (ab xy) = 1 x, y, the space spanned by the correlation vectors ab | ∀ ~p is in fact only 12-dimensional. Moreover, probabilities are always positive and must P thus satisfy the constraints P (ab xy) 0 a, b, x, y. This restricts the set of vectors ~p | ≥ ∀ that correspond to valid correlations P (ab xy) within this 12-dimensional space. Since the | number of positivity constraints is ﬁnite, the set of valid correlation vectors is a polytope (see Appendix A), which we refer to as the positivity polytope. Similarly, no-signalling correlations are normalized and positive. Moreover they satisfy the no-signalling conditions (1.1.2). These linear conditions deﬁne the no-signalling

7 Bell tests in bipartite scenarios

NS Q L

Figure 1.2: Schematic representation of the set of local (L), quantum (Q), and no- signalling (NS) correlations. Note the inclusion L Q NS. ⊂ ⊂ subspace, which is of dimension 8 here. The set of all no-signalling correlations is thus the slice of the positivity polytope with this subspace. This is again a polytope (see Appendix A), which is usually called the no-signalling polytope. The set of local correlations, as deﬁned by (1.1.1), can also be described by a polytope in the space of correlations. Indeed it is known [20] that any local correlation P (ab xy) | can be decomposed as a convex combination

P (ab xy) = p(µ)P (ab xy), p(µ) 0, p(µ) = 1 (1.1.4) | µ | ≥ µ µ X X of deterministic local strategies P (ab xy) = P (a x, µ)P (b y, µ) 0, 1 . On the other µ | A | B | ∈ { } hand any convex combination of deterministic local strategies is also local. The set of local correlations thus corresponds to the convex hull of the deterministic local strategies. Since the number of such strategies is ﬁnite, this set is also a polytope, the local polytope. Whereas any inequality satisﬁed by the local polytope is a valid Bell inequality, the facets of this polytope are tight Bell inequalities. Finally, it is also useful to characterize the set of quantum correlations. These correlations are all the ones which can be obtained by measuring a quantum state ρ with some local measurement operator Ma x and Mb y. They can thus always be written as | |

P (ab xy) = tr(Ma x Mb yρ). (1.1.5) | | ⊗ |

Where ρ 0, trρ = 1, Ma x 0 a Ma x = 11, Mb y 0 b Mb y = 11. ≥ | ≥ | | ≥ | The set of quantum correlations is convex, but admits an infinite number of extremal P P points. It is thus not a polytope. Nevertheless, it can be efficiently characterized by a hierarchy of semi-definite programs [21, 22]. While quantum correlations can violate Bell inequalities, these correlations always satisfy the no-signalling condition (1.1.2). The boundary of this set thus lies between the two preceding sets as represented in Figure 1.2. Note that since quantum correlations can be nonlocal, they can require an exchange of influences between the measurement events. However these influences remains out of reach from us because quantum correlations satisfy the no-signalling condition.

1.1.4 Experimental loopholes Knowing that quantum physics can violate the locality condition (1.1.1) is one thing. Verifying that nature violates it is another, which requires the observation of a faithful

8 1.2 Bell test between an atom and an optical mode

Bell inequality violation. In particular, such an experiment should demonstrate that no locally causal theory is able to reproduce the experimental results. Given the current technological limitations, all Bell inequality violations demonstrated so far suﬀer from at least one of the following two loopholes, which prevents them from strictly concluding about the nonlocal character of nature. The locality loophole. As discussed previously, space-time separation between the measurement events should be guaranteed in order to prevent any communication between the measurement devices. More precisely, one should make sure that the speed of light prevents Alice’s choice of measurement setting to reach Bob’s device before it produces its outcome. And similarly for Bob’s setting. Given the speed of light, this puts stringent constraints on the admissible duration of the measurement processes, or on the distance that should separate them. The detection loophole. If the measurement devices fail to produce outcomes a or b too often, because the systems to be measured are frequently lost along the way for instance, then there is a possibility that discarding the non-detected events can allow for a local model to reproduce the post-selected correlations [14]. The probability that the measurements produce results, given some inputs x and y, should thus not be too low. With the elements we recalled here, we can now present our contributions.

1.2 Bell test between an atom and an optical mode

If Bell experiments conducted so far have always suffered from one of the loophole described above, technological advances suggest that both the locality and the detection loophole might soon be closable within the same experiment. In order to make this happen, novel proposals taking into account present capabilities are highly welcome. Here we describe a proposal for a loophole-free Bell test, and analyse its feasibility. Bell tests with photonic systems are well designed to ensure strict space-like separation between the measurement events, thanks to the high speed at which photons can travel. However optical losses are unavoidable, leaving the detection loophole open. On the other hand, atomic systems can provide very high detection efficiency, but don’t travel well enough to allow for a space-time separation between the measurements. To close both loopholes, we consider here an hybrid entangled system consisting of an atom (which can be detected very efficiently) and a photon (which can travel fast, and thus helps to close the locality loophole). We first describe how entanglement between an atom and a photon can be produced, and then discuss the constraints that an experiment would have to satisfy in order to allow an experiment on this system to demonstrate nonlocality.

1.2.1 Creating atom-photon entanglement Let us consider an atom with a lambda-type level conﬁguration (as depicted in Fig. 1.3), initially prepared in the state g . A pump laser pulse with the Rabi frequency Ω can be | i used to partially excite the atom in such a way that it can spontaneously decay into the level s by emitting a photon. Long after the decay time of the atom, the atom-photon | i state is given by ψφ = cos(θ/2) g, 0 + eiφ sin(θ/2) s, 1 (1.2.1) | i | i | i where θ = dtΩ(t) refers to the area of the pump pulse. The phase term is deﬁned by φ = k r k r where k (k ) corresponds to the wave vector of the pump (the p Rp − s s p s 9 Bell tests in bipartite scenarios

e | !

Ω spontaneous photon

g s | ! | !

Figure 1.3: Basic level scheme for the creation of atom-photon entanglement by partial excitation of an atom. The branching ratio is such that when the atom is excited, it decays preferentially in s . | i spontaneous photon) and rp (rs) is the atom position when the pump photon is absorbed (the spontaneous photon is emitted).

1.2.2 CHSH violation

In order to demonstrate nonlocality with the above state, we propose to test the CHSH Bell inequality [15]: S = E + E + E E 2 (1.2.2) 00 01 10 − 11 ≤ where E = p(a = b xy) p(a = b xy) is the correlation between Alice and Bob’s xy | − 6 | outcomes when they respectively perform measurements x and y. Here we consider that Alice can choose measurement bases for her qubit on the whole Bloch sphere. However, since Bob’s qubit lies in the Fock space spanned by 0 and 1 photons, we let him only choose between two kinds of natural measurements : photon counting and homodyne measurements. Since measurements on the atom can be very efficient, we assume that they always produce an outcome. Similarly, homodyne measurements can be very efficient [16] so that Bob’s homodyne measurement is considered perfectly efficient. However, we let his photon counter have a detection efficiency ηd : when a photon arrives on his detector, it thus produces a click with probability ηd. To analyse the impact of the distance between Alice and Bob, we model the channel through which the photon propagates as a lossy channel with transmission ηt:

0, 0 0, 0 | i → | i (1.2.3) 1, 0 1 η 0, 1 + √η 1, 0 | i → − t| i t| i p here the second qubit is a mode of the environment, which is not observed. Tracing out this mode, we get an eﬀective state after the transmission line of

iφ iφ 2 ρ = (cos θ g, 0 +e sin θ√η s, 1 )(cos θ g, 0 +e− sin θ√η s, 1 )+(1 η ) sin θ s, 0 s, 0 . ηt | i t| i h | th | − t | ih | (1.2.4) Considering this state and equation (1.2.2) together, we optimized the free parameters in the state and measurements to get the largest violation for several choices of ηt and ηd. The result is plotted in Figure 1.4.

10 1.2 Bell test between an atom and an optical mode

Figure 1.4: Amount of CHSH violation achievable in an atom-photon Bell experiment. The dashed-dotted line corresponds to the case in which both measurements on the photonic mode are homodyne measurement. The other curves are for one homodyne measurement and a photon counting. The lowest permissible transmission here is ηt = 61% and the lowest photo-detection eﬃciency is ηd = 39%.

1.2.3 Space-like separation From Figure 1.4, we see that the test above provides a certain robustness with respect to losses and detection ineﬃciency. In order to close both loopholes, these quantities should be compared with the losses expected from an experiment ensuring space-like separation of the measurements. These are ultimately determined by the time needed in order to perform the measurements on the atom or the photon. In our case, we expect the slowest measurement to be the atomic one. Still, the measurement should take about 1 µs [23]. A distance of the order of 300 m would thus be needed to ensure space-like separation. For 800 nm photons, a ﬁber of this length has a transmission of 93%. The scheme with double homodyne measurements is compatible with these requirements (see Figure 1.4), which attests of the potential feasibility of this experiment2.

1.2.4 Conclusion We showed that a sensible violation of the CHSH inequality can be obtained by combining measurements on an atom with photon counting and homodyne measurements on an optical mode. We also argued that the discussed quantum state could be produced with existing technology. Any practical implementation of the above scheme would involve imperfections. For instance, the branching ratio of the atom may not be perfect, meaning that the excited level e in Figure 1.3 could decay to other levels than s , which we didn’t take into | i | i account here. The movement of the atom during the application of the Ω pulse can have

2Note however that the total transmission efficiency also includes the collection efficiency, i.e. the probability with which the photon emitted by the atom is collected into a fibre. Collection efficiencies of the order of 50% have already been demonstrated using a cavity (see paper [K]).

11 Bell tests in bipartite scenarios

M independentAlice’s Bob’s measurement sources measurement ρ *

A *... B n+ n+

order loss * order loss

ρM nA nB − −

Figure 1.5: Setup of a multipair Bell experiment: here a source produces M independent pairs of entangled particles. Since the pairing between Alice’s and Bob’s particles is lost during their transmission, all particles are measured identically by each party. The total number of particles detected in both outcomes + and - are tallied on both sides. an inﬂuence on the phase φ of the produced state as well, and the transmission line of the photon should be stable enough not to loose this phase. All of these aspects can be shown not to threaten directly the main conclusion (see paper [K] for more details). This supports the idea that a Bell experiment closing both the locality and the detection loophole is within technological reach (see also [24, 25] for more proposals along these lines).

1.3 Bell test with multiple pairs

Bell experiments are often realized by measuring one pair of particles at a time. However, there are situations in which the entanglement produced is shared by many particles which cannot be addressed individually. For instance, in [26], many pairs of entangled ultracold atoms have been produced, but they cannot be measured individually. One can thus wonder whether violation of a Bell inequality could in principle be tested in such many-body systems. Here we consider systems consisting of M particles, which cannot be addressed individually. The measurements thus act identically on all the particles that each part receives. For encoding in polarization, this can be modeled by a polarizing beam-splitter (PBS) followed by photon counters (c.f. Figure 1.5). In this situation, the detectors following the PBS can receive diﬀerent numbers of particles. In order to recover binary outcomes allowing to test the CHSH Bell inequality [15], we introduce a post-processing of the outcomes. Namely, whenever the number of photons detected in the ’+’ port n+ is greater than or equal to a given threshold N, the outcome is set to ’+1’, otherwise it is set to ’-1’. Two particular strategies of interest here are the majority voting strategy when N = M/2, and the unanimity voting strategy when N = M.

1.3.1 Two sources Within this measurement setup, we consider two possible sources of entangled particles: a source of distinguishable particles, and a source of indistinguishable ones. The ﬁrst one produces states of the form

M M ρ = ρ⊗ = ( ψ ψ )⊗ (1.3.1) M | ih | 12 1.3 Bell test with multiple pairs

0.2

0.1 Independent pairs 0.05

0.02 Indistinguishable Resistance to noise 0.01 photons M 1 2 5 10 20 50 Number of pairs

Figure 1.6: Maximal resistance to noise in the majority voting scenario (full red lines) and the unanimity scenario (dashed blue lines) for sources producing independent pairs (thick lines) or indistinguishable photons (thin lines). The unanimous vote is more robust with indistinguishable photons, but majority voting on independently produced pairs yields the most persistent violation. where ψ = 1 ( 00 + 11 ). The second source produces indistinguishable particles in | i √2 | i | i the state 1 M ΦM = (a†b† + a†b†) 0 . (1.3.2) | i M!√M + 1 0 0 1 1 | i States of this second form can be obtained for instance by parametric down conversion (PDC), which produce Poissonian distributions of such states.

1.3.2 Noise model

In order to quantify the amount of CHSH violation for each source, we introduce a noisy channel between the source and Alice. This channel consists of a random unitary U = exp( β~n ~σ) applied to the state, where the rotation axis ~n is uniformly distributed on the − · β2 2 Bloch sphere, and the angle β follows a gaussian distribution p(β) = e− 2σ2 (1 e 2σ2 )√2πσ − − centered at the origin and of variance σ2. The state after this channel is given by

ρ = p(β)(U 11)ρ (U † 11)[dU] (1.3.3) out ⊗ in ⊗ ZSU(2) where [dU] is the Haar measure on SU(2). 1 The application of this channel on the maximally-entangled state Φ = (a†b† + | 1i √2 0 0 a†b†) 0 produces the Werner state [19] 1 1 | i 11 ρ = w Φ Φ + (1 w) (1.3.4) | 1ih 1| − 4

1 2σ2 4σ2 6σ2 with w = 3 e− + e− + e− . We thus quantify violations by the largest amount of noise = 1 w 4σ2 which still allows one to ﬁnd a violation of the CHSH inequality. − ' 13 Bell tests in bipartite scenarios

Figure 1.7: Sketch of the experimental setup used to test various Bell inequalities. Alice’s and Bob’s choice of settings are adjusted by rotation of linear polarizers.

1.3.3 Bell violation The best resistance to noise found after optimizing on the settings is represented as a function of the number of photons produced M in Figure 1.6. Interestingly, both sources provide a ﬁnite violation even for fairly large numbers of particles. Still, the maximum 1 violation decreases like M − in both cases, in connivance with the principle of macroscopic locality [27]. Thus, if a Bell violation can be found in multipair systems, it becomes less and less signiﬁcant as the number of particles involved increases.

1.4 Experimental violation of Bell inequalities with a commercial source of entanglement

Back in the 1970-1980s, the first experimenters to test Bell inequalities had to put special efforts in building sources of entangled particles [29]. Since then, a lot of effort has been done to improve these sources. Today, it is possible to buy sources of entangled particles that are ready to test a Bell inequality. Here we demonstrate the violation of several Bell inequalities that we obtained with a commercially available source.

1.4.1 Experimental setup We used the QuTools source [28] to produce pairs of 810nm photons entangled in polarization via spontaneous parametric down-conversion (SPDC). The source consists of a bulk β-barium borate (BBO) crystal, cut for Type II phase-matching, which is pumped at 405nm by a continuous wave diode laser (see Figure 1.7). The two photons produced by the source have orthogonal polarization and are emmited in cones. After selection of the spatial mode corresponding to the intersection of the two cones with pinholes and single-mode ﬁbers, the photons collected can be described by the state

1 iφ ψ = H s V i + e V s H i . (1.4.1) | i √2 | i | i | i | i The setup allows us to measure each photon along any direction lying on the equator of the Bloch sphere. A partial tomography in this x-y plane shows that the state is close to a Werner state 11 ρ = V ψ− ψ− + (1 V ) (1.4.2) | ih | − 4 with visibility V = 94%.

14 1.4 Experimental violation of Bell inequalities with a commercial source of entanglement

1.4.2 Test of several Bell inequalities

Using this source, we tested the CHSH, I3322, AS1 and AS2 Bell inequalities [15, 30, 17, 18]. The results are represented in table 1.1. The values found are in good agreement with the values expected from the partial tomographic knowledge of the source.

exp exp tom I IL IL I I − pnoise(%) (σ units) I 2 2.731 0.015 2.683 49 27 CHSH ± I 4 4.592 0.024 4.769 25 13 3322 ± AS 6 7.747 0.026 7.750 67 23 1 ± AS 10 12.85 0.030 12.819 95 22 2 ± Table 1.1: Measurement of the CHSH inequality and of inequalities inequivalent to CHSH. exp IL is the local bound, I is the value of the Bell parameter obtained experimentally with the optimized settings, Itom is the expected value from the partial tomography, Iexp I − L is the diﬀerence between the obtained value and the local bound in terms of number of standard deviations σ and pnoise(%) is the critical level of white noise that can be added to the system without loosing a violation.

1.4.3 Chained Bell inequality On top of these inequalities, we also tested the N-settings chained Bell inequality, which can be written as

IN = E + E + E + ... + E E 2(N 1) = IN . (1.4.3) 11 12 22 NN − N1 ≤ − L The values obtained experimentally are reported for N 6 in table 1.2. The chained ≤ inequality has a number of applications which we mention below.

exp exp tom I IL N IL I I − pnoise (%) (σ units) 2 2 2.731 0.015 2.683 49 27 ± 3 4 4.907 0.019 4.925 48 18 ± 4 6 7.018 0.023 6.999 44 15 ± 5 8 8.969 0.026 8.996 37 11 ± 6 10 10.91 0.028 10.954 33 8 ± Table 1.2: Measurement of the chained inequalities with N settings per side.

Randomness certiﬁed by the no-signalling principle An interesting property of the chained Bell inequality is that the marginal probabilities P (a x) and P (b y) tend to 1/2 as the violation of the inequality increases. This allows | | one to certify that the outcomes produced by measuring the quantum system must be truly random, in the sense that no algorithm can possibly predict the measurement outcomes [31]. More precisely, the amount of true randomness that could be extracted from the experimental results found by Alice can be evaluated by ﬁnding the largest marginal probability P (a x) which is compatible with the measured Bell inequality violation Iexp. We ∗ | 15 Bell tests in bipartite scenarios

Quantum bound Noïsignaling bound for all N 0.9

0.8 (a|x) * P 0.7

0.6

N=2 N=3 N=4 N=5 N=6

0.5 2Nï2 2Nï3/2 2Nï1 2Nï1/2 2N IN

Figure 1.8: Maximum marginal probability compatible with a violation of the N-settings chained inequality. The bound implied by the no-signalling principle is identical for all N. performed this optimization over the set of quantum correlations as well as among all no- signalling correlations. The result is shown in Figure 1.8 together with the experimentally achieved values Iexp. The strongest bound imposed by the no-signalling principle is P (a x) = 0.7455 ∗ | ± 0.0057, achieved for the inequality with N = 4 settings. This allows one in principle to extract H (a x) = log P (a x) = 0.41 0.01 random bits per run. min | − 2 ∗ | ±

EPR2 local part

N Another property of the chained inequality is that its maximum quantum value IQ = π 2N cos 2N , achievable by measuring a singlet state [32], approaches the no-signalling bound IN = 2N as the number of settings N increases. This allows one to conclude that NS the singlet state has no local part in the sense of EPR2 [33]. Indeed, if a fraction pL of the measured pairs would behave locally during an experimental evaluation of the chained inequality (1.4.3) yielding the value Iexp, the following equation would hold: Iexp = p IN + (1 p )IN , (1.4.4) L L − L NL N N N where IL is the value of I achieved with the local pairs of particles, and INL a value of the same expression achieved on the rest of the particles. Since the following bounds hold: IN 2(N 1) and IN IN = 2N, the local part p of the measured states must L ≤ − NL ≤ NS L be bounded by Iexp p pmax = N . (1.4.5) L ≤ L − 2 N For Iexp = IN , we ﬁnd pmax = N 1 cos π →∞ 0. Thus, for every number of Q L − 2N −→ settings N, testing the chained inequality can provide an upperbound on the local content of the state measured which eventually converges to 0. max In our case, the best bound on pL is found for N = 4 settings, yielding pL = 0.491 0.012. While recent work could demonstrate an even lower value [34], this simple ± 16 1.4 Experimental violation of Bell inequalities with a commercial source of entanglement experiment already shows that at least half of the photon pairs produced by the source are nonlocal.

1.4.4 Conclusion In this experiment, we relied on the fair sampling assumption because the single photons detectors were not eﬃcient enough to close the detection loophole. Moreover the detection events were not space-like separated. Yet, this experiment shows that a simple demonstration of several interesting results of quantum information theory is nowadays possible with modest equipment.

17 Bell tests in bipartite scenarios

18 Chapter 2

Nonlocality with three and more parties

In the precedent chapter, we focused on Bell-type experiments involving two parties only. While this is the simplest case, and indeed the most often discussed one, the idea of local correlations can be extended straightforwardly to multipartite scenarios involving more parties. Labeling the (output,input) of a third party Charly by (c,z), the locality condition (1.1.1) generalises to:

P (abc xyz) = p(λ)P (a x, λ)P (b y, λ)P (c z, λ), (2.0.1) | | | | Xλ and similarly for more parties. Tripartite correlations P (abc xyz) are then referred to as | nonlocal if and only if they cannot be decomposed as (2.0.1).

2.1 Deﬁning genuine multipartite nonlocality

Just like entanglement can have more forms in a multipartite scenario than in the bipartite case [35], it is easy to realize that the deﬁnition (2.0.1) does not capture the whole potential of nonlocality in a tripartite scenario. Consider indeed some bipartite nonlocal correlations P (ab xy) and arbitrary statistics for Charly P (c z). The product of the AB | C | two distributions P (abc xyz) = P (ab xy)P (c z) violates (2.0.1) and is thus nonlocal. | AB | C | However it is clear that Charly plays no role in the nonlocality of these correlations. These correlations are thus not genuinely three-way nonlocal. This observation was ﬁrst made by Svetlichny in 1987 [36], who proposed an inequality capable of certifying (if violated) that correlations cannot be explained by a mechanism involving fewer than 3 parties. This is the Svetlichny inequality

S = E + E + E E + E E E E 4 (2.1.1) 111 112 121 − 122 211 − 212 − 221 − 222 ≤ with E = 1 ( 1)a+b+cP (abc xyz), which is satisfied by all tripartite correlations xyz a,b,c=0 − | of the form P P (abc xyz) = p (λ)P (ab xy, λ)P (c z, λ) | 1 AB | C | Xλ (2.1.2) + p (λ)P (ac xz, λ)P (b y, λ) + p (λ)P (bc yz, λ)P (a x, λ) 2 AC | B | 3 BC | A | Xλ Xλ 19 Nonlocality with three and more parties with p 0 and p (λ) + p (λ) + p (λ) = 1. i ≥ λ 1 2 3 If being unable to decompose some tripartite correlations P (abc xyz) in the form of P | (2.1.2) is sufficient to conclude that none of the parties was separated from the other ones in the process that created these correlations, it was pointed out recently that this condition is not always necessary (see paper [N] and [37]). To understand why this is the case, let us consider the situation in which the three measurement events producing a, b, and c, are not simultaneous but follow an order: Alice measures first, then Bob, and finally Charly (A < B < C). If decompositions of the form (2.1.2) exist for the observed correlations we might want to conclude that these correlations can be reproduced by some interaction between pairs of parties. Yet, this is not possible if every such decomposition happens to contain PAB terms that are signalling from B to A, i.e. such that P (ab xy, λ) depends on y. Indeed, in the considered b AB | configuration (A < B < C), y can always be chosen freely after a. The distribution of a P thus cannot depend on y. It thus seems important, from a physical point of view, to consider decompositions (2.1.2) that are compatible with the situation in which the correlations are produced. In order to conclude something about the nature of correlations that is independent of the situation in which they appear, we suggest to require a consistent decomposition (2.1.2) to exist for all possible measurement situations. Thus, we say that correlations are Svetlichny-sequential iff they can be decomposed as

P (abc xyz) = p (λ)P TAB (ab xy, λ)P (c z, λ) | 1 | C | Xλ (2.1.3) + p (λ)P TAC (ac xz, λ)P (b y, λ) + p (λ)P TBC (bc yz, λ)P (a x, λ) 2 | B | 3 | A | Xλ Xλ for every possible ordering of the measurements. Non-Svetlichny-sequential correlations are then called genuinely tripartite nonlocal. For correlations that are not genuinely tripartite nonlocal in this sense, a biseparable model cannot be constructed coherently for all possible ordering of the measurements. Here P TAB (ab xy, λ) depends on the order | of measurement between Alice and Bob. Namely, P TAB (ab xy, λ) = P A

20 2.2 Multipartite Bell-like inequalities are performed is crucial in several situations, as shown here, and in the last chapter of this thesis.

2.2 Multipartite Bell-like inequalities

If the CHSH inequality (1.2.2) found many applications, it is certainly because of its outstanding properties, but probably also because of its simplicity. Indeed, the CHSH inequality has a lot of symmetries: it is for instance invariant under permutation of parties, so that no party plays a special role, and it only involves correlations between the parties’ outcomes, so that the specific outcomes of a party play no specific role independently of the other party’s outcome. Since the structure of the local polytope quickly gets too complicated to allow a direct computation of its facets [39] when more than two parties, inputs or outputs are considered, it seems natural, when considering such scenarios, to first restrict one’s attention to Bell inequalities having properties similar to the ones we just mentioned. This allows one to simplify the analysis while leaving a hope that the results found in this way can be useful, since the CHSH inequality satisfies these constraints. In this perspective, we note that a complete description of the Bell inequalities which involve only full-correlations terms could be found for all (n, 2, 2) scenarios in [40, 41]. Here we denote by (n, m, k) the Bell scenario involving n parties, each with m possible measurement settings producing one out of k possible outcomes. On the other hand, we describe in paper [E] how the search for Bell inequalities that are symmetric under permutations of the parties can be simplified by considering projections of the local polytope, which are much easier to solve than the full polytopes. This allows us in turn to discover many families of Bell and Svetlichny inequalities for several scenarios (c.f. paper [E] for more details). Here, we consider a special form of Bell expressions that is both symmetric under permutation of the parties and only involves full-correlations. We show that many inequalities presented throughout the years in the literature have this form. This allows us to propose a natural generalization of them to general (n, m, k) scenarios. Moreover, we show that several bounds on these expressions can be easily computed once some bound for the corresponding few-party inequality is known.

2.2.1 A general structure for (n, m, k) scenarios Let us consider the (n, m, k) Bell scenario. Denoting by ~s = (s , . . . , s ) 0, 1, . . . , m 1 n ∈ { − 1 n the settings of all parties, and ~r = (r , . . . , r ) 0, 1, . . . , k 1 n their results, we } 1 n ∈ { − } write the following Bell expression: s Ωn,m,k;f = f [s]m, r P (~r ~s) (2.2.1) − m k | X~s X~r h j ki where s = n s and r = n r are the sums of all parties’ inputs and outputs, x i=1 i i=1 i b c is the integer part of x, [x]y = x y x/y the modulo function, and f : 0, . . . , m 1 P P − b c { − } × 0, . . . , k 1 R is a real-valued function (deﬁned by m k real parameters) that fully { − } → × characterizes Ωn,m,k;f . Clearly, this Bell polynomial is symmetric under permutation of the parties and only involves correlation terms, since only sums of all the parties’ settings and outcomes matter. A choice of function f(s, r) allows one to write an expression for any scenario with n parties, m inputs and k outputs. Table 2.1 shows how diﬀerent choices of this function

21 Nonlocality with three and more parties

m inputs

chained DIEW BKP ?

CHSH Svetlichny CGLMP

n parties k outputs Svetlichny-CGLMP

Figure 2.1: Previously known families of Bell expressions (see [15, 42, 43, 44, 45, 46, 47, 48] and papers [F, L]) are recovered by equation (2.2.1) with the choice of parameter f(s, r) = δ r + δ [ r] . In particular, the CHSH expression is recovered for n = m = k = 2. s,0 s,1 − k This provides a natural way to extend these inequalities to a general scenario (n, m, k) (see also [49]). allow to recover several Bell-like expressions used in the literature. In particular, the choice f(s, r) = δ r + δ [ r] (2.2.2) s,0 s,1 − k allows one to generalize all the expressions represented in Figure 2.1 to the (n, m, k) scenario. Note that the generalization obtained in this way was also discovered independently by [49].

n m k f(s, r) Bell expression 3, odd 2 2 δ r MABK [50] ≥ s,0 3 2 2 δs,0 r DIEW [71] ≥ ≥ ∆ s 3 2 2 cos( − π) r , ∆ R DIEW [71] ≥ ≥ m ∈ 2 2 2 δ r + δ [ r] c.f. Figure 2.1 ≥ ≥ ≥ s,0 s,1 − k

Table 2.1: A summary of some known Bell expressions that can be recovered as special cases of Ωn,m,k;f .

2.2.2 Recursion relation

By performing the change of variable s = [s + s ] , r = [r + r s1+sn ] , equation 10 1 n m 10 1 n − m k (2.2.1) can be rewritten as m 1 k 1 − − Ω = Ω(sn,rn) P (r s ) , (2.2.3) n,m,k;f n 1,m,k;f n| n s =0 r =0 − Xn Xn

(sn,rn) where Ωn 1,m,k;f is equivalent upon permutation of inputs and ouputs to an (n-1)-partite − expression (2.2.1) conditionned on rn and sn. Since the (n-1)-partite polynomial is generated by the same function f(s, r) as the n-partite one Ωn,m,k;f , this provides a way to relate the n-partite expression to polynomials of the same kind involving fewer parties. We describe below how this relation allows one to derive a number of bound for Ωn,m,k;f .

22 2.2 Multipartite Bell-like inequalities

a) b) c) d)

Figure 2.2: In a grouping models, n parties can be shared out into g groups. Arbitrary communication is allowed between parties belonging to the same group, but no communication is allowed between diﬀerent groups. a) With g = 1, every no-signalling correlation can be reproduced by the model. b) and c) g = 2: Parties are shared into two groups. This corresponds to the usual Svetlichny model. Correlations that cannot be reproduced here are genuinely multipartite nonlocal. d) For g = n, the model coincides with the usual local model. Any 2 < g < n allows one to interpolate between the local and usual Svetlichny models.

Tsirelson bounds Given, a Tsirelson bound Ω βT on a bipartite Bell-like expression, equation 2,m,k;f ≥ 2,m,k;f (2.2.3) induces the following Tsirelson bound for the n-partite polynomial:

n 2 T Ω m − β . (2.2.4) n,m,k;f ≥ 2,m,k;f Nontrivial Tsirelson bounds can thus be easily deduced for these multipartite Bell inequalities, thanks to their special structure.

Generalized Svetlichny bounds By considering a scenario in which n parties are gathered into two groups, Svetlichny deduced an inequality which detects when interaction between all parties must have happened [36]. More generally, the amount of interaction needed between n parties in order to reproduce some correlations can be quantiﬁed by the maximal number of groups g into which the parties can be separated while still being able to reproduce these correlations (c.f. Figure 2.2). Within each group, parties are allowed to communicate their inputs to each other, and to agree on which outputs they want to outcome, but no communication is allowed between the diﬀerent groups1. Thanks to relation (2.2.3), the bound of any Ωn,m,k;f polynomial that can be achieved with parties distributed into g groups can be obtained from the local bound of the poly- L nomial with n = g. Denoting this bound by βg,m,k;f gives the following bound for Ωn,m,k;f upon separation of the parties into g groups:

n g L Ω m − β . (2.2.5) n,m,k;f ≥ g,m,k;f Note that similar bounds were already derived for correlations obtained by measuring quantum states that are positive under partial transposition across all partitions of the n systems into g subsystems [51].

Application to quantum states. Violation of (2.2.5) allows one to put an upper bound on the number of groups g into which the parties can be distributed in order for them to

1As discussed in section 2.1, the order in which the different parties are measured should in principle be included in the model. However one can show that this order is not important here, i.e. the different definitions discussed in section 2.1 coincide, because the inequalities we consider here only involve full- correlations.

23 Nonlocality with three and more parties be able to reproduce the observed correlations with local operation and communication within the groups. Here, we consider the above bounds in the case m = k = 2. It turns out that in order to test for an even or odd number of groups it is useful to consider two diﬀerent Bell polynomials (c.f. paper [B]). For g odd, we thus choose f(s, r) = δs,0 r, and when g is even we choose f(s, r) = δ r + δ [ r] . s,0 s,1 − k Considering violations of these inequalities with measurements on the partially-entangled GHZ states GHZ = cos θ 00 ... 0 + sin θ 11 ... 1 (2.2.6) | θi | i | i shows that parties cannot be separated into more that g = 1 2 log sin(2θ) in order − b 2 c to reproduce the achieved correlations (see paper [B]). By considering measurement on the n-partite W state

1 W = ( 10 ... 0 + 01 ... 0 + 00 ... 1 ), (2.2.7) | i √n | i | i | i we could also show that letting two parties interact is not enough to reproduce the correlations that can be observed as soon as n 3. However, we found no violation for the ≥ bounds considered if the parties are separated into less than n 1 groups (see paper [B] − for more details).

Biseparable bounds

Finally, we note that equation (2.2.3) can also be used to deduce bounds on Ωn,m,k;f that are satisﬁed upon measurement of biseparable quantum states. In particular, for k = 2 ouputs and f(s, r) = g(s) r, we show in paper [L] that the following biseparable bound · holds:

m 1 m 1 1 n 2 − ηjπ − s Ωn,m,2;g r m − m g(s) max ηj csc g(s) ωj (2.2.8) · ≥ 2 − j=0,...,m 1 " 2m #! s=0 − s=0 X X

where ηj is the greatest common divisor of 2j + 1 and m, and ωj = exp(iπ(2j + 1)/m). The choice g(s) = δs,0 + δs,1 allows one to write an entanglement witness that is well adapted to detect genuine multipartite entanglement in multipartite GHZ state. We use this witness in chapter 3 of this thesis to detect genuine multipartite entanglement in a system of trapped ions.

2.3 Nonlocality from local marginals

In a multipartite system, every subset of parties constitutes a proper system in itself. The fact that these subsystems describe parts of the same total system requires them to satisfy some compatibility conditions: bipartite correlations P (ab xy) for instance are | compatible with the tripartite ones P (abc xyz) if and only if P (ab xy) = P (abc xyz). | | c | While this condition is easy to check when the global correlations are known, checking P whether several reduced states are compatible with each other is known as the marginal problem [52]. The marginal problem is trivial in the bipartite case: every single-party marginals P (a x) and P (b y) are always compatible with the joint distribution P (ab xy) = P (a x)P (b y). | | | | | However, the situation is less evident in situations involving more parties. For instance, it is known that if PAB violates the CHSH inequality, then it cannot be compatible with

24 2.3 Nonlocality from local marginals

correlations PBC violating CHSH as well; a phenomenon known as the monogamy of nonlocality [53]. Here we want to explore this multipartite constraint further. In particular, we ask whether some local marginals can witness nonlocality (or even genuine multipartite nonlocality), because the only global correlations with which they are compatible is (genuinely multipartite) nonlocal.

2.3.1 An inequality In order to answer this question in the tripartite case, we consider all bipartite correlations PAB, PAC and PBC , which are compatible with Svetlichny-sequential correlations PABC . This set of correlations can be described as

Π = (P ,P ,P ) P s.t. (2.1.3) holds . (2.3.1) { AB AC BC | ∃ ABC } Since Π is convex, it can be characterized with linear Bell-type inequalities. Using the decomposition (2.1.3) and linear programming, one can show that the following inequality is satisﬁed by correlations in Π:

A (1+B +B +C ) A (1+B +C +C ) B +C +B C +B C 4, (2.3.2) −h 0 0 1 0 i−h 1 0 0 1 i−h 0 0 0 0 1 1i ≤ where A = P (0 x) P (1 x), A B = P (00 xy) P (01 xy) P (10 xy) + h xi A | − A | h x yi AB | − AB | − AB | P (11 xy) and similarly for the other parties. Violation of this inequality by some AB | bipartite correlations thus implies that the only tripartite distributions with which they are compatible are genuinely-tripartite nonlocal. Interestingly, this inequality can be violated by measuring a noisy W state 11 W = p W W + (1 p) (2.3.3) | i | ih | − 8 as soon as p > 0.9548, by using the measurement operators

A = cos ασ + sin ασ ,A = cos ασ sin ασ 0 z x 1 z − x B = σ ,B = cos βσ + sin βσ (2.3.4) 0 − z 1 z x C = σ ,C = cos βσ sin βσ 0 − z 1 z − x with α = 3.6241 and β = 2.0221. However, no bipartite correlations achieved by measuring this state can violate a Bell inequality with binary inputs and outputs, because the bipartite reduced states p p 1 p ρ = ( 01 + 10 )( 01 + 10 ) + 00 00 + − 11 (2.3.5) red 3 | i | i h | h | 3| ih | 4 satisﬁes the Horodecki criterion for every 0 < p < 1 [54]. This shows that there exist bipartite quantum correlations that are local, but only compatible with genuinely tripartite-nonlocal correlations.

2.3.2 Conclusion This illustrates the strength of the compatibility constraints that relates diﬀerent parts of a system. Namely, the fact that subsystems are parts of the same system allows one to reveal properties of that system which were not apparent in its individual parts. Note that a similar result can be achieved with respect to entanglement (see paper [P] for more details).

25 Nonlocality with three and more parties

2.4 Tripartite nonlocal boxes

Another way to get some insight into the structure of multipartite correlations is by studying the largest possible set of multipartite correlations. Within the no-signalling subspace, this is deﬁned by all correlations satisfying the positivity conditions

P (abc xyz) 0 a, b, c, x, y, z. (2.4.1) | ≥ ∀ The no-signalling polytope is thus easily characterized in terms of facets. Its extremal points, however, reveal that this polytope has a highly nontrivial geometric structure.

2.4.1 The tripartite nosignalling polytope Using the porta software [105], we could ﬁnd all extremal points of the tripartite no- signalling polytope when all parties have binary inputs and outputs. After sorting these boxes into equivalence classes under permutation of inputs, outputs and parties, we could identify 46 families of extremal no-signalling boxes. A complete description of these families is given in paper [G]. Apart from the deterministic local strategies and the PR boxes, which are also extremal points of the bipartite nosignalling polytope, we found 44 families of truly tripartite boxes. Interestingly, three of them admit a decomposition of the form (2.1.3), and are thus not genuinely 3-way nonlocal in this sense. Using the 46 classes of Bell inequalities that deﬁne the local polytope in this scenario [60], we studied the relation of each box with respect to each Bell inequality (c.f. paper [G]).

2.4.2 Conclusion The reason why so many new no-signalling boxes appear when considering a scenario with more parties is still a mystery, which asks for further investigation. Nevertheless, the observation that the number of extremal boxes coincides with the number of Bell inequalities in this scenario was recently elucidated by Fritz [61]. It seems thus that a strong link exists between the set of nosignalling correlations and that of local correlations.

2.5 A tight limit on quantum nonlocality

As already mentioned in section 1.1, quantum correlations can be nonlocal, but not as much as imposed by the no-signalling conditions (1.1.2) alone. Indeed, while no-signalling correlations can achieve a value of CHSH equal to 4 with a PR box, it is known since Tsirelson [62] that measurements on a quantum system cannot exceed 2√2. Despite recent advances on the subject [63, 8], the question why the nonlocality of quantum correlations is limited the way it is remains open. Here, we introduce a simple game for which quantum correlations provide no advantage, whereas general no-signalling correlations do. The inequality associated to this game is shown to identify part of the boundary separating quantum correlations from more general no-signalling correlations.

2.5.1 Can you guess your neighbour’s input (GYNI)? Let n parties form a circle as represented in Figure 2.3. The game runs as follows: parties are given some input x 0, 1 , and are requested to output their right neighbour’s i ∈ { } 26 2.5 A tight limit on quantum nonlocality

x1 x2 xn

...

a1 a2 an

xn+1 = x1

Figure 2.3: Representation of the GYNI nonlocal game. The goal is that each party outputs its right-neighbour’s input: ai = xi+1. input. The success that the parties have in playing this game can be measured with the following quantity: ω = q(x)P (ai = xi+1 i x) (2.5.1) x ∀ | X where x = (x , x , . . . , x ) 0, 1 n denotes all the parties’ inputs, P (a = x i x) is 1 2 n ∈ { } i i+1∀ | the probability that the players obtain the correct outputs when they received the input string x, and q(x) is the probability that the set of inputs x is given to the parties. One can show that the maximum value of ω that the parties can achieve by using classical or quantum resources is

ωc = max[q(x) + q(x¯)] (2.5.2) x where x¯ = (¯x , x¯ ,..., x¯ ) withx ¯ = x 1 is the negation of x. Since this value is always 1 2 n i i ⊕ achievable classically, it follows that the Bell inequality ω ω can never be violated ≤ c quantum mechanically. One might think that the reason why quantum correlations give no advantage in this game, as compared to local correlations, is that signalling is required in order to guess someone’s input better than classically. But since quantum correlations are no-signalling, they cannot help it. However this is not totally true. Indeed, by choosing the distribution of inputs q(x) adequately, it is sometimes possible to achieve a violation ω > ωc of the inequality without allowing the player to guess anything on the other parties’ inputs. Consider for instance the following input distribution:

21 nˆ if x ... x = 0 q(x) = − 1 ⊕ ⊕ nˆ (2.5.3) (0 otherwise withn ˆ an odd number between 3 and n. In the case n =n ˆ = 3, the inequality ω ω ≤ c can be violated by several tripartite extremal no-signaling boxes (c.f. section 2.4). In particular, two boxes can achieve the value ω = 4/3ωc. It is thus clear that the bound ωc is not a consequence of the no-signaling condition.

2.5.2 Outlook We could check numerically that for the choicen ˆ = 2 n + 1 the inequalities ω ω are b 2 c ≤ c facets of the local polytope up to n = 7. Yet, they can always be violated by no-signalling correlations. This game thus tightly identiﬁes part of the boundary separating quantum from supra-quantum correlations.

27 Nonlocality with three and more parties

2.6 Simulating projective measurements on the GHZ state

One way to study nonlocal correlations is to try to simulate them with a measureable amount of nonlocal resources. This allows one to put an upper bound on the power of these correlations. For instance, it is well known that correlations created upon measurement of a singlet state can be simulated by the use of shared randomness supplemented by 1 bit of communication [55], or by 1 use of a PR box [56]. Thus no correlations found upon measurement of a singlet state can achieve a task that 1 bit of communication, or 1 PR box, cannot. Here, we consider the simulation of the n-partite Greenberger-Horne-Zeilinger (GHZ) state 1 GHZ = ( 00 ... 0 + 11 ... 1 ). (2.6.1) | i √2 | i | i Since this state is genuinely tripartite nonlocal (it can violate the Svetlichny inequality (2.1.1) or its n-partite generalization [46, 47]), it cannot be simulated with just shared randomness and interaction between a subset of the parties. Nevertheless, we consider the task of simulating it with the aid of bipartite resources only.

2.6.1 Nonlocal resources Let us allow the parties to share nonlocal boxes of the following kinds in addition to pre-established randomness. PR box. A Popescu-Rohrlich (PR) box [57] is a nonlocal box that admits two bits x, y 0, 1 as inputs and produces locally random bits a, b 0, 1 , which satisfy the ∈ { } ∈ { } binary relation a + b = xy. (2.6.2) M box. A Millionaire box [58] is a nonlocal box that admits two continuous inputs x, y [0, 1[ and produces locally random bits a, b 0, 1 , such that the following relation ∈ ∈ { } is satisﬁed: a + b = sg(x y) (2.6.3) − where addition is modulo 2 and the sign function is deﬁned as sg(x) = 0 if x > 0 and sg(x) = 1 if x 0. ≤ Note that none of these boxes is signaling: the outcomes produced by the boxes are locally random and thus carry no information on the other party’s choice of input.

2.6.2 Simulation A protocol to simulate measurements on the GHZ state (2.6.1) with nonlocal boxes runs as follows: before letting the parties choose their measurement settings, they are allowed to share any information, plus a number of boxes (as shown in Figure 2.4). The parties can then choose their measurement settings (which we represent by vectors ~a, ~b, ~c on the Bloch sphere). They are allowed to locally process this setting together with the pre-established shared randomness and accesses to their boxes. The parties then output the result of this process, which we denote by α, β, γ 1, 1 . ∈ {− } Using the above boxes, we considered the simulation of the correlations found by measuring the GHZ state in the equatorial plane, i.e. with ~a = (cos φa, sin φa, 0), etc. In this case the correlations take the form

α = β = ... = αβ = ... = 0 (2.6.4) h i h i h i 28 2.6 Simulating projective measurements on the GHZ state

Alice

PR box

M boxM boxM Charlie

PR box Bob

Figure 2.4: Setup for the simulation of the tripartite GHZ state in the x-y plane : two Millionaire boxes are shared between Alice and Bob and each of them shares a PR box with Charlie.

Alice PR box Charlie

PR box

M boxM boxM boxM boxM e

PR box

Bob PR box Dave

Figure 2.5: Distribution of bipartite no-signalling boxes that allows for the simulation of equatorial von Neumann measurement on the 4-partite GHZ state. for all marginal correlations, and

αβ . . . = cos(φ + φ + ...) (2.6.5) h i a b for the full n-partite correlation term. Here are the results that we could show (proofs in paper [C]): Theorem. Equatorial von Neumann measurements on the tripartite GHZ state can be simulated with 2 M boxes and 2 PR boxes distributed as in Figure 2.4. Theorem. Equatorial von Neumann measurements on the 4-partite GHZ state can be simulated with 4 M boxes and 4 PR boxes distributed as in Figure 2.5. Moreover, one can show that a PR box can be simulated with one bit of classical communication transmitted from one end of the box to the other one, and an M box with an average of 4 bits. Each of these protocols can thus be translated into communication models with a ﬁnite-average communication cost. Namely, and average of 10 bits allow for the simulation of equatorial measurements on the tripartite GHZ state, whereas 20 bits suﬃce on average for the four-partite case.

2.6.3 Conclusion We showed that bipartite no-signalling resources are enough to reproduce the nonlocal character of these GHZ correlations, even though these correlations are genuinely multipartite nonlocal. Moreover, we provided models to reproduce these correlations with a ﬁnite amount of communication on average. Note that this latter result was recently improved for the tripartite case [59].

29 Nonlocality with three and more parties

30 Chapter 3

Device-independent entanglement detection

Entanglement is one of the most intriguing feature of quantum physics. It allows several particles to be in a state which cannot be understood as a concatenation of the sate of each particle. Experimental demonstration of entanglement is generally performed with one of the two following techniques: tomography of the full quantum state, or evaluation of an entanglement witness. In the ﬁrst case, the state ρ of the system is characterized by performing a number of complementary measurements on it [64]. For instance, on two qubits, measurement of the product of all Pauli operators σ σ , with j = 0, 1, 2, 3, and σ = 11 allows one in i ⊗ j 0 principle to deduce ρ by solving the set of linear equations

tr(ρ σ σ ) = f (3.0.1) i ⊗ j ij

where fij is the observed frequency for the corresponding measurements. In practice however, experimental imperfections typically lead to a solution for the former set of equations which is unphysical so that more complicated techniques are generally used instead of the linear inversion, like maximum likelihood estimation [65]. Still, once the reconstructed state is found, theoretical analyses can be performed on it to check directly whether the quantum state is entangled or not. In contrast, an entanglement witness is an observable such that tr(ρ ) 0 when- W W ≥ ever ρ is separable [35]. Any decomposition of in terms of local observables allows one W to evaluate it by performing local measurements on the state under consideration. If a value tr(ρ ) < 0 is found, the measured state is then said to be entangled. W 3.1 Imperfect measurements

Any experimental manipulation is affected by imperfections, be it only the finiteness of the number of times measurements are repeated in order to accumulate sufficient statistics. Interestingly, the effect that statistical uncertainties on the frequencies fij can have on tomographically reconstructed states was analysed rigorously only very recently [66, 67]. Still, even in absence of statistical uncertainties, which can in principle be avoided by performing enough measurements in a random order, systematic errors in the measurement process can possibly affect the conclusion of a test. While this problem is known, it is seldom discussed in the literature. Let us show what kind of effects these errors can have in the detection of entanglement.

31 Device-independent entanglement detection

~m3 ~n3

~m2

~n2 ε ε

~m1 ε ~n1

Figure 3.1: Intended and actual measurement directions for tomography on a qubit. The actual measurement directions ~n are distant from the desired ones ~m by an angle smaller than ε.

3.1.1 Effects of systematic errors on tomography In order to evaluate the effect of systematic errors on the process of tomography reconstruction, we consider the situation in which each measurement can be slightly misaligned. Namely, if ~m ~σ denotes the desired measurement on a qubit, the actual measurement · performed can be written as ~n ~σ with ~m ~n cos(ε) (c.f. Figure 3.1). However, since ~n · · ≥ is unknown, results measured along ~n are interpreted during the reconstruction process as coming from measurements along ~m. Considering qubit states ψ , we looked for the maximum effect that these errors could | i have on the reconstructed state ρ by performing the following optimization :

min ψ ρ ψ ψ ,n h | | i | i i (3.1.1) subject to ~m ~n cos(ε) measurement i i · i ≥ ∀ The results of this numerical optimization are shown in Figure 3.2a. For small errors ε in the deﬁnition of the measurement bases, the uncertainty of the reconstructed n-qubit state increases at least as n ε (c.f. paper [O] for more details). Thus, if measurements √2 are done with linear polarizers having a precision of 1o in real space, for instance, the precision of the reconstructed state can decrease by 2.5% per qubit in the system. Interestingly, we found that entangled states are usually more robust to systematic errors than the worst bound shown in Figure 3.2a (see paper [O]). Nevertheless, imperfect measurements on separable states can sometimes lead to an entangled reconstructed state [68]. Entanglement can thus be wrongly witnessed through tomography because of systematic errors.

3.1.2 Eﬀects of systematic errors on entanglement witnesses In a similar fashion, we analysed the witness 1 = 11 GHZ GHZ (3.1.2) W 2 − | ih | which detects genuine multipartite entanglement [35]. For this we used the decomposition of in terms of local operators given in [69]. Allowing again all measurement operators W to diﬀer from the prescribed ones by at most ε, we looked for the smallest value tr(ρ ) W that could be achieved by measuring a biseparable state ρ.

32 3.2 Witnesses insensitive to systematic errors?

min ψ ρ ψ min tr(ρbisep ) h | | i W a) 1 b) 0

0.8 n -0.05 n = 1 = 2 n = 4 0.6 n -0.1 = 2 n n = 6 0.4 = 3 -0.15 n = 8 0.2 n = 4 -0.2 0 -0.25 0 π/18 π/6 π/3 0 0.05 0.1 0.15 0.2 Systematic error ε Systematic error ε

Figure 3.2: a) Minimum ﬁdelity of the tomographically reconstructed n-qubit state when the measurement settings deviate by at most ε from the requested ones. b) Minimum expectation value of tr( ρ) found using imperfect measurements on bispearable states. W

The results are shown in Figure 3.2b. In particular, we note that a negative value can be found, and thus entanglement wrongly detected, as soon as ε > 0. This entanglement witness measured in this manner is thus sensitive to misalignment of the measurements.

3.2 Witnesses insensitive to systematic errors?

If the entanglement detection schemes we just mentioned are sensitive to systematic errors, these kind of errors, unlike statistical errors, can be hard to evaluate in practice: how can one make sure that measurement settings are perfectly aligned? or that they are not more misaligned than some ε? How can one certify that the measurements do not act on a larger Hilbert space than expected? etc. Even in the case that these errors can be reasonably estimated, taking them into account makes the analysis of the situation complicated... It is thus worth asking whether entanglement detection can be made resistant to systematic errors. The answer to this question is already known since several years: Bell inequalities can detect entanglement without relying on any hypothesis about the kind of measurements performed or even about the nature of the measured system. Indeed, since no Bell inequality can be violated by measuring a separable state, violation of a Bell inequality witnesses that the state measured is entangled. All it needs to certify entanglement through the violation of a Bell inequality, in a bipartite scenario for instance, is a proper way of measuring the correlations P (ab xy). That is, the inputs of each party should be indexed by | x and y, their outputs by a and b, and the systems should be well identified from each other. The conclusion is then device-independent as discussed in the introduction. Conversely, if the correlations found during an experiment violate no Bell inequality, then the presence of entanglement cannot be certified based only on the device- independent hypotheses. Indeed, in this case a local strategy can reproduce the correlations, and thus some measurements on a separable state as well. Entanglement can thus be demonstrated in a device-independent manner if and only if measurement statistics can violate a Bell inequality. More precisely, if a Bell inequality β P (ab xy) c is satistifed by all lo- abxy abxy | ≤ cal correlations, then the measurement operators Ma x, Mb y used during the experi- P | | 33 Device-independent entanglement detection ment (which might not be the ones that we think we’re measuring) define the operator

= c11 abxy βabxyMa x Mb y, which is a witness for entanglement: it satisﬁes W − | ⊗ | tr(ρ ) 0 for every state ρ that is separable. A negative value tr(ρ ) < 0 is found as W ≥ P W soon as the Bell inequality is violated. When testing a Bell inequality we thus test an entanglement witness which is adapted to the measurement implemented experimentally, even if these measurements are not known to us. The conclusion is thus independent of these measurements.

3.2.1 Device-independent witnesses for genuine tripartite entanglement The equivalence between Bell inequalities and device-independent entanglement witnesses remains true in scenarios involving more than two parties. However, in these scenarios one is typically interested in witnessing genuine multipartite entanglement [35]. Indeed, it is generally easier to test entanglement between a subset of parties directly on those parties speciﬁcally. If tripartite Bell inequalities can typically be violated with states that are not genuinely multipartite entanglement, and are thus unable to witness this kind of entanglement, violation of a Svetlichny inequality demonstrates genuine tripartite nonlocality (c.f. section 2.1) and is thus suﬃcient to demonstrate genuine tripartite entanglement as well. But it is not necessary [70]. Rather, genuinely tripartite entanglement can be demonstrated from observation of correlations P (abc xyz) as soon as these correlations cannot be obtained by | measuring a biseparable quantum state ρbisep, i.e. if the correlations are not biseparable correlations:

Pbisep(abc xyz) = tr(ρbisepMa x Mb y Mc z) (3.2.1) | | ⊗ | ⊗ | where Ma x, Mb y and Mc z are arbitrary measurement operators. The set| of tripartite| biseparable| correlations can be described in terms of a hierarchy of semideﬁnite programming (SDP) (see Figure 3.3 and paper [J]). It is thus possible to determine directly from the observed correlations whether a genuinely tripartite entangled state was measured, or whether these correlations are compatible with measurements performed on an biseparable state, in which case genuine multipartite entanglement cannot be demonstrated with the device-independent hypotheses only.

3.2.2 A witness for genuine multipartite entanglement Letting ~s 0, . . . , m 1 n be a vector denoting the choice of measurement for the ∈ { − } n parties within m possible ones, and ~r 0, 1 n be the vector of their outcomes, the ∈ { } following inequality is satisﬁed by all biseparable quantum correlations (c.f. papiers [J, L]):

s s 1 − n 2 π I = ( 1)b m cE + ( 1)b m cE 2m − cot = B (3.2.2) n,m − ~s − ~s ≤ 2m n,m [sX]m=0 [sX]m=1 where E = ( 1)rP (~r ~s) is the n-partite correlator, [x] = x x m and s = s , ~s ~r − | m − b m c i i r = ri are the sums of all parties’ inputs and outputs. This inequality can thus be i P P used as a device-independent witness to detect genuine multipartite entanglement. P Measuring the n-partite GHZ state GHZ = 1 ( 0 n + 1 n) with the jth measure- | i 2 | i⊗ | i⊗ ment settings of every party lying in the x-y plane as π π cos(φ )σ + sin(φ )σ , with φ = + j , for j = 0, . . . , m 1, (3.2.3) j x j y j −2mn m −

34 3.3 Experimental demonstration

I3,3 = B3,3

S2/1

Q2/1

Figure 3.3: Particular slice in the space of tripartite correlations with 3 settings and 2 outcomes representing schematically the sets of general quantum correlations (Q3), Svetlichny correlations (S2/1) and biseparable quantum correlations (Q2/1). The inequality (3.2.2) is also represented and detects correlations that are not genuine tripartite-nonlocal.

n 1 π yields the value In,m = 2m − cos 2m > Bn,m. This inequality can thus detect genuine multipartite entanglement in noisy GHZ states ρ = V GHZ GHZ + (1 V )11/2n with π 1 | ih | − visibilities down to Vc = (m sin 2m )− . As the number of settings m increases, this critical visibility decreases, tending to the value of 2/π [71]. While inequality (3.2.2) reduces to the Svetlichny inequality for m = 2, and thus also detects genuine n-partite nonlocality in this case, it does not do so anymore for more inputs. Multipartite entanglement is then detected with a lower visibility than multipartite nonlocality.

3.3 Experimental demonstration

In the group of Prof. R. Blatt in Innsbruck, we tested the inequality (3.2.2) on a system of trapped 40Ca+ ions [72].

3.3.1 Experimental setup and procedure We used a linear trap loaded with n = 3, 4 or 6 ions, the logical states 0 and 1 of each | i | i ions being encoded in the S (m = 1/2) ground state and D (m = 1/2) metastable 1/2 − 5/2 − state, respectively. After initialization of the system in the ground state of the center-of-mass motion by Doppler and sideband cooling and in the logical state 0 n by optical pumping, the ions | i⊗ can be brought to the GHZ entangled state by applying a Mølmer-Sørensen gate [73]. Measurement of the ions in the computational basis is achieved by the electron-shelving technique by scattering light on the S P transition, and detecting the fluorescence 1/2 ↔ 1/2 with a photomultiplier tube. In order to perform measurements of all ions in the x-y plane of the Bloch sphere, we first apply local phase gates exp( i φ σ ) by means of AC-stark- − 2 z shift beams focused on individual ions. The x axis of the Bloch sphere of all ions is then brought to the computational basis by applying a collective π/2 rotation on the qubit transition. While the Mølmer-Sørensen gate can yield maximally entangled states with a high fidelity, the coherence time of the GHZ state produced in this way is of about 2 ms for n = 3 ions, and it decreases quadratically with the number of entangled qubits [74]. The

35 Device-independent entanglement detection duration of the σ rotations, performed sequentially on the ions, taking 100 µs/2π, a z ∼ significant decrease in the quality of the state can take place during the application of these pulses. To avoid this effect, we inverted the state of half of the ions before doing the Mølmen-Sørensen entangling gate for n = 4, 6. This allows one to produce a decoherence- free GHZ state of the form 1 ( 0 n/2 1 n/2 + eiϕ 1 n/2 0 n/2) whose coherence time of 2 | i⊗ | i⊗ | i⊗ | i⊗ 300ms leaves enough time to manipulate all ions. Note that the measurements settings ∼ (3.2.3) need to be adapted to this new state. We thus used the following ones on n = 4, 6 π ions: for m = 2 settings, we used the phases φj = j 2 for the first half of the ions and n+1 1 j n+1 π φj0 = 12n π + −2 π for other ones; for m = 3 we used φj = 12n π + j 3 for the first ions and 2 j φj0 = −3 π for the other ones, where j = 0, . . . , m 1 denotes the different measurement − 5 7n setting of each party. The optimal state then has the phase ϕ = 24−π . Finally, to cancel the effect of eventual drifts during the experiment, measurements sets were taken by blocks of 50 identical measurement chosen in a random order.

3.3.2 Addressing errors In order to show an indisputable violation of (3.2.2), the experiment producing the correlations should close all loopholes that can appear in a Bell experiment (c.f. chapter 1). Of course, this is not the case in the present experiment: even though the detection loophole is closed here, measurements were not performed in a space-like manner. In fact, the different systems are not even totally isolated from each other since they are separated by only 3 5 µm. The measurements performed on the ions might thus not be put in a ∼ tensor form as assumed in section 3.2. If arbitrary joint measurements are allowed in the decomposition (3.2.1) instead of a tensor product of local measurement, any correlation can be obtained by measuring a biseparable state, and thus any value of the inequality (3.2.2) can be reached as well. No interesting bound on an inequality can thus be proven in the presence of arbitrary cross-talks, i.e. without making some assumptions about which cross-talk is present in the system. Note that this situation is similar to the fact that no interesting bound can usually be put on a standard entanglement witness in presence of arbitrary systematic errors. We thus performed a special analysis in order to estimate the amount of cross-talk in our system, and how it could influence the biseparable bound Bn,m of the inequality. In our system, we expect the strongest source of cross-talk between the ions to be due to the imperfect focusing of the AC-stark shift lasers. Indeed, it is the only action which is supposed to act on some ions specifically and which might not: leakage of this laser onto neighbouring ions can cause them to feel part of the rotation imposed on the first ion. The state of an ion, or equivalently the basis in which it is measured, can thus depend on the measurement settings of the other ions. This effect can be modeled by replacing the measurement phases φ by φ = M φ where M = 1, and 0 M if j = k is j j0 k jk k jj ≤ jk ≤ 6 the amount of cross-talk from ion k to ion j. Here is a bound on the worst addressing P error. In order to evaluate the impact that these errors can have on the bound Bn,m, we estimated the amount of addressing errors present in the experiment. This allowed us to determine an upper bound on the addressing errors which was not exceeded in the 6 experiment, except possibly with a probability smaller than 10− . This upper bound is = 0.52% for n = 3, = 5.2% for n = 4, and = 5.4% for n = 6. We then computed numerically the maximum impact ∆IAE = I I=0 that addressing errors bounded by bisep− bisep can have on the biseparable bound for the settings we intended to use in the experiment. Assuming that the maximum contribution of the addressing errors to (3.2.2) is given by

36 3.4 Conclusion

∆IAE, we update the bound B to BAE = B +∆IAE to account for the cross-talks present in the experiment. Note that even though the actual measurement settings might differ from the ones we intended to measure, the modified bound BAE remains valid in the presence of cross-talk if the measurements implemented in the lab differ (not too much) from the ideal ones, =0 because B > Ibisep.

3.3.3 Experimental results The experimental evaluation of the witness are summarized in table 3.1 In all cases the measured values are consistent with a visibility of the state of about 90%, except for the tripartite case in which the GHZ state was not decoherence-free. The inequalities with two inputs (m = 2) coincides with the Svetlichny inequalities and thus detect genuine multipartite nonlocality. The witness with three inputs (m = 3) however, is able to detect genuine multipartite entanglement even in absence of genuine nonlocality. This allows one to demonstrate stronger violations as shown in table 3.1.

Iexp BAE n m BAE Iexp Visibility − (σ units) 2 4.234 4.78(6) 84(1) 9 3 3 10.894 12.39(1) 79.5(1) 150 2 8.832 10.41(6) 92.1(5) 26 4 3 33.513 42.53(8) 90.9(2) 113 2 36.4 40(1) 89(3) 4 6 3 306.8 376(3) 89(1) 23

Table 3.1: Summary of the experimental measurement of (3.2.2). For each scenario considered, the value of (3.2.2) observed is reported as Iexp, together with the associated visibility, i.e. the ratio between this value and the one expected from optimal measurement on a perfect GHZ state. The experimental value should be compared to the bound BAE, which includes a correction due to the addressing errors observed between the ions (c.f. section 3.3.2).

3.4 Conclusion

Any measurement of a device-independent entanglement witness results in the test of a standard entanglement witness which relies on the measurement settings actually implemented in the lab rather than on measurement settings which might not exactly correspond to the experimental situation. This ensures that a violation of the inequality cannot be caused by a miscalibration of the experiment. Device-independent witnesses are thus particularly robust to (possibly unknown) measurement imperfections inherent to every experimental test. Motivated by this perspective, we constructed device-independent witnesses able to detect genuine multipartite entanglement. Since these witnesses can detect genuine multipartite entanglement even in absence of genuine multipartite nonlocality, they can provide larger experimental violations than tests of Svetlichny inequalities, as was demonstrated in the experiment we conducted with the Innsbruck ion group of Prof. R. Blatt. Despite being robust to imperfect measurements, the bounds of these witnesses can be aﬀected by cross-talks between subsystems if these are not perfectly isolated from each

37 Device-independent entanglement detection other. Since this problem is quite generic, and is present in many experimental setups, it deserves further investigation.

38 Chapter 4

Quantum information put into practice

Allowing information to be carried by physical systems described by the rules of quantum physics led to a deep questioning of the theory of information. While many questions remain open, the emerging ﬁeld of quantum information already led to several remarkably concrete applications which would not exist otherwise. Here we present a modest contribution to the analysis of the security of Quantum Key Distribution (QKD), as well as a protocol which can be used to question a database with some level of security.

4.1 Memoryless attack on the 6-state QKD protocol

Quantum key distribution (QKD) allows two parties who share an initial secret key of finite size, to increase its size by exchanging quantum and classical signals through an untrusted environment. The new key generated in this way can then be used for any cryptographic application [80], such as secure transmission of a secret message, a task which is not known to be possible by classical means. Standard security proofs for QKD protocols aim at relying on the weakest possible assumptions. For instance, it is usually admitted that a possible eavesdropper is not constrained by technological limitations but only by the laws of physics. Such assumptions allow one to derive strong security bounds. However, if a particular circumstance happen to restrict further the possible action of an eavesdropper, more refined security analyses taking these limitations into account can allow the trusted parties to improve the efficiency of their protocol. Motivated by the effort put in several groups worldwide [81, 82, 83, 84] to implement quantum memories preserving coherence and population over more than several milisec- onds, we consider the case in which the eavesdropper has no access to a long-lasting quantum memory. Security proofs applicable in this scenario have been presented in [85] for the BB84 protocol, and more recently for the BB84, SARG and 6-state QKD protocols [86]. Here we give a tighter bound than [86] for the achievable secure key-rate of the prepare-and- measure 6-state protocol when the eavesdropper has no access to any quantum memory.

4.1.1 The 6-state protocol The 6-state protocol for quantum key distribution [87] runs in 4 parts.

39 Quantum information put into practice

Alice Eve Bob

ρi F ρ0 { k} i

Figure 4.1: Schematic representation of a Prepare and Measure QKD protocol: Alice prepares a quantum state ρi that she sends to Bob though a public quantum channel, which can be under the control of an eavesdropper.

Distribution : Alice prepares one of the six qubit states ρ = ψ ψ chosen uni- i | iih i| formly at random within

0 + 1 0 1 0 + i 1 0 i 1 ψ1 = 0 , ψ2 = 1 , ψ3 = | i | i, ψ4 = | i − | i, ψ5 = | i | i, ψ6 = | i − | i. | i | i | i | i | i √2 | i √2 | i √2 | i √2 (4.1.1) A i 1 She remembers the basis b = − corresponding to this state as well as the bit X = i 1 b 2 c − mod 2. Alice sends this state to Bob through a public quantum channel. Upon receival of the system from Alice, Bob measures it in either the x, y, or z basis. He remembers his choice of basis bB = 0, 1, 2 as well as the result of his measurement Y = 0, 1. This A step is repeated N times, allowing the parties to accumulate the strings bk , Xk and B { } { } bk , Yk . { } { } A B Sifting : Alice and Bob publicly announce their choice of bases bk and bk . Having learned the other party’s choice of basis, they discard the runs k in which bA = bB (X k 6 k k and Yk are not expected to be correlated in this case), and keep the results from the other A B runs indexed by k0. The basis information is then b = b = b . k0 k0 k0 Error correction : An error correction protocol is run from Alice to Bob1 in order to correct for expected errors between their sifted raw key strings X and Y . This { k0 } { k0 } corrects Bob’s string Y to let him hold the same sifted bit string X as Alice. This { k0 } { k0 } procedure also lets Alice and Bob evaluate the average Quantum Bit Error Rate (QBER): Q = P (X = Y ). k0 6 k0 Secure key extraction : Privacy ampliﬁcation is performed on the corrected bit strings X in order to extract its secret part. { k0 } During the whole protocol, exchanges of classical information are authenticated with the initial secret key shared by the two parties in order to avoid man-in-the-middle attacks.

4.1.2 Secret key rate Here, we consider an eavesdroper, Eve, which can access the quantum channel used by Alice to send the quantum states she prepares to Bob, and which can listen to all classical transmissions taking place during the protocol. However, Eve cannot hold any quantum information. Her most general interaction with the quantum channel can thus be modeled by a POVM acting independently on each of the qubits sent by Alice (c.f. Figure 4.1). Notice that Eve’s power is greatly reduced compared to the case in which she performs a general coherent attack. In particular she cannot use any information about the basis A,B used by Alice or Bob sk to choose how to measure her system. Moreover, since each run k of the protocol is treated independently of the precedent ones by Alice and Bob, the most powerful attack that Eve can perform is an individual attack.

1Note that reverse reconciliation, in which Bob sends information to Alice for her to recover Bob’s key, or two-way reconciliation [88] is also possible, but we don’t consider this case here.

40 4.1 Memoryless attack on the 6-state QKD protocol

0.8 Collective attack Memoryless attack (EB) Memoryless attack (PM)

0.6

0.4 ertkey rate secret

0.2

12.6% 21.0% 0 0% 5% 10% 15% 20.4% 25% QBER

Figure 4.2: Comparison of the secret key rate of the 6-state protocol in diﬀerent situations. The bound for collective attack is as given by [90]. The two bounds against adversary without a quantum memory are in the entanglement-based scheme (EB) as given by [86] and as given by equation (4.1.4) for the prepare and measure scheme (PM).

We thus use the Csisz´ar-K¨orner formula [89], which expresses the secret key rate that Alice and Bob can extract during a realization of the protocol:

r = I(A : B) min(I(A : E),I(B : E)) (4.1.2) − where A represents Alice’s sifted key (i.e. X ), B Bob’s sifted key, and E any system { k0 } hold by Eve. I(X : Y ) here stands for the mutual information between variables X and Y . The mutual information between Alice and Bob is given as usual by:

I(A : B) = 1 h(Q), (4.1.3) − where h(p) = p log p (1 p) log(1 p) is the binary entropy function of p. The following − − − − result provides a lower bound on the key rate r by upper-bounding the maximal mutual information between Alice and Eve as a function of the QBER.

Result. The maximum information that an eavesdropper without quantum memory can have in common with Alice’s bits after sifting is given by:

1 1 3Q(2 3Q) I(A : E) = 1 h − − . (4.1.4) 3 " − p 2 !# (proof in Appendix B)

Note that this bound does not refer to the length N of the raw key produced by Alice and Bob. It is thus only valid in the limit of infinite key length N . → ∞ 4.1.3 Discussion A result similar to the one presented here was recently published by Aurélien Bocquet, Anthony Leverrier and Romain Alléaume in [86]. However, their analysis refers to the entanglement-based realization of the 6-state protocol. In this version, preparation of the state ρi by Alice is realized by letting her measure in the σx, σy or σz basis a maximally entangled state shared with Bob. Since the eavesdropper can interact with the quantum

41 Quantum information put into practice

channel during the distribution of the entangled state ρAB, she can in principle hold a puriﬁcation ψ such that tr ( ψ ψ ) = ρ of this state. | ABEi E | ABEih ABE| AB A comparison between the achievable key rate in the above prepare-and-measure and in the entanglement-based scheme is shown in Figure 4.2. This shows that the key rate is slightly higher in the prepare-and-measure scheme. This contrasts with the same bounds for the BB84 QKD protocol, which are identical for both implementations.

4.2 Private database queries

While QKD allows one to secure the communication between two trustfull parties, many more cryptographic tasks can be considered. Here we consider a situation in which Alice wants to learn about an element of a database held by Bob, without letting Bob know which element she’s interested in. This task is also known as 1 out of N oblivious transfer (for a database of N elements) [75]. The security of the database querries consists of two parts: Database security: Bob wants to bound the information that Alice can access on his • database. Ideally he would like this information to be restricted to 1 bit per querry. User privacy: Alice wants to bound the probability that Bob can learn which item • of his database she is interested in. Ideally, he should get no information about it. Even though it was proved that both aspects of the security cannot be fully satisﬁed at the same time [76, 77], Giovannetti, Lloyd and Maccone [78] recently proposed a quantum protocol that could provide a reasonable level of security for both the user and the database provider. However, in lossy situations the security of their protocol is compromised: since it requires Alice to send her question to Bob before knowing whether her system will come back with an answer or not, Bob can take advantage of the losses by requiring Alice to send her question several times, and thus learning what her question is with high probability. Here we propose a protocol for private database queries based on the SARG QKD protocol [79], which is fundamentally noise-resistant. After presenting the main ideas of our protocol, we argue about the partial security it provides to both the database provider, and the user.

4.2.1 Sketch of the protocol The protocol for private database queries presented here is based on the SARG04 QKD protocol [79], and only diﬀers in the classical processing. We summarize here the main steps of the protocol. Distribution : Bob uniformly chooses one of the four qubit-states , , , | ↑i | →i | ↓i and sends it to Alice. Alice measures the quantum system she received from Bob | ←i either in the σx or in the σz basis and records the measured state. Sifting : If Alice didn’t receive some systems from Bob, due to losses, she tells so to Bob which discards these runs. This allows the protocol to be loss resistant since at this stage, no information about the database, or about Alice’s question has been exchanged. For the systems that Alice received, Bob announces a pair of states within the following ones which contains the state he prepared: , , , , , , {| ↑i | →i} {| →i | ↓i} {| ↓i | ←i} , . {| ←i | ↑i} Transcoding : Bob translates the state and to bits 0, and and | ↑i | ↓i | ←i | →i to bits 1. On her side, knowing her measurement results as well as the sifting sets, Alice

42 4.2 Private database queries tries to guess the bit that Bob computed. This can be summarized in the following table if her measurement result is : | ↑i

Alice’s measurement result sifting set guess of Bob’s state guess of Bob’s bit , ? ? | ↑i {| ↑i | →i} , 1 | ↑i {| →i | ↓i} | →i , 1 | ↑i {| ↓i | ←i} | ←i , ? ? | ↑i {| ←i | ↑i} Information reduction : The bit string of length k N is divided into k substrings × of length N. These substrings are added bitwise, yielding a string of length N. The bitwise addition is commutative and acts as + + = = +, + = , + ? = ? =? (c.f. ⊕ −⊕− ⊕− − ⊕ −⊕ Figure 4.3). If Alice is left with a string of question marks ?, the protocol is restarted. If this happens too often, Bob aborts the protocol to avoid that Alice keeps only cases where she has few question marks.

Figure 4.3: Alice’s information on the key is reduced by xor-ing several keys together.

Database access : Alice announces the number s = j i where j is the item of the − database that she’s interested in and i is an item of the xor-ed key Kf that she knows. Bob announces the N bits C = X Kf where X are the elements of his database. i i ⊕ i+s n Alice deduces the element she’s interested in X = C Kf . i i ⊕ j

4.2.2 Discussion As mention above, a private database query protocol must provide two kinds of security. First, the database holder needs some guarantees that little information about his full database is revealed during the protocol. To see why this is the case here, we realize that the only way for Alice to know elements of Bob’s database is by guessing bits in the key Kf . But the states that Alice needs to discriminate for this, even after having learned the sifting sets, are not orthogonal to each other. An individual attack thus never allows her to learn Bob’s bit with certainty. There is thus a bound on how much information on the database Alice has access to in this case. In paper [H] we discuss in more details how the reduction step ensures that the key hold by Alice contains many question marks, so that she cannot learn many elements of Bob’s database. Second, the user needs to make sure that the database holder has little chance of guessing the element of the database that she is interested in. In order to learn the item of the database that Alice is interested in, Bob needs to guess j, the item of Alice’s final key that is different from a question mark. He thus needs to learn about the conclusiveness of Alice’s transcoding. But the choice of Alice’s measurement bases is unknown to Bob, he can thus never be sure whether she translated her result to a question mark or not. This remains partly true even in the case that Bob sends different states than the ones prescribed by the protocol (c.f. paper [H] for more details).

43 Quantum information put into practice

The above protocol for database queries thus provides some level of security for both the user and the database provider, while being resistant to losses. The exact amount of security provided is however not very clear yet. In particular, we only considered here speciﬁc individual attacks. It would thus be interesting on one side to study more general attacks, and on the other side to develop security proofs for given classes of attacks.

44 Chapter 5

Finite-speed hidden inﬂuences

The violation of a Bell inequality with space-like separated measurements precludes the explaination of nonlocal correlations in terms of causal influences propagating slower than light. Yet, these correlations can still be explained in a causal manner if one gives up Bell’s locality condition. Indeed, this is the explanation followed when one says something like “A measurement on the singlet state ψ = 1 ( 01 10 ) yielding result ‘0’ in the | −i √2 | i − | i computational basis of Alice prepares the state 1 for Bob”. With a slightly different | i taste, Bohmian mechanics also provides a causal explanation for quantum correlations, which does not rely on quantum steering or collapse of the wavefunction. However, both of these explanations are much more nonlocal than a simple violation of Bell’s local causality condition implies: not only do they involve faster-than-light influences at a distance, but these influences also have immediate effects on distant particles no matter how far away they are. Here we question whether such a strong violation of the notion of locality is necessary or not.

5.1 Finite-speed propagation and v-causal theories

One way of violating Bell’s local causality condition while still keeping a notion of“locality” is to allow causal influences to propagate faster than light, but only up to some finite speed v < . In this way, instantaneous influence at a distance is avoided, and causal ∞ influences can still be understood as propagating in spacetime, i.e. acting locally, “de proche en proche”. Since the advent of special relativity, it might seem uncalled-for to consider faster- than-light propagations in space-time1. Indeed, it is well-known that faster-than-light information transmission in a Lorentz-invariant theory can generate temporal paradoxes [9]. However, this needs not be the case if the theory describing the interaction with supra- luminal transmissions is not Lorentz-invariant. For instance, if the speed of every faster- than-light communication is defined in a unique reference frame, then a temporal order is restored. Considering thus a preferred reference frame for definiteness, we can formalize the idea of finite-speed causal influences as follows: to every event K, a past and a future v-cone can be associated in the preferred frame (c.f. Figure 5.1a). What happens at K can only influence other events lying in the future v-cone of K, and K can only depend on what is contained within its past v-cone. We denote by A < B configurations in which

1As a matter of fact, the same remark applies to instantaneous inﬂuences of the kind we just mentioned, which is rarely mentioned.

45 Finite-speed hidden inﬂuences

a) b) c) future B time

K A A B

past space

Figure 5.1: Space-time diagram in the preferred reference frame a) The past and future v- cones (hatched areas) define the sets of events that can influence, or that can be influenced by K within a v-causal theory. b) A < B: finite-speed influences can propagate from A to B. c) A B: no influence can be directly exchanged between two events which are ∼ not in each other’s v-cones.

A lies in the past v-cone of B, and A B those in which A and B lie outside each-other’s ∼ v-cones (c.f. Figure 5.1b-c). Any theory satisfying these constraints is referred to as being v-causal. Note that Bell’s condition of local causality is recovered for v = c.

5.1.1 v-causal models and experimental limitations Clearly, v-causal theories, just like locally causal ones, are fundamentally incompatible with quantum physics. Indeed, they don’t allow two parties to violate a Bell inequality if their measurements are performed simultaneously in the preferred frame, whereas quantum physics predicts that such inequalities can be violated independently of the space-time location of the measurements. Provided that correlations in nature agree with the quantum predictions, one could thus expect to be able to rule out v-causal models experimentally. However this is not directly possible. Indeed, due to the finite accuracy inherent to every experimental manipulation, a v-causal model with sufficiently large speed v can always explain the experimental violation of a Bell inequality. Moreover, since quantum correlations are no-signalling, they can always be reproduced with the aid of the one-way communication available to v-causal models. Thus, quite on the contrary, if all correlations that can be observed in Bell-like experiments agree with the quantum predictions, then they can also be explained by a v-causal theory. Experiments performed so far have thus only been able to put a lower bound on the speed v that is needed for the viability of v-causality. For instance, Salart et al. [91] and Cocciaro et al. [92] have shown that, if the speed of the earth in the preferred frame is 3 less than 10− c, then v-causal theories must have a speed v larger than 10000 times the speed of light c. Given that experimental results cannot rule out v-causal models directly, we examine below in more details the potential physical consequences of these theories. In the following we distinguish between several kinds of correlations. First, correlations are referred to as easily accessible in an experiment if they don’t require very good synchronization between any measurements. All correlations that a v-causal model can freely choose because influence was able to propagate through all parties are of this kind. Second, hardly accessible correlations are those which require nearly perfect synchronization, the degree of synchronization required depending on the speed v of the model. Since some measurements are too simultaneous to allow influences to propagate between them in this case, v-causal models cannot produce all possible correlations of this kind. Finally,

46 5.2 The hidden inﬂuence polytope correlations are said to be not directly accessible if they require perfect synchronization between some measurements. In this case at least part of these correlations involve simultaneous measurements and are thus even inaccessible in principle. A v-causal model is then said to be quantum if every time its correlations are easily accessible they are also in agreement with the quantum prediction. v-causal models which are not quantum can in principle be detected experimentally, whereas quantum v-causal models are experimentally indistinguishable from quantum physics without extraordinary synchronization capabilities. Even though both easily accessible and hardly accessible correlations are in principle measurable, we would like to say something about v-causal models independently of their typical speed v based only on the measurement of easily accessible correlations.

5.1.2 Inﬂuences without communication?

As mentioned in chapter 1, the fact that faster-than-light influences be needed in order to reproduce some correlations does not necessarily allow these correlations to be used to signal faster than light. Rather, a violation of the no-signalling conditions (1.1.2) is needed to allow correlations to be used for communication. The superluminal influences of a v-causal model can thus remain hidden from observers having only access to the produced correlations if these correlations are no-signalling. In particular, as long as the correlations produced agree with the quantum predictions, they are no-signalling and thus cannot be used to communicate. Since all easily accessible correlations produced by a quantum v-causal model are quantum, simultaneous measurements must be considered in order to allow quantum v- causal models with arbitrary speed v to produce correlations diverging from the quantum prediction. It was suggested in [93, 94] that the correlations predicted by a v-causal model in this situation could become signalling and allow for faster-than-light communication. Here, we investigate this question in more detail, and show that it is indeed possible to communicate faster than light in a v-causal world in which all easily accessible correlations are in agreement with quantum physics. Note that a first example of situation in which a v-causal model was shown to allow for faster than light communication was put forward recently in [95]. However, this example requires the observation of supra-quantum correlations in order to conclude. It thus doesn’t apply to quantum v-causal models, and is not likely to lead to an experimental application. We present below a general approach which allows one to test if the existence of signalling correlations can be deduced from the knowledge of potentially accessible correlations. We then examine whether such a test can be expected to be conclusive if correlations observed experimentally are assumed to be the ones predicted by quantum theory.

5.2 The hidden inﬂuence polytope

Following the above discussion, we consider a space-time conﬁguration in which some measurements are simultaneous in order to open the possibility for quantum v-causal models to produce non-quantum correlations. We then examine whether the correlations

47 Finite-speed hidden inﬂuences

time B0 C0

A space

Figure 5.2: In the four-partite Bell-type experiment characterized by the space-time ordering R = (A < D < (B C)), no inﬂuence can be exchanged between Bob and Charly. ∼ However, if Charly delays his measurement, he can allow the conﬁguration to recover a complete order T1 = (A < D < B < C0). Similarly, Bob can delay his measurement in order to obtain the order T2 = (A < D < C < B0). .

produced by the model in this configuration can remain no-signalling or not2. For definiteness, let us consider here the 4-partite space-time configuration R = (A < D < (B C)) shown in Figure 5.2. A v-causal model in this situation must produce ∼ BC correlations that are local, even after conditioning on what happened at A and D (see paper [M] for more details). The correlations P (bc yz, axdw) must thus satisfy all | bipartite Bell inequalities βi P (bc yz, axdw) βi (5.2.1) bcyz | ≤ 0 bcyzX where (βi , βi ) denote the coefficients of all Bell inequalities that are relevant given { 0 bcyz }i the number of inputs and outputs of each party. On the other hand, the correlations P (abcd xyzw) are no-signalling if and only if they | satisfy the 4-partite no-signalling conditions:

P (abcd xyzw) = P (bcd yzw) , P (abcd xyzw) = P (acd xzw) | | | | a X Xb (5.2.2) P (abcd xyzw) = P (abd xyw) , P (abcd xyzw) = P (abc xyz). | | | | c X Xd No-signalling correlations produced by a v-causal model in the R configuration must thus satisfy both condition (5.2.1) and (5.2.2). Since these form a finite set of linear conditions, they define a polytope in the space of correlations (c.f. Appendix A). We refer to this polytope as the hidden influence polytope associated to R. In order to test whether a v-causal model satisfies the above conditions, we need to know which correlations the model produces in the R configuration. But since B and C are measured simultaneously in R, the correlations P (abcd xyzw) are not directly accessible: | their observation requires perfect synchronization between some of the measurements. Still, given the properties of v-causal models, one can show that some parts of the 4- partite distribution P (abcd xyzw) can be deduced indirectly. | 2Note that signalling could be activated in cases where the model only produces no-signalling correlations as well. Indeed, if a marginal probability distribution can have different (no-signalling) values depending on the time chosen by some other party to perform his measurement, in the fashion of [93, 94], this change in the correlation can allow to guess the time of measurement chosen by a distant party. However we don’t consider this possibility here.

48 5.2 The hidden inﬂuence polytope

To see this, consider that Charly, in the experiment of Figure 5.2, could perform his measurement at C as planed initially or choose to delay it to C0 (or even to never do it). In any case, since he can in principle make his choice outside of the past v-cone of A, B and D, his choice cannot affect what happens at A, B and D. Thus, the ABD marginal produced by the model must be the same in the R configuration as in the T1 = (A < D < B < C0) configuration. Since correlations in the T1 configuration are easily accessible, the ABD marginal in the R configuration can be determined through measurements in T1. Similarly, one can show that the ACD marginal in the R configuration must equal that in the T2 = (A < D < C < B0) configuration. It is thus also easily accessible, and both the ABD and the ACD marginal in the R configuration can in principle be known. The BC marginal is however clearly inaccessible experimentally since it explicitely requires measurements to be performed simultaneously. These two 3-partite marginals thus constitute the maximum amount of information that one can hope to infer about the R configuration. In order to reach a conclusion without making assumptions on the value of the unknown marginals, we project the hidden influence polytope onto the subspace spanned by the ABD and ACD marginals. This allows to deduce the conditions that are satisfied by the v-causal model in the situation R, in terms of the known marginals only. Note that whenever an inequality satisfied by this projected polytope is violated, one of the two conditions (5.2.1) or (5.2.2) must be violated as well. Since (5.2.1) cannot be violated in the R configuration, by the definition of v-causality, the model must violate condition (5.2.2) in this configuration, i.e. produce signalling correlations. Using techniques described in Appendix A, we could find several inequalities of the projected hidden influence polytope in configuration R when all parties use binary inputs and outputs. We present one of them below.

5.2.1 Quantum violation and faster-than-light communication The following inequality is satisﬁed by all no-signalling correlations produced by a v-causal model in the R conﬁguration (c.f. Figure 5.2):

S = 3 A B B C 3 D − h 0i − h 0i − h 1i − h 0i − h 0i A B A B + A C − h 1 0i − h 1 1i h 0 0i + 2 A C + A D + B D h 1 0i h 0 0i h 0 1i B1D1 C0D0 2 C1D1 − h i − h i − h i (5.2.3) + A B D + A B D + A B D h 0 0 0i h 0 0 1i h 0 1 0i A B D A B D A B D − h 0 1 1i − h 1 0 0i − h 1 1 0i + A C D + 2 A C D 2 A C D h 0 0 0i h 1 0 0i − h 0 1 1i 7, ≤ where A = P (0 x) P (1 x), A B = ( 1)a+bP (ab xy) and so on. h xi A | − A | h x yi ab − AB | Recall that by construction this inequality only involves correlations that are easily P accessible through some experiment. A quantum v-causal model can thus reproduce any value of S that is achievable with quantum correlations. Interestingly, this inequality can be violated by measuring a 4-qubit state (c.f. paper [M]). We can thus deduce that the corresponding quantum v-causal model must produce signalling correlations in the R conﬁguration.

49 Finite-speed hidden inﬂuences

time

B C D

space A Figure 5.3: In the conﬁguration of Figure 5.2, letting the parties B and C broadcast (at light-speed) their measurement results allows one to evaluate the marginal correlations BCD at the point D0, which lies outside of the future light-cone of A (shaded area). If this marginal depends on Alice’s input, it can thus be used for superluminal communication from A to D0. Similarly, if the ABC marginal correlations depend on the measurement w made at D, superluminal communication is possible from D to the point A0.

Thanks to the geometry of this conﬁguration, any signalling obtained in the correlations can be used to communicate faster than light as soon as v > c. Indeed, by deﬁnition of v-causal models, signalling can neither happen from B to ACD, nor from C to ABD, which lie in the past of C. It must thus happen either from A to BCD or from D to ABC. In both cases, this signalling in the correlations can be used to send signals faster than light (see Figure 5.3). Finite-speed v-causal models for quantum correlations can thus be used to communicate faster than light.

5.3 Experimental perspectives

By construction, inequality (5.2.3) can be evaluated without requiring perfect synchronization between any of the four parties measured. It thus opens the possibility to test v-causal models experimentally in a way that is independent of the speed v, unlike precedent approaches. Namely, an experimental violation of equation (5.2.3) would allow to conclude that if a v-causal model is responsible for the observed correlations, then it must also allow to communicate faster than light in some situations. Note that this conclusion is also valid if other systems than the four of interest happen to be measured during the experiment, even if this makes some measurements happen simultaneously in the privileged reference frame. One way to evaluate the quantity S experimentally could be by performing measurements in the T1 and T2 conﬁgurations of Figure 5.2. This is possible in principle if one knows how the preferred reference frame moves with respect to earth. A simpler demonstration of the violation of S could also be performed without closing the locality loophole, as is common in many Bell experiments. Indeed, by performing measurements in a time-like manner, it is easy to ensure that the measurements are performed according to the T1 and T2 orders. However, this experiment would not be as strong as the previous ones as it would rely on a proper shielding of the measured systems

50 5.4 Conclusion in order to ensure that no communication between them happened by an exchange of some physical systems. Also, it would not allow to conclude directly (i.e. without invoking further assumptions) about the possibility to communicate faster-than-light. Rather, it would allow to conclude that slower-than-light signalling (communication without a physical support) is possible.

5.4 Conclusion

Thanks to inequality (5.2.3), we proved that the nonlocality of quantum correlations cannot be explained by superluminal ﬁnite-speed causal inﬂuences without opening the possibility to communicate faster than light. If one rejects this possibility, then one should also reject v-causal models as an attempt to keep a form of locality in causal explanations of quantum correlations. Moreover, we argued that extraordinary synchronization between measurements is not necessary in order to reach this conclusion, for all v-causal model. This contrasts with previous approaches to v-causal models which could only test models with a speed v limited by technological constraints. It thus opens the possibility for new experimental approach to these models.

51 Finite-speed hidden inﬂuences

52 Conclusion and outlook

In this thesis we presented several studies related to correlations in the context of quantum physics. First, we focused on general properties of correlations, the most important of which being the nonlocality of quantum correlations, i.e. the ability for results obtained upon measurement of a quantum system to violate a Bell inequality. While nonlocality has attracted a lot of attention in the bipartite case, our results indicate that the situation changes dramatically when a third party is considered. Indeed, several of the results presented here don’t have a bipartite analogue: it is impossible in the bipartite case to deduce that some global correlations are nonlocal by only studying their marginals (c.f. section 2.3), no tight Bell inequality for two parties is known to be impossible to violate with quantum correlations (c.f. section 2.5), and bipartite Bell experiments can only test v-causal models with a bounded speed v (c.f. chapter 5). This seems to indicate that much is still to be discovered in multipartite systems. For instance, it would be interesting to explore further the role that the relation between different marginals of a system can play. Also, since several results obtained in the bipartite case don’t extend straightforwardly to more parties, it could be interesting to look at these in more detail. Not much is known for instance about the possibility to simulate entangled states with classical resources in multipartite scenarios. Also, some physical principles like information causality, which generalizes the no-signalling principle to situations in which physical supports with bounded capacity are allowed to carry information, have not yet found a good way to be expressed in multipartite scenarios [8, 96, 97]. Further investigation on these topics can give hints as to whether the difficulty encountered here is simply technical or whether a more fundamental reasons is responsible for them. At a more technical level, given the important role played by polytopes in the characterization of correlations, the development of new tools to work with them would be very helpful. For instance, better ways to deal with symmetries of polytopes are highly desirable. Indeed, symmetries typically induce a high level of redundancy in the description of polytopes, which makes several tasks on them highly inefficient.

In this thesis we also showed how working with correlations can provide robust conclusions in practical situations subject to uncertainties. The device-independent assumptions are indeed weak enough to potentially allow for their implementation in practice, and yet strong enough to allow for the demonstration of interesting results, like the existence of genuinely multipartite entanglement. In other words, the ability to properly separate subsystems under study, and to identify their different possible inputs and outputs can be sufficient to obtain significant results. However, experimental systems need not always meet these requirements. For instance, ions sharing the same trap can be hard to address individually, leading to an imperfect separation between subsystems (c.f. section 3.3). Other systems, like super-

53 Conclusion and outlook conducting qubits [98, 99], are subject to similar limitations. While this mismatch with the working assumptions could be seen as invalidating any possible conclusion, it also seems natural to expect small amounts of cross-talks between subsystems to have limited consequences. A proper way to estimate these cross-talks as well as a careful analysis of their possible impact would be welcome as it would allow one to easily apply the device- independent approach to many practical systems. However it remains to be found.

Finally, we also used correlations in this thesis as tools to study fundamental properties of nature. In particular, following Bell, we questioned the emergence of quantum nonlocal correlations in space-time. What our result suggests is that instead of asking how faster- than-light causal influences can coexist with the theory of relativity, we might just have to wonder about how infinitely-fast causal influences are at all compatible with relativity. Making the meaning of any of these questions more precise would already be a significative step forward.

54 Acknowledgements

First of all, I would like to thank Prof. Nicolas Gisin without which none of the work presented here would have been possible. I am very grateful for the opportunity he gave me to join his group, as well as for his availability for discussions, and in general for his constant support.

A long time ago, Cyril Branciard, Nicolas Brunner and Christoph Simon accompanied my ﬁrst steps in the ﬁeld of quantum information. Thank you!

I owe Stefano Pironio a great deal for all he gave me, from an ounce of mathematical rigor to advices on belgium chocolate.

Thanks to Yeong-Cherng Liang who has always been of great support to me, and whose complementary point of view on many subjects I very much appreciated.

My thanks also go to Nicolas Sangouard for all these ‘short’ discussions...

It was both a privilege and a pleasure to work with the Innsbruck team. Thanks a lot to Prof. Rainer Blatt for making this possible, and to Julio Barreiro for the correspondences.

I’m very grateful to Enrico Pomarico for sharing with me concerns that an physicist can face during an experiment, and for his company during conferences which I enjoyed a lot.

Thank you Bruno Sanguinetti for this marvelous time in Prague.

I am also thankful to Antonio Ac´ın who invited me several times to Castelldefels, and who suggested subjects to work on with some of his coworkers. Thanks also to Mafalda Almeida and Lars Wurﬂinger¨ for the nice collaborations.

I would also like to thank Tam´asV´ertesi for the work we did together, as well as the many other visitors that came to the GAP for a day or more of exchange.

Thank you Clara for your cheerfulness ;-)

Thanks also to Michael Afzelius, Denis Rosset, Charles Lim Ci Wen, Tomy Barnea, Basile Grandjean, Pavel Sekatski, Raphael Ferretti-Sch¨obitz, Markus Jakobi, Keimpe Nevenzeel and all members of the GAP which I had the chance to meet; thanks for your friendliness.

Finally, I am very grateful to my friends and family for their support. Thank you!

55 Acknowledgements

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62 Appendix A

Polytopes

A.1 Deﬁnition and terminology

d A polytope R is the convex hull of a finite number of points Vi = (Vi,1,Vi,2,...,Vi,d) P ⊂ ∈ Rd d = x = (x1, x2, . . . , xd) R s.t. x = qiVi, qi 0 (A.1.1) P { ∈ ≥ } Xi where we denote by x = (1, x1, x2, . . . , xd) the points x completed by an extra component to fit in a space of dimension d + 1 for convenience [100]. In general, several sets of points V can describe the same polytope through { i}i P (A.1.1). For instance, if a point V is not an extremal point of , i.e. if q 0 such that i P ∃ i ≥ Vj = i=j qiVi for some j, then Vi i=j describes the same polytope . On the other 6 { } 6 P hand, if Vj is an extremal point of , then no set of points V Vj can describe the P P { i0} 6⊃ same polytope . The description of a polytope through (A.1.1) is thus minimal when P all point in V are extremal points of the polytope, i.e. vertices. We refer to this as the { i}i extremal points description of a polytope, or V -representation. Notice that the condition for extremality of a point V is linear. The minimal set V can thus be found from i { i}i,min V with the help of linear programming. { i}i The dimension of a polytope dim( ) is given by the dimension of the smallest vector P space that contains . It can be computed from the rank of its extremal points as P rk(V ) if q 0 s.t. q V = (1, 0, 0,..., 0) dim( ) = ij ∃ i ≥ i i i (A.1.2) P (rk(Vij) 1 else. − P The main theorem on polytopes [100] tells that any polytope can also be described P as the intersection of finitely many half-spaces x H H , namely as: j j jk ≥ − 0,k

d P d = x R s.t. xjHj,k 0 k (A.1.3) P { ∈ ≥ ∀ } Xj=0 As in the extremal point description of a polytope (A.1.1), the half-spaces description of a polytope can be made unique and minimal by requiring its inequalities to be irredundant, i.e. such that no qk 0 can satisfy Hj,k = qk Hj,k . ≥ k0=k 0 0 When an inequality is irredundant, it is called a facet of the6 polytope. Its intersection P with is then of dimension dim( ) 1. An inequality satisﬁed by the polytope which P P − is not a facet might still have a non-null intersection with the polytope. The intersection

63 Polytopes of this inequality with the polytope is a often called a face of the polytope, and has a dimension strictly less than dim( ) 1. P − A polytope can thus be described in two equivalent ways (A.1.1), (A.1.3). Transform- ing one representation of a polytope into its dual one is in general a diﬃcult task [101]. Nevertheless, when the polytopes are not too complicated, it can be possible to perform this transformation exactly with the aid of a computer. Several open-source softwares are available for this, like lrs [102], cdd [103], skeleton [104] or porta [105].

A.2 Some operations on polytopes

Polytopes can be manipulated in several ways. Here we describe some of these operations. See also the appendix of [61] for more examples.

A.2.1 Projection

One way to reduce the dimensionality of a polytope is to project it onto a subspace S Rd. For this, consider the linear projection operator Π : Rd S. Without loss of ⊂ →1 generality, Π can be taken to act as Π(x) = (x1, x2, . . . , xs, 0,..., 0) , where s = dim(S) is the dimension of the projected space. The projection of a polytope described in terms of extremal points is easily computed by projecting its extremal points:

d 0 = Π( ) = x R s.t. x = qiΠ(Vi), qi 0 . (A.2.1) P P { ∈ ≥ } Xi Note that all projected vertices Π(V ) need not be extremal points of the projected { i }i polytope anymore. However all extremal points of are necessarily projections of some P0 extremal points of the original polytope. They are thus necessarily contained in the set Π(V ) . Projection of a polytope can thus only reduce the number of its extremal { i }i vertices. When a polytope is specified in terms of half-spaces, finding the H-representation of its projection is more difficult. The Fourier-Motzkin algorithm achieves this without requiring to first transform the description of the polytope into its V-form, but it becomes quickly unpractical for larger problems because of its double-exponential computational complexity.2 Still, if one is not interested in the full set of inequalities describing the projected polytope , a number of its facets can be found heuristically. Here we describe two P0 linear programs that can be used for this. The first linear program for finding facets of a projected polytope is similar to the shooting oracle described in [106]. The idea is, starting from a point that belongs to the interior of , to travel as far as possible in one direction of the subspace S, until touching P0 the boundary of . The point then reached must generically belong to a facet of . P0 P0

Find a facet of a projected polytope. Let x0 belong to the interior of , and P0 1A change of variables can be performed on order to let Π take this form if necessary. 2Note that other algorithms have been proposed, such as the ESP one [106] which is sensitive to the number of facets in the projected polytope 0 rather than in the number of facets of the full polytope . P P

64 A.2 Some operations on polytopes

let v S be a direction in the projected subspace. The linear program ∈ s 0 min Ho,kzk + xj Hj,kzk zk Xk Xk Xj=1 s subject to v H z = 1 j j,k k − (A.2.2) Xj=1 Xk H z = 0 j = s + 1, . . . , d j,k k ∀ Xk z 0 k ≥ provides the inequality s I + x I 0 (A.2.3) 0 j j ≥ Xj=1 s 0 with coeﬃcients I0 = k Ho,kzk + k j=1 xj Hj,kzk, Ij = k Hj,kzk which is satisﬁed by all points x belonging to the projection of the polytope . To P P P P P0 see this one can check that the dual of this program is

max µ µ,ys+1,...,yd s d (A.2.4) subject to H + (x0 + µv )H + y H 0 k 0,k j j j,k j j,k ≥ ∀ Xj=1 j=Xs+1 which computes the largest value of µ such that x0 + µv . ∈ P0

In order to ﬁnd a facet with the above linear program, one needs to choose a direction v S. Choosing an interesting direction is not necessarily obvious when nothing or just ∈ little about the projected polytope is known. Here is a linear program which can provide an interesting direction v to look for a facet when some facets of the projected polytope are known. The idea is that an interesting direction to look for a new facet of the P0 projected polytope is in the direction of a vertex of the polytope described by the known facets of this polytope: if there is a diﬀerence between the known polytope and , then P0 some of its extremal points need to be outside , otherwise the two polytopes are equal P0 and there is nothing left to be found.

Find an extremal point of a polytope. Let Hk k be a set of inequalities deﬁning d { } a polytope 00, and let w R be a direction in space. The following linear P ∈ program yields a point x which lies on the boundary of : P00

min wjxj xj j X (A.2.5) subject to H + H x 0 k 0k jk j ≥ ∀ Xj If w is chosen at random, the point x is generically an extremal point of . P00

65 Polytopes

Application to polytope representation conversion It is possible to find the H-representation of a polytope from its V -representation by performing a polytope slice followed by a polytope projection. Indeed, the conditions in equation (A.1.1) can be understood as defining a polytope ˜ Rd+n, where n is the P ∈ number of extremal points of , and ˜ is defined by the following H-representation: P P ˜ d+n = (x, q) R s.t. x = qiVi, qi 0 (A.2.6) P { ∈ ≥ } Xi ˜ d Elimination of the variables qi, i.e. projection of onto the subspace S = R thus defines P the set of x such that the conditions in (A.1.1) hold. This projected polytope is thus , described in terms of inequalities. This is the basic technique used by the software P porta [105] to solve a polytope.

A.2.2 Slice Another way of reducing the dimensionality of a polytope is to consider a slice of it, that is, the intersection of the polytope with a subspace S Rd of dimension s. S can be P d ⊂ defined by a set of linear equations: S = x R s.t. xjEjl = 0 l = 1 . . . d s . The { ∈ j ∀ − } slice of which belongs to S is easily defined if is specified in the H-representation: P P P d 0 = x R s.t. xjMkj 0 k, xjEjl = 0 l (A.2.7) P { ∈ ≥ ∀ ∀ } Xj Xj Computing the slice of a polytope in its V -representation is more complicated. In fact, on can show through polytope duality [100] that this task is equivalent to a projection of the polytope dual to [107]. It can thus be done with the techniques described in the P precedent section.

A.2.3 Another tasks : finding facets lying under an inequality An inequality satisfied by a polytope is not a facet of the polytope if its dimension is lower than d 1. When this is the case, one can show that the inequality can be expressed as − the convex combination of a number of tighter inequalities, the tighter of which are facets of the polytope, sharing an intersection of dimension d 1 with the polytope. Thus, if a − point violates a non-facet inequality, it can only violate some of these tighter ones by a larger amount. Given a non-facet inequality, it can thus be interesting to look for these tighter facets. A method for finding the facets underlying an inequality given a V -description of the polytope is presented in paper [E]. The idea is that the rank of the set of extremal points saturating the inequality can be augmented by adding more extremal points of the polytope to this set. When the achieved rank is sufficient, and if the hyperplane passing through these points does not cut the polytope into two parts, then it describes a facet of the polytope. By construction, the intersection of this facet with the polytope coincides with the intersection of the original inequality with the polytope since it is generated by the same set of extremal points.

66 Appendix B

Memoryless attack on the 6-state protocol – proof

Here we provide a proof for the bound (4.1.4) used in the main text.

Proof. Without loss of generality, Eve’s POVM elements can be written as:

F = A† A = a 11 + (b a )P (B.0.1) k k k k k − k k

where Ak = √akUkPk + √bkUkPk is a Kraus operator associated to the element Fk, ak, bk 0, Uk is a unitary operator, Pk = 11 Pk and Pk is a one-dimensional projector.≥ − Eve’s information on Alice’s bit. We consider a given run k of the protocol for which Alice and Bob used the same basis b = bk, and denote by A Alice’s bit, B Bob’s bit and E the result of Eve’s POVM measurement. The total information gained by Eve on Alice’s bit after the sifting procedure is given by I(A :(E, b)). Since b is independent of A and E, and since the state produced by Alice ρ = ρ(A, b) is a function of A and b, we have:

I(A :(E, b)) = H(A) + H(E, b) H(A, E, b) − = H(A) + H(E) + H(b) H(A, E, b) − (B.0.2) = H(A, b) + H(E) H(A, E, b) = I((A, b): E) − = I(ρ : E) = log(6) I(ρ E) − | Where H is Shannon’s entropy, and we assumed that the six states are chosen by 1 Alice with the same probability 6 . We thus need to bound the quantity I(ρ E) which expresses the information that Eve is missing after she learns the result of| her measurement, to know which state ρ was prepared by Alice. Using the fact that Prob(E = k) = tr(Fk)/2, this quantity can be expressed as

I(ρ E) = I(E ρ) + log(tr(F )) + log(3) (B.0.3) | | k Xk where I(E ρ) = Prob(A = a, b = s, E = k) log(Prob(E = k A = a, b = s)). | − a,s,k | For every statePρ that Alice can produce, ρ = 11 ρ can also be produced by Alice. An attack described by the POVM elements F −thus provides Eve with the same { k} information as the attack F˜k where F˜k = Ak11 + (bk ak)Pk. Indeed, as shown above, this information is a{ symmetric} function of P (E =−k ρ = ρ(a, b)) and |

tr(ρFk) = ak + (bk ak)tr(ρPk) − (B.0.4) = a + (b a )tr(ρF˜ ) = tr(ρF˜ ). k k − k k k 67 Memoryless attack on the 6-state protocol – proof

Moreover, the information that can be extracted from a mixture of measurements M1 applied with probability p1 and M2 applied with probability p2 is

I([(p1,M1), (p2,M2)]) = p1I(M1) + p2I(M2), (B.0.5) where I(M1,2) is the information that can be extracted by using measurement M1,2 only. We can thus write

1 1 I( F ) = I F F˜ { k} 2 k ∪ 2 k (B.0.6) a + b F F˜ = k k I k , k 2 "(ak + bk ak + bk )# Xk where the factor 1/(ak + bk) is a normalization coeﬃcient. Thus, the information gained by performing an arbitrary POVM measurement can also be achieved by mixing measurement strategies consisting of only two POVM elements. Let us thus consider a POVM measurement for Eve consisting only of the two elements Fk ak+bk ˜ and Fk . ak+bk By direct computation one ﬁnds that 1 I(E ρ) = (h(c0 ) + h(d0 ) + h(e0 )) (B.0.7) k| 3 k k k

ak+tr(ρ2Pk)(bk ak) ak+tr(ρ4Pk)(bk ak) ak+tr(ρ6Pk)(bk ak) with c0 = − , d0 = − , e0 = − . Since k ak+bk k ak+bk k ak+bk this function is convex in c0, d0, e0, its minimum lies on the boundary of the admissible region

1 2 1 2 1 2 1 tr(ρ P ) + tr(ρ P ) + tr(ρ P ) . (B.0.8) 2 k − 2 2 k − 2 2 k − 2 ≤ 4 More precisely, this is found for tr(ρ P ) 0, 1 , tr(ρ P ) 0, 1 , or tr(ρ P ) 2 k ∈ { } 4 k ∈ { } 6 k ∈ 0, 1 . In this case, since log tr Fk = 0, we ﬁnd { } ak+bk bk 2 + h a +b I(ρ E ) = k k + log(3) (B.0.9) | k 3

ak where k = . bk All in all, this gives the following bound on Eve’s information about Alice’s bit:

2 + h 1 a + b 1+k I(A :(E, b)) 1 k k . (B.0.10) ≤ − 2 · 3 Xk

Perturbation on Bob’s system. The attack of Eve delivers the state ρi0 = k AkρiAk† to Bob instead of the expected ρi. This creates some errors in the outcomes of Bob, which are measured by the QBER: P

6 1 Q = 1 P (A = i, B = i) = 1 tr(ρ0 E ) (B.0.11) − − 6 i i i=1 i X X where Ei, i = 1,..., 6 are the six possible measurement operators of Bob and

ρ0 = a b U ρ U †+√a (√a b )U P ρ P U †+ b ( b √a )U P ρ P U †. i k k k i k k k− k k k i k k k k− k k k i k k k X p p p p (B.0.12)

68 Note, that the attack A has the same eﬀect as A˜ for A˜ = √a U P + { k} { k} k k k k √bkUkP k. Indeed, for all i there is a j such that ρi = ρj and Ei = Ej, and one can check that:

tr(UkρiU †Ei) = tr(Ukρ U †Ei), k i k (B.0.13) tr(UkPkρiPkUk†Ei) = tr(UkP kρiP kUk†Ei).

So we can assume that both Ak and A˜k are present in the attack. In this case the perturbed state is

2 (√ak √bk) ρ0 = a b U ρ U † + − (U P ρ P U † + U P ρ P U †). (B.0.14) i k k k i k 2 k k i k k k k i k k k X p To bound the second part of this expression, one can show by direct computation that

tr(UkP kρiP kUk†Ei) + tr(UkPkρiPkUk†Ei) i X (B.0.15) = tr(P kρi)tr(P kUk†EiUk) + tr(Pkρi)tr(PkUk†EiUk) i X = 2((1 c)(1 c˜) + cc˜ + (1 d)(1 d˜) + dd˜+ (1 e)(1 e˜) + ee˜) − − − − − − ˜ where c = tr(Pkρ2), d = tr(Pkρ4), e = tr(Pkρ6),c ˜ = tr(UkPkUk†E2), d = tr(UkPkUk†E4) ande ˜ = tr(UkPkUk†E6). The maximum value of (B.0.15) under the constraints 2 2 2 2 2 2 c 1 + d 1 + e 1 1 , c˜ 1 + d˜ 1 + e˜ 1 1 can be − 2 − 2 − 2 ≤ 4 − 2 − 2 − 2 ≤ 4 checked to be 4.

Finally, the ﬁrst part of (B.0.14) can also be bounded since tr(UkρiUk†Ei) 1. Thus we ﬁnd that ≤ a + b (1 √ )2 Q k k − k . (B.0.16) ≥ 2 · 1 + k Xk Putting the two bounds together. Let us consider equations (B.0.10) and (B.0.16) together. Keeping the sum ak + bk constant for all k, we choose two values of k if possible: k1 and k2 such that k1 < k2. Following [85] one can show that increasing k1 in such a way that keeps the bound on the QBER (B.0.16) unchanged, can only decrease k2 and increase Eve’s information as given by (B.0.10). It is thus always better to have = k. In this case both bounds become: k ∀ 1 1 h 1+ (1 √)2 I(A :(E,S)) − ,Q − (B.0.17) ≤ 3 ≥ 3(1 + ) which can be summarized as

1 1 3Q(2 3Q) I(A :(E,S)) 1 h − − . (B.0.18) ≤ 3 " − p 2 !#

Tightness of the bound To show that the above bound is tight, consider the attack 1 γ in which Eve uses the two POVM elements Fk = −2 11 + γ k k for k = 0, 1. This gives P (ρ = z k) = (1 ( 1)kγ)/6, P (ρ = x k) = P (ρ|=ih |y k) = 1/6, and so 2 ± | 1 ±1 −γ ± | ± | I(A E) = 3 + log 3 + 3 h −3 . Since I(A) = 1 + log 3, this attack proved Eve with | 1 1 γ a mutual information with Alice of I(A : E) = I(A) I(A E) = 1 h − . − | 3 − 2 1 √1 γ2 − − Moreover, the QBER induced by this attack is Q = 3 . Thus Eve can choose an attack that saturates equation (4.1.4): the bound is tight.

69 Memoryless attack on the 6-state protocol – proof

70 Papers

List of published papers:

[A] Testing a Bell inequality in multipair scenarios, J.-D. Bancal, C. Branciard, N. Brun- ner, N. Gisin, S. Popescu and C. Simon, Phys. Rev. A 78, 062110 (2008).

[B] Quantifying Multipartite Nonlocality, J.-D. Bancal, C. Branciard, N. Gisin and S. Pironio, Phys. Rev. Lett. 103, 090503 (2009).

[C] Simulation of Equatorial von Neumann Measurements on GHZ States Using Non- local Resources, J.-D. Bancal, C. Branciard and N. Gisin, Adv. in Math. Phys., vol. 2010, Article ID 293245 (2010).

[D] Guess Your Neighbor’s Input: A Multipartite Nonlocal Game with No Quantum Ad- vantage, M. L. Almeida, J.-D. Bancal, N. Brunner, A. Ac´ın, N. Gisin and S. Pironio, Phys. Rev. Lett. 104, 230404 (2010).

[E] Looking for symmetric Bell inequalities, J.-D. Bancal, N. Gisin and S. Pironio, J. Phys. A: Math. Theor. 43, 385303 (2010).

[F] Detecting Genuine Multipartite Quantum Nonlocality: A Simple Approach and Gen- eralization to Arbitrary Dimensions, J.-D. Bancal, N. Brunner, N. Gisin and Y.- C. Liang, Phys. Rev. Lett. 106, 020405 (2011).

[G] Extremal correlations of the tripartite no-signaling polytope, S. Pironio, J.-D. Bancal and V. Scarani, J. Phys. A: Math. Theor. 44, 065303 (2011).

[H] Practical private database queries based on a quantum-key-distribution protocol, M. Jakobi, C. Simon, N. Gisin, C. Branciard, J.-D. Bancal, N. Walenta and H. Zbinden, Phys. Rev. A 83, 022301 (2011).

[I] Various quantum nonlocality tests with a commercial two-photon entanglement source, E. Pomarico, J.-D. Bancal, B. Sanguinetti, A. Rochdi and N. Gisin, Phys. Rev. A 83, 052104 (2011).

[J] Device-Independent Witnesses of Genuine Multipartite Entanglement, J.-D. Bancal, N. Gisin, Y.-C. Liang and S. Pironio, Phys. Rev. Lett. 106, 250404 (2011).

[K] Loophole-free Bell test with one atom and less than one photon on average, N. San- gouard, J.-D. Bancal, N. Gisin, W. Rosenfeld, P. Sekatski, M. Weber and H. Wein- furter, Phys. Rev. A 84, 052122 (2011).

[L] A framework for the study of symmetric full-correlation Bell-like inequalities, J.- D. Bancal, C. Branciard, N. Brunner, N. Gisin and Y.-C. Liang, J. Phys. A: Math. Theor 45, 125301 (2012).

71 Papers

Preprints:

[M] Quantum nonlocality based on ﬁnite-speed causal inﬂuences leads to superluminal signaling, J.-D. Bancal, S. Pironio, A. Ac´ın, Y.-C. Liang, V. Scarani and N. Gisin, arXiv:1110.3795

[N] The deﬁnition of multipartite nonlocality, J. Barrett, S. Pironio, J.-D. Bancal and N. Gisin, arXiv:1112.2626

[O] Imperfect measurements settings: implications on quantum state tomography and entanglement witnesses, D. Rosset, R. Ferretti-Sch¨obitz, J.-D. Bancal, N. Gisin and Y.-C. Liang, arXiv:1203.0911

[P] Useful multipartite correlations from useless reduced states, L. E. Wurﬂinger,¨ J.- D. Bancal, T. V´ertesi, A. Ac´ın and N. Gisin, arXiv:1203.4968

72 Paper A

Testing a Bell inequality in multipair scenarios

J.-D. Bancal, C. Branciard, N. Brunner, N. Gisin, S. Popescu and C. Simon

Physical Review A 78, 062110 (2008)

PHYSICAL REVIEW A 78, 062110 ͑2008͒

Testing a Bell inequality in multipair scenarios

Jean-Daniel Bancal,1 Cyril Branciard,1 Nicolas Brunner,1 Nicolas Gisin,1 Sandu Popescu,2,3 and Christoph Simon1 1Group of Applied Physics, University of Geneva, 20 rue de l’Ecole-de-Médecine, CH-1211 Geneva 4, Switzerland 2H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom 3Hewlett-Packard Laboratories, Stoke Gifford, Bristol BS12 6QZ, United Kingdom ͑Received 8 October 2008; published 15 December 2008͒ To date, most efforts to demonstrate quantum nonlocality have concentrated on systems of two ͑or very few͒ particles. It is, however, difﬁcult in many experiments to address individual particles, making it hard to highlight the presence of nonlocality. We show how a natural setup with no access to individual particles allows one to violate the Clauser-Horne-Shimony-Holt inequality with many pairs, including in our analysis effects of noise and losses. We discuss the case of distinguishable and indistinguishable particles. Finally, a comparison of these two situations provides insight into the complex relation between entanglement and nonlocality.

DOI: 10.1103/PhysRevA.78.062110 PACS number͑s͒: 03.65.Ud

I. INTRODUCTION way to tell which particle is entangled with which. The corresponding loss of entanglement has been derived in Ref. ͓5͔. Entanglement is the resource that allows one to establish In the second case, the pairs are indistinguishable; so in some quantum nonlocal correlations ͓1͔. These correlations have sense the information about the pairing is here lost in a co- been the center of a wide interest, because of their fascinat- herent way. ing nature and of their impressive power for processing in- Reid et al. ͓6͔ have considered the case of indistinguish- formation. Experimentally, quantum nonlocality has been able pairs ͑with global measurement͒ in optics. More specifi- demonstrated in so-called Bell experiments, which have to cally, these authors, extending on a previous work of Drum- date all confirmed the quantum predictions ͓2͔. mond ͓7͔, showed how Bell inequalities can be tested ͑and Most theoretical works on Bell experiments and Bell in- violated͒ when many pairs are created via PDC. In this case equalities have focused on the case where the source emits a the pairs are indistinguishable because of the process of single entangled pair of particles at a time. Indeed, this is the stimulated emission. In Ref. ͓8͔, Jones et al. have considered simplest situation to study. From the experimental point of a related scenario; there, entangled pairs are delivered via an view, most experiments have been designed in order to inept delivery service, but at the end only a single pair is match this theoretical model. For example, in photonic ex- measured. Also considering multiparticle entanglement in periments, the source, usually based on parametric down- such scenario is an interesting problem: see, for example, conversion ͑PDC͒, is set in the weak regime; i.e., when the Refs. ͓9,10͔. source emits something, it is most likely a single pair of In this paper we will study the violation of Bell inequali- entangled photons. ties in a general multipair scenario. We start by treating the However, there are experimental situations, such as in case of independent pairs ͑Sec. II͒. We argue that the resis- many-body systems, where producing single entangled pairs tance to noise is here the relevant measure of nonlocality, is rather difficult. For instance, in Ref. ͓3͔ many entangled evaluating it. The consequences of particle losses are also pairs ͑Ӎ104͒ of ultracold atoms have been created, but can- investigated. Next, we move to the case of indistinguishable not be addressed individually. So, while entanglement has pairs ͑Sec. III͒ after a brief review of the results of Ref. ͓6͔, definitely been created in this system, one still lacks an effi- we present an analysis of the influence of noise and losses in cient method for demonstrating its quantum nonlocality this case. In Sec. IV, we compare the entanglement and non- through the violation of some Bell inequality. The goal of the present paper is to discuss techniques for testing Bell inequalities in such multipair scenarios, where the particles on Alice’s and Bob’s side cannot be individually addressed, and must therefore be measured globally ͑see Fig. 1͒. What we mean here by global measurements is that each particle is submitted to the same measurement. Note that the case of more general measurements ͑collective measurements on all particles͒ has been considered in Ref. ͓4͔. Basically one should distinguish two cases: independent FIG. 1. Setup: a source produces M independent pairs ͑or pairs and indistinguishable pairs. In the first case, the pairs equivalently M independent sources each produce a pair͒, the pair- are created independently, but cannot be addressed individu- ing between Alice’s and Bob’s particles is lost during their trans- ally; therefore, they must be measured globally ͑on both Al- mission, and each party measures all their incoming particles in the ͒ ϩ͑Ϫ͒ ice’s and Bob’s sides . During this global measurement, the same basis. The total number n+͑−͒ of particles detected in the classical information about the pairing is lost: there is no outcome is tallied on both sides.

1050-2947/2008/78͑6͒/062110͑8͒ 062110-1 ©2008 The American Physical Society BANCAL et al. PHYSICAL REVIEW A 78, 062110 ͑2008͒ locality in both cases. This leads us to a surprising result: threshold N=M /2,2M /3,...,M such that the outcome is ജ while the state of indistinguishable pairs contains more en- + iff n+ N. tanglement than the state of independent pairs ͑after the clas- At this point the two relevant questions are the following: sical mixing͒, the latest appears to be more nonlocal. In other ﬁrst, is it possible to violate the CH inequality with any of words, the incoherent loss of information provides more non- these voting procedures? Second, if yes, which strategy locality, but less entanglement, than the coherent loss of in- yields the largest violation? To address these questions one formation ͑indistinguishable pairs͒. This provides a novel ex- must compute the joint and marginal probabilities entering ample ͑here in the case of multipairs͒ of the complex relation the CH inequality for each procedure. between entanglement and nonlocality. Finally we provide some experimental perspectives ͑Sec. V͒ and conclusions. B. Pure states

II. INDEPENDENT PAIRS Let us ﬁrst consider the pure entangled states ͉␺͘ ␪͉ ͘ ␪͉ ͘ ͑ ͒ We consider a source emitting M entangled pairs, each of = cos 00 + sin 11 , 4 them being in the same entangled two-qubit state ␳. Thus the ͑ ͒ ␳ ͉␺͗͘␺͉ ͉⌿ ͘ so that in Eq. 1 , = . We will also write M global state is =͉␺͘M. For detectors with a perfect efﬁciency ␩=1, all M ␳ = ␳M = ␳ ␳ ¯ ␳. particles are detected on both Alice’s and Bob’s sides. The M marginal and joint probabilities entering the CH inequality M times ͑1͒ for a vote with given threshold N are

Each pair being independent, Alice and Bob receive M M uncorrelated particles. Since Alice and Bob are unable to M A͑ ͒ ͩ ͪ ͑ ͒n+ ͑ ͒M−n+ ͑ ͒ P+ A = ͚ p+ A p− A , 5 address single particles in their ensemble, they perform a n =N n+ global measurement on their M particles; i.e., all M particles + ͑ are measured in the same basis we shall consider here only n␣␤ p␣␤͑A,B͒ ͒ ͑ ͒ ͑ ͒ von Neumann measurements or, equivalently, in the same P++ A,B = M! ͚ ͟ , 6 direction on the Bloch sphere. After the measurement appa- n␣␤෈⌶ ␣,␤ n␣␤! ratus, two detectors count the number of particles, n and n , + − where p ͑A ,B ͒= 1 tr(͑ +A ͒ ͑ +B ͉͒␺͗͘␺͉) and in each output mode ͑see Fig. 1͒. If the detectors are per- ++ i j 4 1 i 1 j p ͑A /B ͒ 1 (͑ A ͒ / ͑ B ͉͒␺͗͘␺͉) fectly efﬁcient ͑␩=1͒, one has M =n +n . + i j = 2 tr 1+ i 1 1 1+ j are the quantum + − joint and marginal probabilities for a single pair. Alice and Bob’s outputs are denoted ␣,␤෈͕+,−͖, n␣␤ is the number of A. Testing the CH inequality pairs which gave detections ␣ and ␤, and ⌶=͕n␣␤ ෈ ͉͚ A ജ B ജ ͖ Our goal is to test a Bell inequality. Here we shall focus N+ n␣␤=M ,n+ =n+++n+− N,n+ =n+++n−+ N is the on the simplest Bell inequality, the Clauser-Horne-Shimony- set of all events yielding the result “ϩϩ” after voting. Holt ͑CHSH͒ inequality ͓11͔, which involves two inputs on Next, one can choose the state ͉␺͘ and the measured set- ͑␪ Alice and Bob’s sides, A1, A2 and B1, B2, and two outputs tings. For the maximally entangled state of two qubits ␣,␤෈͕+,−͖. For convenience we write it under the CH form =␲/4͒ one may choose the standard optimal ͑for the case ͒ ␴ ␴ ͓12͔ M =1 settings for the CH inequality—i.e., A0 = z, A1 = x, ␴x+␴z −␴x+␴z B = ͱ , and B = ͱ . Doing so with majority voting =−PA͑A ͒ − PB͑B ͒ + P ͑A ,B ͒ + P ͑A ,B ͒ 0 2 1 2 ICH + 1 + 1 ++ 1 1 ++ 1 2 ͑N=M /2͒, the CH inequality can be violated for any value ͑ ͒ ͑ ͒ ഛ ͑ ͒ + P++ A2,B1 − P++ A2,B2 0, 2 of M; the maximal amount of violation is numerically found to decrease with the number of emitted pairs as M−1. where P ͑A ,B ͒ is the probability for both Alice and Bob to ++ i j Remarkably, a higher violation is found for different mea- output “ϩ” when performing measurements A and B , re- i j surement settings, given by spectively. Recall that under the hypothesis of no-signaling ͓ ͔ ␴ both CH and CHSH inequalities are equivalent 13 . Now, in A0 = z, order to test inequality ͑2͒, Alice and Bob must transform their data, basically n and n , into a binary result “ϩ”or ␣␴ ␣␴ + − A1 = sin 2 x + cos 2 z, “Ϫ”. A natural way of doing it is by invoking a voting procedure: for instance, ␣␴ ␣␴ B0 = sin x + cos z, ͑ ͒ ജ i Majority voting: if n+ n− “+”; → B = − sin ␣␴ + cos ␣␴ . ͑7͒ otherwise “− ”, 1 x z → ␣ϳ ␲ −1/2 With 2ͱ2 M , those settings are numerically found to be ͑ ͒ −1/2 ͓ ii Unanimous voting: if n+ = M “+”; optimal. In this case the decrease of CH is only M see → Fig. 2͑a͔͒. The state leading to the largestI violation is always otherwise “− ”, ͑3͒ → the maximally entangled one ͑␪=␲/4͒ for majority voting. or any intermediate possibility—for instance, 2/3or3/4 ma- The one-parameter planar settings ͑7͒ were already used jority. For each voting method and given M corresponds a by several authors ͓6,7͔; for example, in Bell experiments

062110-2 TESTING A BELL INEQUALITY IN MULTIPAIR SCENARIOS PHYSICAL REVIEW A 78, 062110 ͑2008͒

noise can be modeled at the level of the source, supposing that the produced pairs are not in the pure state ͉␺͗͘␺͉, but instead in a Werner state of the form

1 ␳ = w͉␺͗͘␺͉ + ͑1−w͒ . ͑8͒ 4

The resistance to noise of a given violation is then defined by the maximal amount ⑀=1−w of white noise that can be added to the pure state ͉␺͗͘␺͉ such that the resulting state ␳ still violates a Bell inequality ͑CH here͒. Considering now that the sources of Fig. 1 produce the state ͑8͒, we look for the largest value of ⑀ which still gives ͑ ͒ a positive value of CH, using settings of the form 7 and optimizing on the stateI ͑␪͒. For all voting strategies we find a resistance to noise decreasing like ⑀ϳM−1, the majority voting being still the best choice ͓see Fig. 2͑b͔͒. Unlike when maximizing CH in the absence of noise, here the optimal state is alwaysI the maximally entangled one ͑␪=␲/4͒, even for intermediary voting strategies, for which the CH violation with this state decreases exponentially withI M. This shows that appropriate figures of merit need to be used when examining practical situations. These results are encouraging, but just as detectors might not be perfect, maybe the source cannot guarantee an exact number of pairs, M, as needed here. To show that these vio- ͑ ͒ FIG. 2. Color online CH violation and resistance to noise for a lations are relatively robust towards this issue, we now look source producing M independent pairs. The states and settings used ͑ ͒ at the case of sources producing a number of entangled pairs are discussed in the text. a Maximal CH values for various thresh- which follow a Poissonian distribution. olds: majority voting ͑solid red line͒,3/4 voting ͑dotted green line͒, and unanimity ͑dashed blue line͒. The decrease is as M−1/2 for the first two and exponential for the last one. The highest violation is D. Poisson sources thus reached using a majority vote. ͑b͒ Resistance to noise for the ␳ different thresholds ͑same colors͒. All curves decrease as M−1. The A Poissonian source produces a state M of M pairs with most resistant violation is that achieved by using majority voting. a Poissonian probability

͑ ͒ ␮M using a state we shall look at 10 later with the unanimous p͑M͒ = e−␮ , ͑9͒ vote for any M. M! Performing numerical optimizations, we also found that a violation can be obtained for any voting strategy with any where ␮ is the mean number of photon pairs. With such a number of emitted pairs M ͓see Fig. 2͑a͔͒. We optimized the source, a different number of pairs is created every time. So state ͑␪͒ and the four measurement settings, each time find- for a chosen voting assignment ͑3͒ the threshold N varies ͑ ͒ ͑ ing optimal settings of the form 7 . For the unanimous vote, with the total number of photons detected, M =n+ +n− we for instance, the optimal state is less and less entangled as the still consider perfect detectors͒, according to each realiza- number of emitted pairs, M, increases, as described in ͓14͔, tion. and the violation decreases exponentially with M. Thus for Using settings of the form ͑7͒, we optimized numerically pure entangled two-qubit states, the largest amount of viola- ␣ and the state ͑␪͒ for several votes, in a situation where the tion is obtained with majority voting. source is Poissonian. Doing so in order to get the largest CH violation and the highest resistance to noise, we obtained results very similar to that of the fixed M case, verifying in C. Resistance to noise ␮−1/2 particular a decrease of CH as for the majority vote We now compute the resistance to noise that these viola- and of the resistance to noiseI as ␮−1 ͑see Fig. 3͒. Similarly, tions could bear, which is the relevant measure of nonlocality the states yielding the largest CH values are the maximally considering experimental perspectives—the amount of viola- entangled one for the majorityI vote and partially entangled tion being basically just a number, without much significance ones for the two other votes. A difference, however, is that in the present case as we shall see. CH is found to decrease slower than exponentially for the In a practical Einstein-Podolsky-Rosen ͑EPR͒ experiment, Iunanimity vote. Ͼ imperfect detectors, noisy sources, or disturbing channels in- Note also that since it is possible to find CH 0 and the troduce noise in the measurement results. To first order, this probability to get a ϩ result vanishes for ␮ I 0, there exist → 062110-3 BANCAL et al. PHYSICAL REVIEW A 78, 062110 ͑2008͒

FIG. 4. ͑Color online͒ CH violation with inefficient detectors as a function of the probability for a photon to be detected. The upper thin curve shows the traditional case M =1 with known critical efficiency ␩=2/3 ͓15͔. The two other curves are for M =5 pairs with majority ͑solid red line͒ and unanimity voting ͑dashed blue line͒. find a Bell violation in such circumstances. Figure 4 shows the maximal CH violations obtained ͑optimizing on states and settings͒ as a function of the detection efficiency for M =1 and M =5 with majority and unanimity voting. The required detector efficiency increases with the number of pairs, leaving no chance to find a violation at high M with ␩Ӷ1. One way to deal with detector inefficiencies consists in post-selecting events in which exactly M photons are detected on both sides. In this way, cases in which particles FIG. 3. ͑Color online͒ Maximal Bell violation and resistance to were not detected are neglected and the Bell violation is ͑ noise with a Poissonian source of independent pairs. The red lines recovered independently of the losses. If detectors also have represents the majority voting, the green dotted lines the 3/4 vot- dark counts, noise will appear in the statistics, which can be ing, and the blue dashed lines the unanimity voting. Settings are treated with Werner states as presented in the “resistance-to- ͒ chose in the form ͑7͒; optimal states are discussed in the text. ͑a͒ noise” section. This approach is, however, not perfect as it is subject to the so-called detection loophole ͓16͔: there exist For a large mean photon number, the decrease of CH goes like ␮−1/2 for the majority and 3/4 vote, just like for theI fixed M case. local models, exploiting detectors inefficiencies, that can violate a Bell inequality ͓17͔. But also, one needs to know ex- Concerning the unanimity vote, CH decreases faster than a polynomial, but slower than an exponential.I ͑b͒ Resistance to noise is very actly the number of pairs, M, created before measuring them. similar for all strategies, decreasing as ␮−1 just like with as source This last condition might not be guaranteed, for example, of fixed pairs number. with Poissonian sources where knowledge of M is often in- ferred from the number of detected particles. ␮Ӎ To estimate the impact of losses, we consider the case in an optimal 1.2–1.8 yielding a maximum CH violation. which exactly 1 of the M photons flying to Alice and 1 going But this feature is not found in the resistance to noise. to Bob are not detected. As the number of created pairs increases, this is a situation that must happen frequently even E. Inefficient detectors with very efficient detectors. Using the majority and unanimity vote in this situation, we numerically verified that the CH We now consider detectors with finite efficiency ␩Ͻ1 inequality could not be violated, at least for M ഛ50. and look in what circumstance a Bell violation can still be ␩ A way to understand this result is by noting that the sets observed in a multipair scheme with such detectors. is to of events yielding results ϩ and Ϫ are separated by only one be understood here as the probability for a particle to be photon number. Thus, removing one photon mixes the two detected. sets. It should thus be advantageous to separate these two In general, in the presence of detector inefficiencies ͑or ജ ജ cases such that, for instance, n+ N +, n− N −, M −N particle losses͒ the total number of particles detected by Al- Ͻn ϽN . Using this particular post-selection,→ → we could ͑ A A B B͒ + → ice and Bob are different n+ +n− n+ +n− . Thus, for a given find a Bell violation in the case of one photon loss on both voting strategy, the thresholds N applied by Alice and Bob sides, with N=M −1 ͑unanimity voting͒, starting at M =5. might be different for the same event, since it depends on the For details on this post-selection, see Ref. ͓14͔. total number of photons detected by each party. Testing a Bell inequality in this situation without appealing to post- III. INDISTINGUISHABLE PHOTONS selection introduces no detection loophole, but it is not a In the first part of this work we showed how, using mul- surprise to find that high efficiencies are needed in order to tiple independent pairs together with independent global

062110-4 TESTING A BELL INEQUALITY IN MULTIPAIR SCENARIOS PHYSICAL REVIEW A 78, 062110 ͑2008͒

measurement on all the photons produced, one could ﬁnd a substantial CH violation, even in the presence of lots of pairs. But how good is this compared to a source producing the 2M photons altogether? For the sake of comparison we now consider a speciﬁc example, commonly produced in many laboratories. By the same occasion it will uncover some aspects of the relation between entanglement and nonlocality. ␳ The state we are discussing now can be written as M ͉⌽ ͗͘⌽ ͉ = M M with 1 ͉⌽ ͘ ͑ † † † †͒M͉ ͘ ͑ ͒ M = a0b0 + a1b1 0 , 10 M!ͱM +1

where a0 and a1 are orthogonal modes on Alice’s side and b0 ͑ and b1 orthogonal modes on Bob’s side for instance, horizontal and vertical polarization modes͒. A way to produce this state is with a PDC source, which gives a Poissonian distribution of such states. The same global measurements as previously performed on M photons can be realized here by just using the same setup as before: a polarizer followed by two photon counters on each side ͑same setup as represented in Fig. 1, but with a different source͒. Considering the state ͑10͒, we make a similar analysis as previously, briefly reviewing the results of ͓6,7͔ for the amount of violation achievable and presenting our own analysis for the resistance to noise. We computed the new probabilities entering the CH expression for this specific state and, choosing variousI voting procedures, numerically optimized the settings according to FIG. 5. ͑Color online͒ Comparison between sources producing ␣, Eqs. ͑7͒, in order to get the largest violation. Surpris- independent pairs or indistinguishable photons, using settings of the CH ͑ ͒ ͑ ͒ ingly, for any number of photons, M,I all voting procedures form 7 . a Maximal CH violation achieved with a source of indistinguishable photons for various voting procedure ͑superposed yield approximately the same maximum violation of CH, −1 ͓ ͑ ͔͒ I black dots͒. Compared to the previously calculated violations for decreasing as M see Fig. 5 a . This is even more surpris- ͓ ͑ ͔͒ ing as the settings needed for that are not the same for all independent pairs same curves as in Fig. 2 a , unanimity voting ͑lower blue dashed line͒ yields less violation, while majority voting voting methods. ͑upper red line͒ yield the highest values. Note that the maximal Note that Reid et al. ͓6͔ used another figure of merit: S ICH +B violation with indistinguishable photons almost does not depend on ICH ͓ A͑ ͒ B͑ ͔͒ = B with B= P+ A1 + P+ B1 , which gives different re- the voting procedure used. ͑b͒ Maximal resistance to noise in the sults for the different voting strategies. Recalling the artifacts majority voting scenario ͑solid red lines͒ and the unanimity sce- we already found in the amount of CH violation for Poisso- nario ͑dashed blue lines͒ for sources producing independent pairs nian sources, we choose to look now at an experimentally ͓thick line, same curves as in Fig. 2͑b͔͒ or indistinguishable photons meaningful figure of merit: namely, the resistance to noise. ͑thin line͒. The unanimous vote is more robust for indistinguishable photons, but majority voting on independently produced pairs yields the most persistent violation. A. Noisy symmetric state

Unlike for distinguishable photons, the effect of a noise The state after the noisy channel is thus given by map on a symmetric M-photon state does not affect each photon independently. We thus need here a more precise ␳ ͵ ͑␤͒͑ ͒␳ ͑ † ͓͒ ͔͑͒ noise model. For the sake of simplicity, we put ourselves in out = p U 1 in U 1 dU 11 ͑ ͒ an asymmetric setting, modeling the noise observed in the SU 2

state measured by Alice and Bob as coming from the imper- ϱ ␲ ␲ fection of the channel linking the source and Alice. Because 1 2 = ͵ d␤͵ d␪͵ d␾sin2 ͑␤͒sin ͑␪͒p͑␤͒ the channel slightly deteriorates the systems passing through ␲ 4 −ϱ 0 0 it, but has no preferred basis, we model it, by an average ϫ͓ ͑␤ ␪ ␾͒ ͔␳ ͓ †͑␤ ␪ ␾͒ ͔ ͑ ͒ over all rotation axes nជ =͑sin ␪ cos ␾,sin ␪ sin ␾,cos ␪͒ in U , , 1 in U , , 1 , 12 the Bloch sphere of rotations U by an angle 2␤, with ␤ following a properly normalized Gaussian distribution p͑␤͒ where we have used the appropriate Haar measure of 2 2 SU͑2͒ in terms of the Euler angles: ͓dU͔ 2 −␤ /2␴ ␴ជ ͑ជ ជ ជ ͒ = 2 e . For any representation = J ,J ,J of 1 2 ͑ −2␴ ͒ͱ ␲␴ x y z ͑␤͒ ͑␪͒ ␤ ␪ ␾ 1−e 2 = 4␲ sin sin d d d . We introduced the Haar mea- SU͑2͒ generators, the rotation operator is U=exp͑−␤nជ ·␴ជ ͒. sure here because it is the only measure which is invariant

062110-5 BANCAL et al. PHYSICAL REVIEW A 78, 062110 ͑2008͒ under group operations. It thus treats every rotation the same ͉0͘ ͉0͘ + ͉1͘ ͉1͘ M ͉⌿ ͘ ͩ A B A B ͪ way, reﬂecting the fact that the noise has no preferred rota- M = ͱ2 tion axis. ␳ ͉⌽ ͗͘⌽ ͉ M This channel applied to a single pair state in= 1 1 ͉⌽ ͘ 1 ͑ † † † †͉͒ ͘ ͑ ͒ =2−M/2͚ ͚ ␲͉0͘i͉1͘M−i ␲͉0͘i͉1͘M−i with 1 = ͱ2 a0b0 +a1b1 0 produces a Werner state 8 , al- A A B B i=0 ␲෈⌸M lowing one to make a correspondence between the usual i noise model used in the previous part of this work, in terms sym M of Werner states and this one: −M/2 ͉ ͘ ͉ ͘ ϳ͉⌽ ͘ —— 2 ͚ i,M − i A i,M − i B M , → i=0

␴2 ␴2 ␴2 ͑16͒ 3w = e−2 + e−4 + e−6 . ͑13͒ ⌸M ͑ M ͒ where i is the set of all i possible arrangements of i “0” ͑ †͒i͑ †͒M−i ͉ ͘ a0 a1 ͉ ͘ This relation allows us to interpret the amount of white noise and M −i “1”, and i,M −i A = ͱi!͑M−i!͒ 0 is the Fock state ⑀=1−w as being, to ﬁrst order, the variance of the random describing i of Alice’s M photons in the “0” state and M −i in rotation angle: the “1” state. ͉⌿ ͘ ͉⌽ ͘ So the only difference between M and M is the dis- tinguishability of the M photons ﬂying to Alice or Bob. But ⑀ =4␴2 + O͑␴4͒. ͑14͒ in the setup we considered ͑as described in Fig. 1͒,wedid not take advantage of the particular pairing between some of Applying this channel to the state ͉⌽ ͘ for various M and M Alice’s photons with some of Bob’s ones. Because we ap- performing the majority and unanimity votes with settings in plied a global measurement, we could even suppose that all ␣, Eqs. ͑7͒, we found that the unanimity procedure is more photons on Alice’s ͑Bob’s͒ side were mixed before reaching robust to noise than the majority vote, scaling like ϳM−1 the beam splitter. In other words, we classically lost trace of ͓see Fig. 5͑b͔͒. the pairing between Alice’s and Bob’s photons. We are thus comparing a situation in which one explicitly chose not to B. Particle losses distinguish between photons belonging to a given set, with another one for which these photons are intrinsically indis- To compare indistinguishable and independent pairs in the tinguishable. case of losses, we consider the case in which one particle is Let us now compare the entanglement present in both lost on each side, yielding a total number of detections, states. Eisert et al. ͓5͔ calculated the amount of entanglement ͉⌿ ͘ 2͑M −1͒. In terms of modes, the state measured after the loss present in the state of distinguishable particles M after of particles can be written having forgotten the pairing of Alice’s photons with Bob’s ones. For M even,

M/2 ␳ ϳ a b ͉⌽ ͗͘⌽ ͉a†b† + a b ͉⌽ ͗͘⌽ ͉a†b† ͑2j +1͒2 M +1 M−1 0 0 M M 0 0 0 1 M M 0 1 E = E͉͑⌿ ͒͘ = ͚ ͩ ͪlog ͑2j +1͒. d M M͑ ͒ / 2 ͉⌽ ͗͘⌽ ͉ † † ͉⌽ ͗͘⌽ ͉ † † j=0 2 M +1 M 2−j + a1b0 M M a1b0 + a1b1 M M a1b1. ͑15͒ ͑17͒ ͉⌽ ͘ Concerning the state of indistinguishable particles M , ͑ ͒ Using such a state, we could find a violation for sufficiently writing it in terms of modes as in Eq. 16 , we see that its many pairs M ജ10, starting with majority voting. Thus, there entanglement is given by is no need for additional post-selection here. ͉͑⌽ ͒͘ ͑ ͒ ͑ ͒ Ei = E M = log2 M +1 , 18 since it is a maximally entangled state of two systems of IV. DISTINGUISHABLE VERSUS dimension M. Evaluating these two quantities, we find Ei Ͼ ∀ / M ϱ INDISTINGUISHABLE PAIRS Ed M, and more precisely Ei Ed ——→ 2. So more entanglement is present in the state where→ photons are quan- In the last sections we examined how two different mul- tumly indistinguishable, while a larger violation of the CH ͉⌿ ͘ ͑ ͒ ͉⌽ ͘ ͑ ͒ tiparticle bipartite states M , Eq. 4 , and M , Eq. 10 , inequality can be observed using a natural setup if the pho- could be used to show nonlocality using a natural setup pro- tons are in principle distinguishable, but we choose not to ducing binary outcomes. These two states are actually re- make any difference between them. Looking at how resistant lated: if one were to produce the state of independent pairs these violations are with respect to noise confirms this order. ͉⌿ ͘ M with fundamentally indistinguishable photons on both It should only be noted that compared to particle losses, the Alice and Bob’s sides, then the state created would be sym- indistinguishable case looks more resistant, since no addi- metric with respect to permutations between Alice’s photons tional post-selection was necessary to find a violation when or Bob’s ones, and we would actually have produced state both Alice and Bob lost a particle during the experiment. ͉⌽ ͘ ͉⌿ ͘ M . This can be seen by projecting M onto the corre- This is in agreement with other results ͓18͔, showing that sponding symmetric subspaces: entanglement and nonlocality are different measures.

062110-6 TESTING A BELL INEQUALITY IN MULTIPAIR SCENARIOS PHYSICAL REVIEW A 78, 062110 ͑2008͒

TABLE I. Summary of the main results of this work.

Indistinguishable Independent pairs photons

͉0͘ ͉0͘ + ͉1͘ ͉1͘ M 1 ͫ A B A B ͬ ͓a†b† + a†b†͔M͉0͘ 0 0 1 1 State ͱ2 M!ͱM +1

Entanglement 1 after particles ϳ ͑ ͓͒ ͔ log ͑M +1͒ log2 M 5 2 order loss 2

Largest CH ϳM−1/2 ϳM−1 violation ͑majority voting͒ ͑any voting procedure͒

Highest ϳM−1 ϳM−1 resistance ͑majority voting͒ ͑unanimous voting͒ to noise

Loss of one No violation without post- Violation possible particle on selection, at least for M ഛ50 for M ജ10 each side

V. EXPERIMENTAL PERSPECTIVES entanglement using PDC sources was demonstrated in ͓10͔. A careful analysis of post-selection might thus open the pos- In this section we give a brief overview of experimental sibility to feasible experiments. situation where our techniques might be applied. As mentioned previously, the experiment of Ref. ͓3͔ VI. CONCLUSION shows evidence for entanglement in ensembles of ultracold We considered Bell experiments on multiple pairs of par- atoms of 87Rb in an optical lattice. Entanglement between ticles, where the two parties are not able to address each two atomic levels is generated via a partial swap gate, an particle separately and thus call upon global measurements, entangling operation. In order to apply our techniques, the projecting all of their incoming particles in the same basis. atoms of each level should be addressed separately; that is, Votes were introduced as a natural way to produce binary Alice should hold all atoms in the ground state and Bob all outcomes from two detection numbers. This allowed us to atoms in the excited state. Note that in this experiment the test the CH inequality in the presence of both a source of M pairs are independent because they are located in different independent pairs and of M indistinguishable pairs, high- regions of the optical lattice. lighting a violation of the CH inequality for any number of Another experimental situation invoking Bose-Einstein pairs, M. Considering the resistance to noise of such viola- condensates where our techniques might be useful is super- tions, modeled as a noisy channel, we could provide an ex- radiant scattering ͓19͔. It has been argued that this process perimentally meaningful measure of nonlocality. The impact generates entanglement between the emitted photons and the of losses was also evaluated for the two situations, showing atoms of the condensates. In that case the particles would be that indistinguishable pairs are more robust against losses. indistinguishable. More detailed results are summarized in Table I. Finally, a A third possibility is the experiment discussed in Ref. comparison of the nonlocality observed for each source with ͓20͔, which is a proposal for energy-time entanglement of the entanglement of their respective states provided another quasiparticles in a solid-state device. This experiment is an example of the nonmonotonicity between these two quanti- adaptation of the Franson-type experiment ͓21͔ with en- ties. tangled electron-hole pairs. Finally it is also worth mentioning quantum optics. How- ACKNOWLEDGMENTS ever, it is not clear that our techniques will turn out useful in We thank F.S. Cataliotti for pointing out the potential ap- this field, since they require high detection efficiencies, a plication of our results to superradiant scattering. We ac- feature that still lacks generally in optics. Still, sources pro- knowledge financial support from the EU project QAP ͑No. ducing independent entangled pairs, or indistinguishable IST-FET FP6-015848͒ and the Swiss NCCR Quantum photons via PDC, are already well understood. Multiphoton Photonics.

062110-7 BANCAL et al. PHYSICAL REVIEW A 78, 062110 ͑2008͒

͓1͔ J. Bell, Physics ͑Long Island City, N.Y.͒ 1, 195 ͑1964͒. ͓11͔ J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. ͓2͔ A. Aspect, Nature ͑London͒ 398, 189 ͑1999͒. Rev. Lett. 23, 880 ͑1969͒. ͓3͔ M. Anderlini et al., Nature ͑London͒ 448, 452 ͑2007͒. ͓12͔ J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526 ͑1974͒. ͓ ͔ 4 Y.-C. Liang and A. C. Doherty, Phys. Rev. A 73, 052116 ͓13͔ D. Collins and N. Gisin, J. Phys. A 37, 1775 ͑2004͒. ͑ ͒ 2006 . ͓14͔ N. Brunner, C. Branciard, and N. Gisin, Phys. Rev. A 78, ͓5͔ J. Eisert, T. Felbinger, P. Papadopoulos, M. B. Plenio, and M. 052110 ͑2008͒. Wilkens, Phys. Rev. Lett. 84, 1611 ͑2000͒. ͓15͔ P. H. Eberhard, Phys. Rev. A 47, R747 ͑1993͒. ͓6͔ M. D. Reid, W. J. Munro, and F. De Martini, Phys. Rev. A 66, ͓16͔ P. Pearle, Phys. Rev. D 2, 1418 ͑1970͒. 033801 ͑2002͒. ͓ ͔ ͑ ͒ ͓7͔ P. D. Drummond, Phys. Rev. Lett. 50, 1407 ͑1983͒. 17 N. Gisin and B. Gisin, Phys. Lett. A 260, 323 1999 . ͓ ͔ ͓8͔ S. J. Jones, H. M. Wiseman, and D. T. Pope, Phys. Rev. A 72, 18 A. A. Méthot and V. Scarani, Quantum Inf. Comput. 7, 157 ͑ ͒ 022330 ͑2005͒. 2007 . ͓9͔ G. Toth, C. Knapp, O. Guhne, and H. J. Briegel, Phys. Rev. ͓19͔ S. Inouye et al., Science 285, 5427, ͑1999͒. Lett. 99, 250405 ͑2007͒. ͓20͔ V. Scarani, N. Gisin, and S. Popescu, Phys. Rev. Lett. 92, ͓10͔ H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D. 167901 ͑2004͒. Bouwmeester, Phys. Rev. Lett. 93, 193901 ͑2004͒. ͓21͔ J. D. Franson, Phys. Rev. Lett. 62, 2205 ͑1989͒.

062110-8 Paper B

Quantifying Multipartite Nonlocality

J.-D. Bancal, C. Branciard, N. Gisin and S. Pironio

Physical Review Letters 103, 090503 (2009)

week ending PRL 103, 090503 (2009) PHYSICAL REVIEW LETTERS 28 AUGUST 2009

Quantifying Multipartite Nonlocality

Jean-Daniel Bancal, Cyril Branciard, Nicolas Gisin, and Stefano Pironio Group of Applied Physics, University of Geneva, 20 rue de l’Ecole-de-Me´decine, CH-1211 Geneva 4, Switzerland (Received 16 March 2009; published 26 August 2009) The nonlocal correlations of multipartite entangled states can be reproduced by a classical model if sufficiently many parties join together or if sufficiently many parties broadcast their measurement inputs. The maximal number m of groups and the minimal number k of broadcasting parties that allow for the reproduction of a given set of correlations quantify their multipartite nonlocal content. We show how upper bounds on m and lower bounds on k can be computed from the violation of the Mermin-Svetlichny inequalities. While n-partite Greenberger-Horne-Zeilinger states violate these inequalities maximally, we find that W states violate them only by a very small amount.

DOI: 10.1103/PhysRevLett.103.090503 PACS numbers: 03.67.Mn, 03.65.Ud

By performing local measurements on an n-partite en- cal communication models in the manner of Svetlichny tangled state, one obtains outcomes that may be nonlocal, [10–13], where the n parties are divided into m disjoint in the sense that they violate a Bell inequality [1]. Since the subgroups. Within each group, the parties are free to seminal work of Bell, nonlocality has been a central sub- collaborate and communicate with each other but are not ject of study in the foundations of quantum theory and has allowed to do so between distinct groups. The idea is that a been supported by many experiments [2,3]. More recently, given set of correlations contains more multipartite non- it has also been realized that it plays a key role in various locality if more parties need to join to be able to reproduce quantum information applications, where it represents a these correlations (see Fig. 1). The second measure of resource different from entanglement [4]. multipartite nonlocality that we introduce is based on While nonlocality has been extensively studied in the models where k parties broadcast their measurement inputs bipartite (n ¼ 2) and to a lesser extent in the tripartite (n ¼ to all others. The idea again is that correlations that require 3) case, the general n-partite case remains much unex- more broadcasting parties to be simulated contain more plored. The physics of many-particle systems, however, multipartite nonlocality. The maximal number m of groups is well known to differ fundamentally from the one of a few and the minimal number k of broadcasting parties that particles and to give rise to new interesting phenomena, allow for the reproduction of a given set of correlations such as phase transitions or quantum computing. Entangle- thus represent two simple ways of quantifying their multi- ment theory, in particular, appears to have a much more partite nonlocal content. complex and richer structure in the n-partite case than it Given an arbitrary set of correlations, it may in general has in the bipartite setting [5,6]. This is reflected by the fact be difficult to determine the corresponding values of m and that multipartite entanglement is a very active field of k. To evaluate these quantities, we introduce a family of research that has led to important insights into our under- Bell tests based on the Mermin-Svetlichny (MS) inequal- standing of many-particle physics (see, e.g., [7,8]). In view ities [10,14]. Specifically, we compute the maximal value of this, it seems worthy to investigate also how nonlocality of the MS expressions achieved by models where n parties manifests itself in a multipartite scenario. What new fea- form m groups and where k parties broadcast their inputs. tures emerge in this context, and what are their funda- By comparing the amount by which quantum states violate mental implications? How does one characterize the non- the MS inequalities with our bounds, one thus obtains locality of experimentally realizable multiqubit states, constraints on the values of m and k necessary to reproduce such as W states, for instance? What role do n-partite their nonlocal correlations. Since these criteria are based nonlocal correlations play in quantum information proto- on Bell-like inequalities, they can be tested experimentally. cols, e.g., in measurement-based computation [9]? The vision behind the present Letter is that, in order to answer such questions and make further progress on our understanding of multipartite nonlocality, one should first find ways to quantify it. Motivated by this idea, we intro- FIG. 1. Different groupings of n ¼ 4 parties into m groups. duce two simple measures that quantify the multipartite Within each group, every party can communicate to any other extent of nonlocality. party, as indicated by the arrows. (a) If all parties join into one A natural way to characterize nonlocality is to attempt to group (m ¼ 1), they can achieve any correlations. (b),(c) If they replicate it using models where some nonlocal interactions split into m ¼ 2 groups, they can realize some nonlocal corre- (such as communication) are allowed between some par- lations but not all. (d) If they are all separated (m ¼ n), they can ties. The first measure that we consider is based on classi- only reproduce local correlations.

0031-9007=09=103(9)=090503(4) 090503-1 Ó 2009 The American Physical Society week ending PRL 103, 090503 (2009) PHYSICAL REVIEW LETTERS 28 AUGUST 2009

A Bell-like test for a given number of groups could with each other but are not allowed to do so between a priori depend on how the groups are formed, e.g., 2 þ distinct groups. 2 in Fig. 1(b) or 1 þ 3 in Fig. 1(c), and on which party (ii) Broadcasting.—Out of the n parties, k of them can belongs to which group. But the tests that we present here broadcast their input to all other parties. The remaining depend only on the total number m of groups and not on m ¼ n k parties cannot communicate their input to any how the parties are distributed within each group. other party. Furthermore, in the measurement scenario that we consider In the framework of these two communication models, in this work (restricted to ‘‘correlation functions’’), a com- the values that can be reached by the MS polynomials are munication model with m disjoint groups is less powerful bounded as follows. than a communication model with k ¼ n m broadcast- Theorem.—For both the grouping and the broadcasting ing parties. Yet we find that the bounds on the MS expres- models, sions are identical in both cases. m ðnmÞ=2 jSn j2 : (3) As mentioned above, our results can be used to estimate the multipartite nonlocal content of quantum states. We Moreover, this bound is tight; i.e., for each model, there m ðnmÞ=2 carry out this analysis for Greenberger-Horne-Zeilinger exists a strategy that yields jSn j¼2 (in the case of (GHZ)-like and W states in the last part of this Letter. the grouping model, this is true for any possible grouping Definitions.—We consider a Bell experiment involving of the n parties into m groups). n parties which can each perform one out of two measure- Before proving our theorem, let us elaborate on some ments. The outcomes of these measurements are written aj comments. First of all, let us mention that, for m ¼ 2, the 0 and aj and can take the values 1. Letting M1 ¼ a1,we results obtained in Refs. [11,12] for the grouping model are define recursively the MS polynomials [10,11,14,15]as recovered. Note also that, since we consider correlation functions M ¼ 1ða þ a0 ÞM þ 1ða a0 ÞM0 ; n 2 n n n1 2 n n n1 (1) only, the grouping model is weaker than the broadcasting 1 model. Indeed, in each group, one can assume that all 0 Mn ¼ pffiffiffi ðMn MnÞ; (2) 2 parties send their inputs to one singled-out party, which decides for the correlation function of the whole group. 0 where Mn is obtained from Mn by exchanging all primed The broadcasting model clearly allows more communica- þ and nonprimed aj’s. Mn and Mn are equivalent under the tion than this. 0 0 exchange faj;ajg$faj;ajg for any single party j, which The fact that the same bounds hold for the two models is corresponds to a relabeling of its inputs and outputs. The not trivial and is actually a special property of the MS MS polynomials are symmetric under permutations of the expressions. Indeed, we have been able to construct in- parties. equalities that distinguish between these models. We interpret these polynomials as sums of expectation A more technical remark.—As observed in Ref. [13] for values by identifying each term of the form a1 ...an with the case m ¼ 2, the structure of the MS inequalities allows the correlation coefficient ha1 ...ani, which is the expec- one to detect a stronger form of nonlocality than the one tation value of the product of the outputs a1 ...an. The induced by grouping. It is interesting to identify precisely above polynomials can thus be interpreted as Bell inequal- the most general communication model associated with M 1 ities. Their localpffiffiffi bounds are known [11]tobej nj this stronger form of nonlocality. and jMn j 2, while the algebraic bounds (the maximal The common feature of the two above models that we value achieved by an arbitrary nonlocal model) are easily exploit in our proof (see below) and that fundamentally bn=2c bðn1Þ=2cþ1=2 Sm found to be jMnj2 and jMn j2 . limits the values of the n expressions is that there exists a In the remainder of this Letter, we shall be interested in special subset of m parties such that none of the n parties the following family of polynomials: knows more than one input from this subset. This is obvious in the broadcasting model; in the grouping model, Mn for n m even; Sm ¼ simply pick one party in each of the m groups. Let us n Mþ n m : n for odd therefore define the most general (but less natural) com- Quantifying multipartite nonlocality through communi- munication model with this property. cation models.—In a classical communication model, the n Restrained-subset model.—Among the n parties, there is parties have access to shared randomness and are allowed a subset of m parties, such that none of the n parties to communicate their inputs to some other parties. Given receives more than one input from this subset. The other the information available to them, each party then produces parties are free to communicate as they wish. Note that the a local output. Here, as explained in the introduction, we parties within the special subset of m parties cannot receive define two families of models that depend on a parameter inputs from any other party in the subset, as they already m (or k ¼ n m) which quantify the extent of multipartite know their own input. nonlocality. This model also satisfies the bound (3); for the case m ¼ (i) Grouping.—The n parties are grouped into m sub- 2, the results of Ref. [13] are recovered. This model is sets. Within each group, the parties are free to collaborate optimal for the MS expressions, in the sense that any 090503-2 week ending PRL 103, 090503 (2009) PHYSICAL REVIEW LETTERS 28 AUGUST 2009 additional communication between the parties allows them 2ðni1Þ=2 at the same time. This is because the (tight) to violate (3). Mþ 2ni=2 algebraic bound for Gi is , and Eq. (2) tells us that Proof of (3).—It is sufficient to prove (3) for our stron- 0 in order to achieve it both MG and MG must reach their gest model, i.e., for the restrained-subset model. Since the i i algebraic limit. Similarly, there exists a strategy for groups MS inequalities are symmetric under permutations of the þ with an even number of parties ni such that jM j¼ parties, we can assume without loss of generality that the Gi jM j¼2ðni1Þ=2 fM; M0g parties 1; ...;mare the ones in the restrained subset. Gi . We shall thus associate polyno- m þ Consider first the case of even n m, for which Sn ¼ mials to odd groups and fM ;M g to even ones. Mn. Applying twice the recursive definition (1), we get Consider two groups Gi and Gj and their union Gij ¼ Gi [ Gj. From the definitions (1) and (2), one can derive M 1 a a M0 a a0 M n ¼ 2ð n n1 n2 þ n n1 n2 the decompositions: 0 0 0 0 þ anan 1Mn 2 anan 1Mn 2Þ: (4) M 1 M M M0 M0 M M0 ; Gij ¼ 2½ Gi ð Gj þ Gj Þþ Gi ð Gj Gj Þ ð0Þ ð0Þ Using again twice (1) for Mn 2, we can replace Mn 2 as a 1 þ þ þ MG ¼ ½MG ðMG þ MG ÞþMG ðMG MG Þ; ð0Þ nm ij 2 i j j i j j function of Mn 4 in (4). Iterating this process 2 times, 1 0 0 M ¼ ½M ðMG þ M ÞM ðMG M Þ: we end up with the following expression for Mn: Gij 2 Gi j Gj Gi j Gj

X1 0 0 1 s s s ...s M M n mþ1 n mþ1 Similar relations are also obtained for Gij since ð G Þ ¼ Mn ¼ an ...am 1Mm ; (5) 2ðnmÞ=2 þ M. Now inserting in the above relations the value sn;...;smþ1¼0 G attained by the strategies that we just mentioned for the 0 1 0 where ai ¼ ai and ai ¼ ai and where, depending on the two initial groups, one finds that their combined strategy sn;...;sm 1 s ; ...;s M þ 0 ðni1Þ=2 ðnj1Þ=2 value of ( n mþ1), m is equal to one of the can achieve jMG j¼jMG j¼2 2 or M ; M0 ij ij polynomials f m mg. Mþ M 2ðni1Þ=22ðnj1Þ=2 sn...sm 1 j G j¼j G j¼ , depending on which The MS polynomial Mm þ is a function of the out- ij ij s ...s s ...s 0 0 n mþ1 n mþ1 set of polynomials is associated to the two initial groups. puts fa1;a1 ...;am;amg, i.e., Mm ¼ Mm 0 0 Iterating this construction by joining groups succes- (a1;a1 ...;am;am). Among the parties fm þ 1; ...;ng, Q M m 2ðni1Þ=2 there exists a (possibly empty) subset fj1; ...;jlg that do sively 2 by 2, we find j nj¼ i¼1 when there Mþ not receive any input from parties 2; ...;m but possibly isQ an even number of even groups and j n j¼ 0 m 2ðni1Þ=2 from party 1. Define two effective outputs A1 and A1 as i¼1 otherwise. Since the parity of the number s s s s j1 jl 0 0 j1 jl n m A1 ¼ a1a ...a and A ¼ a a ...a , respectively. of even groups is the same as the parity of , there j1 jl 1 1 j1 jl S There also exist similar disjoint subsets for parties Qmust exist a strategy which achieves j nj¼ m 2ðni1Þ=2 2ðnmÞ=2 j 2; ...;m, for which we also define effective outputs i¼1 ¼ . 0 0 Nonlocality of quantum states.—Suppose that one ob- A2;A2; ...;Am;Am. Then we can write m ðnmÞ=2 jSn j2 s s s ...s s ...s serves a violation of the inequality . One a n ...a mþ1 M n mþ1 M n mþ1 A ;A0 ...;A ;A0 : n mþ1 m ¼ m ð 1 1 m mÞ can then conclude that, in order to reproduce the corre- s ...s sponding nonlocal correlations in the framework of our M n mþ1 A ;A0 ...;A ;A0 Formally, m ( 1 1 m m) is a MS polyno- communication models, the parties cannot be separated in m mial that involves parties isolated from each other, since more than m 1 groups or that at least k þ 1 ¼ n m þ A A0 j the outputs j and j of party do not depend on the input 1 parties must broadcast their input. Thus, the above m 1 m of any of the other parties. It can therefore not bounds on Sn give us bounds on the multipartite character exceed its local bound 1. Inserting this bound in (5), we of the observed nonlocal correlations (an upper bound on ðnmÞ=2 find jMnj2 . m or a lower bound on k). For odd values of n m, we have to consider the poly- Here we discuss the violation of the inequalities (3) for m þ nomials Sn ¼ Mn . Using the definitions (1) and (2), one n-partite GHZ-like and W states. States in the GHZ family þ can show that Mn has a similar decomposition as Mn in are defined as jGHZi¼cosj00...0iþsinj11...1i. þ (5). The same reasoning as before then leads to jMn j The maximal value of Mn for these states was conjectured ðnmÞ=2 ðn1Þ=2 2 . j in Ref. [16]tobeMn ¼ maxf1; 2 sin2g. Numerical Proof of the tightness of (3).—To prove that (3) is a tight optimizations (see Fig.pffiffiffi2) induce us to conjecture that þ ðn1Þ=2 bound, it is sufficient to prove that it can be reached by our similarly Mn ¼ maxf 2; 2 sin2g. Upon compari- weaker communication model, i.e., the grouping model son with the bound (3), we conclude that all n-partite (for any possible distribution of the n parties into m GHZ states with >=8 are maximally nonlocal accord- groups). ing to our criterion (i.e., all parties must be grouped to- Let Gi (i ¼ 1; ...;m) denote the m groups into which gether or n 1 parties must broadcast their input to the n parties are split. For all groups Gi having an odd reproduce their correlations). Less entangled GHZ states, number ni of parties, there exists a strategy for the parties on the other hand, cannot be simulated if the parties are G M M0 m 1 k in i to reach both algebraic bounds j Gi j¼j Gi j¼ separated in more than groups or if fewer than þ 090503-3 week ending PRL 103, 090503 (2009) PHYSICAL REVIEW LETTERS 28 AUGUST 2009

þ FIG. 3. Maximal values of Mn (solid line) and Mn (dashed þ FIG. 2. Maximal values of Mn and Mn for partially entangled line) for n-partite W states. The curves were obtained by a GHZ states for 3 n 6. The dots are values found by nu- general numerical optimization for n 9 and under the hy- merical optimization, and the solid lines are the conjectured pothesis that all parties use identical measurement settings for M Mþ 2ðn1Þ=2 sin2 M violationpffiffiffi n ¼ n ¼ (valid only above 1 for n 10 n 19. The asymptotic values for n !1 computed as þ and 2 for Mn ). explained in the text are also shown.

1 ¼ n m þ 1 parties broadcast their inputs whenever ity of W states. Finding which one of these possibilities is ðm1Þ=2 >c with sin2c ¼ 2 . Interestingly, c is the the correct one is an interesting problem for future re- n search. Also, it would be interesting to analyze the non- same for all . pffiffiffi Consider now the W states jWni¼ð1= nÞ locality of other multipartite quantum states with our ðj10...0iþ...þj0...01iÞ. Numerical optimizations criteria. suggest that the maximal values of the MS polynomials As suggested by the situation in entanglement theory, we for these states are upper bounded by a small constant for do not expect our measures to be the only ways to quantify all n (see Fig. 3). To convince ourselves that this is indeed the multipartite content of nonlocality. It would thus be of the case, we analyzed analytically the case where all pairs interest to look for different measures, based on other of measurement settings are the same for all parties. This is nonlocal models than the ones considered here. justified by the results of our numerical optimizations up to Finally, let us stress that the criteria that we presented in n ¼ 9, for which the optimal measurement settings can this Letter can be tested experimentally. It would thus be always be of this form. We thus introduce for all n parties worth (re)considering experiments on multipartite non- two measurement operators A0 and A1 represented by locality in view of our results. vectors a~i ¼ðsini cosi; sini sini; cosiÞ. One can We acknowledge support by the Swiss NCCR Quantum show that, as n increases, the maximal value of jMnj or Photonics and the European ERC-AG QORE. þ jMn j can be reached for i ¼ 0 and i ! 0. Assuming a n power lawpffiffiffi for ið Þ, one finds that it should be given by c = n i i at the maximum. After optimization of the [1] J. Bell, Speakable and Unspeakable in Quantum c c M Mþ constants 0 and 1 for both n and n , we found that Mechanics (Cambridge University Press, Cambridge, the asymptotic maximal values of the MS polynomials England, 1987). (under our assumptions, which we believepffiffiffiffiffiffiffiffi are not restric- [2] M. Genovese, Phys. Rep. 413, 319 (2005). þ n1 [3] A. Aspect, Nature (London) 398, 189 (1999). tive) are jM1j’1:62 and jM1j¼2 2=e. Since Sn > 1 for n 3, letting one party broadcast his input, or letting [4] J. Barrett et al., Phys. Rev. A 71, 022101 (2005); C. two parties join to form a group, is not sufficient to repro- Brukner et al., Phys. Rev. Lett. 92, 127901 (2004); A. K. W Ekert, Phys. Rev. Lett. 67, 661 (1991); J. Barrett et al., duce the correlations of the state. However, we cannot Phys. Rev. Lett. 95, 010503 (2005); A. Acıń et al., Phys. reach the same conclusion if more than two parties join or Rev. Lett. 98, 230501 (2007); L. Masanes, Phys. Rev. Lett. if k ¼ n m 2 parties broadcast their inputs, since the 102, 140501 (2009). criterion (3) is not violated in this case. [5] W. Du¨r et al., Phys. Rev. A 62, 062314 (2000). Conclusion.—We proposed in this Letter two simple [6] R. Horodecki et al., Rev. Mod. Phys. 81, 865 (2009). measures of multipartite nonlocality and introduced a se- [7] A. Osterloh et al., Nature (London) 416, 608 (2002). ries of Bell tests to evaluate them. This represents a pri- [8] G. Vidal, Phys. Rev. Lett. 93, 040502 (2004). mary step towards a quantitative understanding of quantum [9] R. Raussendorf et al., Phys. Rev. Lett. 86, 5188 (2001). nonlocality for an arbitrary number n of parties. [10] G. Svetlichny, Phys. Rev. D 35, 3066 (1987). While GHZ states exhibit a strong form of multipartite [11] D. Collins et al., Phys. Rev. Lett. 88, 170405 (2002). [12] M. Seevinck et al., Phys. Rev. Lett. 89, 060401 (2002). nonlocality according to our criterion, we found that W k [13] N. S. Jones et al., Phys. Rev. A 71, 042329 (2005). states violate our inequalities only for small values of . [14] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990). This suggests that W states exhibit only a very weak form [15] R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 of multipartite nonlocality. Or it might be that other in- (2001). equalities are necessary to quantify properly the nonlocal- [16] V. Scarani et al., J. Phys. A 34, 6043 (2001).

090503-4 Paper C

Simulation of Equatorial von Neumann Measurements on GHZ States Using Nonlocal Resources

J.-D. Bancal, C. Branciard and N. Gisin

Advances in Mathematical Physics, volume 2010, Article ID 293245 (2010)

Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2010, Article ID 293245, 14 pages doi:10.1155/2010/293245

Research Article Simulation of Equatorial von Neumann Measurements on GHZ States Using Nonlocal Resources

Jean-Daniel Bancal, Cyril Branciard, and Nicolas Gisin

Group of Applied Physics, University of Geneva, 20 rue de l’Ecole-de-Medecine,´ 1211 Geneva 4, Switzerland

Correspondence should be addressed to Jean-Daniel Bancal, [email protected]

Received 31 August 2009; Accepted 11 December 2009

Academic Editor: Shao-Ming Fei

Copyright q 2010 Jean-Daniel Bancal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reproducing with elementary resources the correlations that arise when a quantum system is measured quantum state simulation allows one to get insight on the operational and computational power of quantum correlations. We propose a family of models that can simulate von Neumann measurements in the x − y plane of the Bloch sphere on n-partite GHZ states. For the tripartite and fourpartite states, the models use only bipartite nonlocal boxes; they can be translated into classical communication schemes with ﬁnite average communication cost.

1. Introduction

Understanding the nonlocal correlations created upon measurement of some entangled quantum system is a problem which runs up against our common representation of the world, by the very definition of nonlocality, i.e. violation of a Bell inequality 1. Indeed, no explanation one would reasonably accept as possible, like agreement prior to measurement or subluminal communication of inputs, seems to be used by nature in order to create these correlations see the numerous experimental violations of Bell inequalities 2. Still, some insight on the power of such correlations was gained when people came out with models able to reproduce them in terms of classical resources. For instance, Toner and Bacon 3 showed how to simulate von Neumann measurements on a singlet state with one bit of communication. Such a result puts an upper bound on the required amount of nonlocal resources needed for the reproduction of singlet correlations; it guarantees also that the corresponding correlations are not a stronger resource of nonlocality than 1 bit of classical communication. Adifferent kind of resources that was also considered are the so-called nonlocal boxes 4: these are simple nonlocal correlations which do not allow signaling. Successful 2 Advances in Mathematical Physics simulation schemes using nonlocal boxes as unique nonlocal resources include the simulation of the singlet 5 and of partially entangled two-qubit states 6. Concerning multipartite systems, communication models reproducing Pauli measurements on n-partite GHZ or on graph states have also been proposed 7, 8. For arbitrary possible measurements on the tripartite GHZ state, previous studies suggested that its simulation with bounded communication might be impossible, taking as an example correlations corresponding to measurements of this state in the x − y plane of the Bloch sphere 9. In this paper, we construct a model which analytically reproduces these equatorial correlations, and whose only nonlocal resources are Popescu-Rohrlich PR boxes 10 and Millionaire boxes 11. Thus a finite number of bipartite nonlocal boxes are proven to be sufficient to reproduce these genuinely tripartite nonlocal correlations. Note also that even though our model does not give an upper bound on the worst-case communication cost, it does provide a communication model with finite expected communication cost, simulating for instance the tripartite GHZ state with an average total of 10 bits of communication between the parties c.f. Appendix B. The paper is organized as follows: first, we recall the correlations of the GHZ state that we want to simulate. We then present a model for the 3-partite case, and generalize it to more parties. We discuss the construction and then conclude.

2. GHZ Correlations

Consider the n-partite GHZ state

1 ··· ··· |GHZn √ |00 0 |11 1. 2.1 2

Our goal is to reproduce the correlations which are obtained when von Neumann measurements are performed on this state, by using other nonlocal resources such as nonlocal boxes possibly supplemented with shared randomness. For n 2, the protocol presented in 5 for the singlet state allows one to reproduce the correlations for any measurement settings, using one PR box. Here we recall the deﬁnition of a PR box:

PR Box

A Popescu-Rohrlich PR box is a nonlocal box that admits two bits x, y ∈{0, 1} as inputs and produces locally random bits a, b ∈{0, 1}, which satisfy the binary relation

a b xy. 2.2

Going to n ≥ 3, we shall only consider measurements in the x − y plane equatorial measurements, which have the nice feature of producing unbiased marginals: all correlation terms involving strictly fewer than n parties vanish. We write each party’s measurement operator as: A cos φaσX sin φaσY , B cos φbσX sin φbσY ,.... Denoting the binary result of each measurement by α,β,...∈{−1, 1}, the correlations we are interested in are given by

α β ··· αβ ··· 0 2.3 Advances in Mathematical Physics 3

φb φa φc

Alice PR Charlie Bob

α γ β | Figure 1: Simulation of GHZ3 in a Svetlichny scenario: Alice and Bob form a group and can share their information with each other, while Charlie is separated from them. In this scenario 1 PR box allows one to reproduce the equatorial correlations. for all sets of fewer than n parties, and ··· ··· αβ ω cos φa φb φz 2.4 for the full n-partite correlation term. In other words, outcomes appear to be random except when all of them are considered together, in which case their correlation takes a form reminiscent of the singlet state. To simulate such correlations, nonlocal boxes similar to the Millionnaire box will be useful, so let us recall what a Millionnaire box is.

MBox

A Millionaire box is a nonlocal box that admits two continuous inputs x, y ∈ 0, 1 and produces locally random bits a, b ∈{0, 1}, such that a b sg x − y , 2.5 where the sign function is deﬁned as sgx0ifx>0andsgx1ifx ≤ 0. It is worth mentioning that even though we restrict the set of possible measurements on the GHZ states, the correlations we consider can still exhibit full n-partite nonlocality. Indeed, the Svetlichny inequality for n parties can be maximally violated with settings in the x − y plane 12, 13. This implies that in order to simulate these correlations, any model must truly involve all n parties together 14.

3. Simulation Model for the 3-Partite GHZ State

Let us consider the above correlations for n 3 parties, for which the outcomes of all parties need to be correlated according to αβγ cos φa φb φc . As a ﬁrst step towards the simulation of these correlations, let us relax some of the constraints and allow two parties to cooperate in a Svetlichny-like scenario 15see Figure 1: for instance Alice and Bob would be allowed to communicate with each other, but not with Charlie who is kept isolated from them. In such a scenario, the three parties could create correlations of the desired form with one PR box by using the protocol of 5 to generate outputs α and γ that have a cosine correlation of the form αγ cos φab φc , with a ﬁctitious measurement angle φab φa φb. By then setting either α α, β 1or 4 Advances in Mathematical Physics

α −α, β −1 each with probability 1/2,andγ γ, they would recover the desired tripartite correlations αβγ cos φa φb φc . Of course, letting Alice and Bob share their inputs is not satisfactory yet, as this would require signaling between them. We shall now see that it is actually possible to reseparate them, while keeping the tripartite correlation term unchanged. In order to do so, let us recall that the model used above to create the bipartite cosine correlation with a PR box works by asking the parties here, Alice-Bob together and Charlie to input in the box terms of the form 5

− − x sg cos φab ϕ1 sg cos φab ϕ2 , 3.1 − − z sg cos φc ϕ sg cos φc ϕ− ,

where ϕ1,ϕ2,ϕ ,ϕ− are hidden variables shared by all parties that we shall deﬁne later . One can see that in the Svetlichny scenario, Alice and Bob do not really need to share their − measurement angles, but only the terms sg cos φa φb ϕλ . Hopefully, there is a way for Alice and Bob to compute this function nonlocally by using the forementioned Millionaire box Mbox. For convenience, let us deﬁne the following nonlocal box.

Cosine Box ∈ A bipartite Cosine box Cbox is a nonlocal box that admits two angles φa,φb 0, 2π as inputs and produces locally random binary outcomes a, b ∈{0, 1}, correlated according to

a b sg cos φa φb . 3.2

We show in Appendix A that a bipartite Cboxis equivalent to an Mbox. C boxes are exactly what we need for our problem, as the following result shows.

Result 1. Equatorial von Neumann measurements on the tripartite GHZ state can be simulated with 2 C boxes and 2 PR boxes.

Proof. The simulation can be realized with the following model; we refer to Figure 2 for the distribution of the nonlocal boxes between the three parties Alice, Bob, and Charlie, and for the numbering of their inputs denoted xi,yi and zi for each party, resp. and outputs denoted ai,bi and ci . →− →− Let Bob and Charlie share two→− independent→− →− random vectors λ 1, λ 2 uniformly ± distributed on the sphere S2. We deﬁne λ ± λ 1 λ 2 and refer to ϕ1,ϕ2,φ ,ϕ− for their phase angle in polar coordinates. Let the parties input the following variables into their boxes:

x1 φa,x2 φa,x3 a1 a2, y1 φb ϕ1,y2 φb ϕ2,y3 b1 b2, 3.3 − − z1 z2 sg cos φc ϕ sg cos φc ϕ− . Advances in Mathematical Physics 5

Alice 3 1 2 PR 1

Charlie cbox cbox 2 1 2 PR 3 Bob | Figure 2: Setup for the simulation of GHZ3 in the x-y plane. The Alice-Bob group was split by using two Cboxesand a second PR box.

The three parties then output

− A α 1 , with A a1 a3 − B β 1 , with B b1 b3 3.4 − C − γ 1 , with C c1 c2 sg cos φc ϕ .

The output of each party is the XOR of outputs received from nonlocal boxes shared with all other parties. Since these boxes are no-signaling, a single output of any nonlocal box is necessarily random. The only way as not to get a random average correlation is thus to consider all parties together since missing one produces a random term. All correlations involving fewer than 3 parties thus average to zero. Concerning the 3-party correlations, we have − A B C a1 b1 a3 c1 b3 c2 sg cos φc ϕ − a1 b1 x3z1 y3z2 sg cos φc ϕ − a1 b1 a1 b1 a2 b2z1 sg cos φc ϕ − a1 b1 sg cos φc ϕ 3.5 sg cos φa φb ϕ1 sg cos φa φb ϕ2 × − − sg cos φc ϕ sg cos φc ϕ− →− →− →− · →− · sg v ab λ 1 sg c λ →− →− →− →− →− · →− · →− · →− · sg v ab λ 1 sg v ab λ 2 sg c λ sg c λ − ,

→− − − − − →− where we deﬁned v ab cos φa φb , sin φa φb , 0 and with c cos φc, sin φc, 0 being Charlie’s setting. Following the proof of 5 see→− also →3− , we ﬁnd that the average of this quantity over the values of the hidden variables λ 1 and λ 2 is →− →− − 1 − v · c 1 cos φa φb φc A B C ab 3.6 2 2 6 Advances in Mathematical Physics which leads, as requested, to αβγ cos φa φb φc . 3.7

Coming back to the Svetlichny construction, we see that it was indeed possible to split the Alice-Bob group by allowing them to share two C boxes. Concerning the PR box,ithadto be split also, into two new PR boxes, in order to recover the desired result: the computation made by the PR box in the Svetlichny setup is now performed nonlocally, by the 2 PR boxes, using inputs distributed over the 3 parties. We restricted here to measurements in the x-y plane, but with a slight modification, Charlie could actually simulate any measurement basis. Indeed a way to understand the appearance of the model for the singlet state the bipartite cosine correlation,inthe Svetlichny scenario, is to realize that the fictitious measurement angle φab φa φb that Alice and Bob used above corresponds to the direction in which they would prepare a state for Charlie if they were to measure their part of the original GHZ state in their respective bases. In other words, in the quantum scenario, when Alice and Bob measure the GHZ state, they prepare one of the two states 1 −iφ φ |z± √ |0 ± e a b |1 , 3.8 2 for Charlie. But a way for them to prepare one of these states, if they share a singlet or rather, a bipartite GHZ or |Φ state with Charlie, is by measuring their part of the |Φ ff state along φab, which is what they e ectively do in our model. So in fact they prepare a state |z± for Charlie, which he can measure in the direction he wants in particular, outside the − x y plane . The→− only modification→− in the model needed for→− that is that Charlie should use →− · →− · →− · z1 z2 sg c λ sg c λ − and C c1 c2 sg c λ to allow his measurement to point outside the x − y plane. We do not claim that the above model is optimal. It could be that strictly fewer nonlocal boxes are actually enough to reproduce the same correlations. It is nonetheless remarkable that truly tripartite correlations can be simulated with bipartite nonlocal resources only. It is also quite surprising that the model we presented here does not need more shared randomness than in the bipartite case. It might possibly be that a model that would use fewer nonlocal resources would require more shared randomness.

4. Simulation Model for the 4-partite GHZ State In the previous section, we showed how to split φab from 3.1 into two phases φa,φb,in order to reseparate the group formed by Alice and Bob in the Svetlichny scenario. It is in fact similarly possible to split φc in order to have a total of 4 parties into play.

Result 2. Equatorial von Neumann measurements on the 4-partite GHZ state can be simulated with 4 C boxes and 4 PR boxes.

Proof. The simulation can be realized with the following model, analogous to the previous one; we now refer to Figure 3 for the distribution of the nonlocal boxes between the four Advances in Mathematical Physics 7

3 3 Alice PR Charlie 4 4 1 2 1 2 PR cbox cbox cbox cbox

PR 1 2 1 2 4 4 3 3 Bob PR Dave

| Figure 3: Setup for the simulation of GHZ4 in the x-y plane. Notice that if any of the 4 parties is taken away together with the nonlocal boxes it shares with the other parties, we recover a setup with 2 PR and | 2 Cboxes, which corresponds to the simulation setup for GHZ3 ,asinFigure 2.

parties Alice, Bob, Charlie, and Dave, and for the numbering of their inputs xi,yi,zi, and wi and outputs ai,bi,ci, and di . →− →− Bob and Charlie still share two independent random vectors λ 1, λ 2 uniformly distributed on the sphere S2. With the same notations as before, let the four parties now input the following variables into their boxes:

x1 φa,x2 φa,x3 x4 a1 a2, y1 φb ϕ1,y2 φb ϕ2,y3 y4 b1 b2, 4.1 − − z1 φc ϕ ,z2 φc ϕ−,z3 z4 c1 c2, w1 φd,w2 φd,w3 w4 d1 d2.

The parties should then output

− A α 1 , with A a1 a3 a4 − B β 1 , with B b1 b3 b4 4.2 − C γ 1 , with C c1 c3 c4 − D δ 1 , with D d1 d3 d4.

For the same reason as in the tripartite case, all correlations of fewer than four parties vanish. For the 4-partite correlation term, the calculation of A B C D is straightforward, following similar lines as in the tripartite case. It leads to a similar expression as in 3.5, →− →− except that c should now be replaced by v cd cos φc φd , sin φc φd , 0 . This leads to the requested 4-partite correlation term: αβγδ cos φa φb φc φd . 4.3

Again, there is no claim of optimality for the above model, but it is also remarkable that truly 4-partite correlations can still be simulated with bipartite nonlocal resources only, and no more shared randomness than for the bipartite case. 8 Advances in Mathematical Physics

5. Going to More Parties 5.1. Possible Extension of the Model to Any Number of Parties

In the last two sections, we showed how to construct models for the simulation of GHZ states involving n 3, 4 parties by splitting the n parties into two groups. Each group then had to Σ Σ calculate functions of the form sg cos φai ϕλ with for instance φai φa φb, ϕλ ϕ1. Now, if we consider more parties, splitting them into two groups necessarily results in at least one of the groups having more than two parties. One could for instance have n − 1 parties on one side and 1 party on the other side. The sign function that each group has to calculate thus involves in general more than two phase angles. This motivates the deﬁnition of a generalization of the Cboxto n parties.

Multipartite Cosine Box ∈ An n-partite Cboxis a nonlocal nonsignaling box that admits n angles φi 0, 2π as inputs ∈{ } and produces binary outcomes ai 0, 1 , correlated according to

ai sg cos φi . 5.1 i i

The outcomes of the box are locally random. Also, all correlations involving fewer than n outputs vanish. Multipartite C boxes allow one to generalize our model to the simulation of multipartite GHZ state with any number of parties, by separating the n parties into two groups, consisting of k parties on one side and n − k parties on the other.

Result 3. Equatorial von Neumann measurements on n-partite GHZ states can be simulated with 2 k-partite Cboxes 2 n − k-partite Cboxes kn − k PR boxes for any 0

Sketch of the Proof

Following the previous constructions, the group with k parties needs to calculate nonlocally ··· two terms of the form sg cos φ1 φk ϕλ , which can be done by using two k-partite C boxes, and the other group can similarly do its job with two n−k-partite C boxes.Asitwasthe case for the 4-partite case, each party from the ﬁrst group also needs to share a PR box with each other party in the second group. We thus understand that by separating the n parties into these two groups, a total of 2 k-partite C boxes 2 n − k-partite C boxes kn − k PR boxes is suﬃcient to simulate the correlations of the n-partite GHZ state measured in the x − y plane. Interestingly again, no more shared randomness than for the bipartite case is required.

5.2. A Simpler Model

If we allow the parties to share nonlocal boxes involving more than two parties, then there is actually a simpler model which uses a single n-partite Cboxto reproduce the equatorial GHZ correlationsas deﬁned by 2.3 and 2.4. Advances in Mathematical Physics 9

Result 4. Equatorial von Neumann measurements on n-partite GHZ states can be simulated with a single n-partite C box. ∈ Proof. Consider indeed the following strategy: Alice generates a random variable ϕλ − π/2,π/2 according to the distribution ρ ϕλ 1/2 cos ϕλ . She inputs φa ϕλ in the n-partite Cbox, while all other n−1 partners simply input their measurement angle. From the outputs a,b,...of the box, each party can compute the ﬁnal outputs α −1a,β−1b,... All correlations between the outputs that involve fewer than n parties vanish, while for the n-partite correlation term, they get, as requested: π/2 ··· − sg cos ϕλ φa φb φz αβ...ω 1 ρ ϕλ dϕλ −π/2

⎧ −Σ ⎪ π/2 φi π/2 ⎪1 − 1 Σ ⎪ cos ϕλ dϕλ cos ϕλ dϕλ if 0 < φi <π ⎨ − −Σ 2 π/2 2 π/2 φi 5.2 ⎪ − −Σ ⎪ π/2 φi π/2 ⎪−1 1 ⎩ cos ϕλ dϕλ cos ϕλ dϕλ else 2 −π/2 2 −π/2−Σφi ··· cos φa φb φz .

Note that in the bipartite case, this model gives a new, simplified, way of simulating the equatorial correlations of the singlet state with a single Millionaire box. It is worth noting that it does not require any shared randomness. It uses however a strictly stronger nonlocal resource than the model with one PR box 5, since an Mboxcannot be simulated with one PR box c.f. Appendix B. Compared to this last simple model, our previous construction allows one to reduce the multipartiteness of the nonlocal boxes used to simulate the same correlations. Finitely many nonlocal boxes involving no more than n/2 parties are sufficient to reproduce n- partite equatorial GHZ correlations. In particular, for n ≤ 4, bipartite resources are sufficient. If one really wants to use only bipartite nonlocal boxes, we show in Appendix C that multipartite nonlocal boxes with continuous inputs, binary outputs, and only fully n-partite nonvanishing correlations can always be simulated with bipartite boxes, as it is the case for boxes with a finite number of inputs 16. However, the construction we use is quite special, as the boxes we need can have inputs or outputs that cannot be written as real numbers.

6. Conclusion

We proposed models reproducing the correlations of the tripartite and 4-partite GHZ states measured in the x − y plane, with a finite number of bipartite nonlocal boxes. Extending our results to n-partite GHZ states was possible after releasing the requirement that the nonlocal boxes had to be bipartite. We believe that our results give a new motivation for finding whether or not the GHZ correlations can also be simulated in a bounded communication scheme. Note that our models can be translated into finite expected communication schemes, since a PR box can be replaced by 1 bit of communication and an Mboxa bipartite Cbox by 4 bits in average, as 10 Advances in Mathematical Physics we show in Appendix B. This gives a model with an average of 10 bits of communication between the parties. Note that this model with finite expected communication could also be recast as a detection loophole model. More generally, it would be interesting to know whether the simulation of n-partite GHZ states can always be achieved with a finite amount of bipartite resources only for instance a finite number of M boxes. Considering also measurements outside of the x − y plane seems quite challenging because the marginals do not vanish anymore, but it would certainly be of interest too. Finally, it would be worth studying other multipartite quantum correlations. The W state for instance, seems to be a good candidate for this, when measurements are again restricted to the x − y plane, because of the simplicity of its correlations. Indeed they only − consist of bipartite correlation terms of the form αβ 2/n cos φa φb , all other correlation terms being 0 for any number of parties n. It is not known whether these correlations are nonlocal for all n, but in the case n 4 there exists a Bell inequality which allows one to show that these correlations are indeed nonlocal 17.

Appendices A. The Bipartite Cosine Box is Equivalent to a Millionaire Box

Here we show that in the bipartite case, the Cosine box is equivalent to a Millionaire box,up to local operations on the inputs and outputs. The general n-partite C boxes can thus also somehow be seen as a generalization of an Mboxto more parties. Let us ﬁrst give the intuition, it is indeed clear that a bipartite Cboxis equivalent to ∈ − a“sine box”, that would take two angles φa,φb π, π as inputs and would output two locally random bits a, b ∈{0, 1} with correlations satisfying

− a b sg sin φa φb . A.1

∈ − ∈ Now, if φa π, 0 , Alice can input φa π 0,π in the sine box instead of φa, and ﬂip her output so that A.1 is still satisﬁed. This also holds for Bob; we can thus assume that ∈ − − φa,φb 0,π . In that case, sg sin φa φb sg φa φb .Thesine box thus compares the values of the two real numbers φa,φb; this is exactly what an Mboxwould do! More precisely, to construct a Cboxfrom an Mbox2.5, Alice and Bob can input − x 1/π φa mod π and y 1/π φb π/2 mod π .Fromtheoutputsaandbof − the Mbox, they can calculate a a φa/π and b b φb π/2 /π , which satisfy a b sg cos φa φb , as requested. This construction is illustrated on Figure 4. Reciprocally, the Mboxcan trivially be reproduced with a Cbox, if Alice inputs φa x − − and Bob inputs φb y π/2.

B. Expected Communication Cost of a Millionaire Box

We show in this Appendix that a Millionaire box cannot be simulated with ﬁnite communication. We propose however a scheme to simulate it with 2-way communication, unbounded in the worst case, but with a ﬁnite expected number of bits. Advances in Mathematical Physics 11

φa φb

1 1 x φ mod π y −φ π/2 mod π π a π b

M-box

φa φb π/2 a a b b − π π

Figure 4: How to realize a bipartite Cosine box from a Millionaire box

Suppose first that Alice and Bob have a finite number 2k of possible inputs k bits. We show with a crossing sequence argument 18 that any communication scheme that can simulate the outputs of a Mboxfor this number of possible inputs necessarily uses at least k bits. This shows that in the limit of infinitely many inputs i.e., for the general Mbox, unbounded communication is required. Indeed suppose that a scheme using k0

1 Alice sends her nth digit to Bob. 2 Bob compares the bit he received with his nth digit and answers 0 if they are the same and 1 if they are diﬀerent. 3 If Alice receives a 0, she iterates n and goes back to step number 1. If however she receives a 1, then they both know which one of them has the largest input number. Alice can output a predetermined random bit, and Bob a bit correctly correlated to Alice’s, so as to reproduce the behavior of the Mbox.

The average number of communication cycles needed in this scheme depends on the probabilistic distribution of x and y. In particular if these distributions are independent and 12 Advances in Mathematical Physics

− uniform on the interval 0, 1, the probability that the protocol stops at the nth step is 2 n,and therefore the expected number of rounds is

∞ − 1/2 n2 n 2. B.1 − 2 n1 1 1/2

Since each round uses 2 bits of communication one in each direction, a total of 4 bits of communication is needed on average. Similar ideas can also be used to simulate n-partite C boxes with ﬁnite expected communication.

C. Simulation of n-partite Correlations with Bipartite Nonlocal Resources ∈ Consider an n-partite probability distribution for continuous inputs xi R and binary { } − outputs ai 0, 1 , which contains vanishing correlations for up to n 1 parties. We show that it can be simulated with only bipartite nonlocal boxes. This can be seen as a generalization of 16, in which a similar decomposition was constructed for distributions with ﬁnitely many inputs in terms of PR box. ﬃ To show this result, it is su cient to concentrate on the total correlation term i ai f xi involving all parties’ outputs ai, because all other correlation terms can then be put to zero by letting all pairs of parties decide randomly to permute their output or not. Consider thus the n-partite correlation term. We proceed by recursion: starting with the case n 2, in which it is obvious that any bipartite no-signaling correlation can be produced by a bipartite nonlocal box satisfying

a1 a2 fx1,x2. C.1

Now let us suppose that we have a model which can reproduce any correlation term for n−1 parties. Any n-partite correlation term can then be simulated in the following way: for − each value z that the nth party’s input xn can take, deﬁne the following function for the n 1 ≡ ﬁrst parties: fz x1,...,xn−1 f x1,...,xn−1,z . Each of these functions can be implemented by the scheme reproducing the n − 1-partite correlation functions. Now each of the n − 1 parties can collect the outputs αi z it received for each possible value of z, and plug them into a special kind of bipartite nonlocal box it shares with the last party. This box takes as inputs a function f : 0, 1 →{0, 1} on one side i.e., a continuous number of binary inputs and a real number x ∈ 0, 1 on the other side, and produces binary outputs a and b, such that

a b fx. C.2

If all parties input f αi z into such a box they share with the last party, and this last i party inputs xn into all of these boxes, writing the outputs of each of these boxes ai and an, Advances in Mathematical Physics 13 n−1 i we can set the last party’s output to be an i1 an to get a correlation term:

n n−1 n−1 n−1 i ai ai an ai an αi xn i1 i1 i1 i1 C.3 fxn x1,...,xn−1 f x1,...,xn , as required. Note that this construction needs continuously many nonlocal boxes. To avoid that, one could collect all the boxes C.1 that calculate fz xi,xj for all z into a single one that would output all the values at the same time. Note however that such a box would actually ℵ output continuous outputs of cardinality 2 i.e., binary functions deﬁned on R .Notethat the other boxes C.2 also admit such inputs on one of their side.

Acknowledgments

The authors thank Stefano Pironio for useful discussions. They acknowledge support by the Swiss NCCR Quantum Photonics and the European ERC-AG QORE.

References

1 J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge, UK, 1987. 2 A. Aspect, “Bell’s inequality test: more ideal than ever,” Nature, vol. 398, no. 6724, pp. 189–190, 1999. 3 B. F. Toner and D. Bacon, “Communication cost of simulating Bell correlations,” Physical Review Letters, vol. 91, no. 18, Article ID 187904, 4 pages, 2003. 4 J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts, “Nonlocal correlations as an information-theoretic resource,” Physical Review A, vol. 71, no. 2, Article ID 022101, 2005. 5 N. J. Cerf, N. Gisin, S. Massar, and S. Popescu, “Simulating maximal quantum entanglement without communication,” Physical Review Letters, vol. 94, no. 22, Article ID 220403, 4 pages, 2005. 6 N. Brunner, N. Gisin, S. Popescu, and V. Scarani, “Simulation of partial entanglement with nonsignaling resources,” Physical Review A, vol. 78, no. 5, Article ID 052111, 2008. 7 T. E. Tessier, C. M. Caves, I. H. Deutsch, B. Eastin, and D. Bacon, “Optimal classical-communication- assisted local model of n-qubit Greenberger-Horne-Zeilinger correlations,” Physical Review A, vol. 72, no. 3, Article ID 032305, 5 pages, 2005. 8 J. Barrett, C. M. Caves, B. Eastin, M. B. Elliott, and S. Pironio, “Modeling Pauli measurements on graph states with nearest-neighbor classical communication,” Physical Review A, vol. 75, no. 1, Article ID 012103, 2007. 9 A. Broadbent, P. R. Chouha, and A. Tapp, “The GHZ state in secret sharing and entanglement simulation,” in Proceedings of the 3rd International Conference on Quantum, Nano and Micro Technologies (ICQNM ’09), pp. 59–62, Cancun, China, February 2009. 10 S. Popescu and D. Rohrlich, “Quantum nonlocality as an axiom,” Foundations of Physics,vol.24,no.3, pp. 379–385, 1994. 11 A. C. Yao, “Protocols for secure computations,” in Proceedings of the 23rd Annual Symposium on Foundations of Computer Science (Chicago, Ill., 1982), pp. 160–164, IEEE, New York, NY, USA, 1982. 12 D. Collins, N. Gisin, S. Popescu, D. Roberts, and V. Scarani, “Bell-type inequalities to detect true n- body nonseparability,” Physical Review Letters, vol. 88, no. 17, Article ID 170405, 4 pages, 2002. 13 M. Seevinck and G. Svetlichny, “Bell-type inequalities for partial separability in N-particle systems and quantum mechanical violations,” Physical Review Letters, vol. 89, no. 6, Article ID 060401, 4 pages, 2002. 14 J.-D. Bancal, C. Branciard, N. Gisin, and S. Pironio, “Quantifying multipartite nonlocality,” Physical Review Letters, vol. 103, no. 9, Article ID 090503, 2009. 14 Advances in Mathematical Physics

15 G. Svetlichny, “Distinguishing three-body from two-body nonseparability by a Bell-type inequality,” Physical Review D, vol. 35, no. 10, pp. 3066–3069, 1987. 16 J. Barrett and S. Pironio, “Popescu-Rohrlich correlations as a unit of nonlocality,” Physical Review Letters, vol. 95, no. 14, Article ID 140401, 4 pages, 2005. 17 J.-D. Bancal, S. Pironio, and N. Gisin, in preparation. 18 E. Kushilevitz and N. Nisan, Communication Complexity, Cambridge University Press, Cambridge, UK, 1997. Paper D

Guess Your Neighbor’s Input: A Multipartite Nonlocal Game with No Quantum Advantage

M. L. Almeida, J.-D. Bancal, N. Brunner, A. Ac´ın, N. Gisin and S. Pironio

Physical Review Letters 104, 230404 (2010)

105

week ending PRL 104, 230404 (2010) PHYSICAL REVIEW LETTERS 11 JUNE 2010

Guess Your Neighbor’s Input: A Multipartite Nonlocal Game with No Quantum Advantage

Mafalda L. Almeida,1 Jean-Daniel Bancal,2 Nicolas Brunner,3 Antonio Acıń,1,4 Nicolas Gisin,2 and Stefano Pironio5 1ICFO-Institut de Ciencies Fotoniques, E-08860 Castelldefels, Barcelona, Spain 2GAP-Optique, Universite´ de Gene`ve, CH-1211 Geneva, Switzerland 3H. H. Wills Physics Laboratory, University of Bristol, Bristol, BS8 1TL, United Kingdom 4ICREA-Institucio´ Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain 5Laboratoire d’Information Quantique, Universite´ Libre de Bruxelles, 1050 Bruxelles, Belgium (Received 10 April 2010; published 9 June 2010) We present a multipartite nonlocal game in which each player must guess the input received by his neighbor. We show that quantum correlations do not perform better than classical ones at this game, for any prior distribution of the inputs. There exist, however, input distributions for which general no- signaling correlations can outperform classical and quantum correlations. Some of the Bell inequalities associated with our construction correspond to facets of the local polytope. Thus our multipartite game identifies parts of the boundary between quantum and postquantum correlations of maximal dimension. These results suggest that quantum correlations might obey a generalization of the usual no-signaling conditions in a multipartite setting.

DOI: 10.1103/PhysRevLett.104.230404 PACS numbers: 03.65.Ud, 02.50.Le, 03.65.Ta, 03.67.Hk

In recent years, the study and understanding of quantum that each participant provides an output bit ai 2f0; 1g nonlocality—the fact that certain quantum correlations equal to its right-hand neighbor’s input bit: violate Bell inequalities [1]—has benefited from a cross- a ¼ x 1 for all i ¼ 1; ...;N; (1) fertilization with information concepts. i iþ N On one hand, nonlocality has been identified as a key where xNþ1 x1. The 2 possible input strings x ¼ resource for quantum information processing. It allows, for ðx1; ...;xNÞ are chosen according to some prior distribu- instance, the reduction of communication complexity [2], tion qðxÞ¼qðx1; ...;xNÞ, which is known to the parties. and in the device-independent scenario, where one wants to The figure of merit of the game is given by the average achieve an information task without any assumption on the winning probability devices used in the protocol, it can be exploited for secure X x a x x key distribution [3], state tomography [4], and randomness ! ¼ qð ÞPð i ¼ iþ1j Þ; (2) x generation [5]. On the other hand, information concepts have provided a where Pðai ¼ xiþ1jxÞ¼Pða1 ¼ x2; ...;aN ¼ x1jx1; ...; deeper understanding of the nature of quantum nonlocality. xNÞ denotes the probability of obtaining the correct outputs It is known, in particular, that the no-signaling principle (1) when the players have received the input string x.Of (no arbitrarily fast communication between remote parties) course, players are not allowed to communicate after the is compatible with the existence of correlations more non- inputs are distributed. Thus, their performance depends local than those allowed in quantum theory [6,7]. However, only on the initially agreed-upon strategy and on the shared recent works have shown that the existence of such physical resources. stronger-than-quantum correlations would have deep The GYNI game captures a particular notion of signal- information-theoretic consequences: they would, for in- ing: if the players were able to win with high probability, stance, collapse communication complexity [8] and allow their output would reveal some information about their perfect nonlocal computation [9]. In a related direction, it neighbor’s input. We therefore expect that the nonlocal has been realized that quantum correlations actually obey a strengthened version of no-signaling, the principle of information causality [10]. Up to now, such questions have been almost exclusively considered in the bipartite scenario. Here our aim is to investigate the separation between quantum and no- signaling correlations in a multipartite scenario. For this, we introduce and study a simple multipartite nonlocal game, guess your neighbor’s input (GYNI). FIG. 1. Representation of the GYNI nonlocal game. The In GYNI, N distant players are arranged on a ring and goal is that each party outputs its right-hand neighbor’s input: 0 1 each receive an input bit xi 2f ; g (see Fig. 1). The goal is ai ¼ xiþ1.

0031-9007=10=104(23)=230404(4) 230404-1 Ó 2010 The American Physical Society week ending PRL 104, 230404 (2010) PHYSICAL REVIEW LETTERS 11 JUNE 2010 correlations of quantum theory cannot be exploited by the parties share an entangled state jc i and perform pro- noncommunicating observers to perform better at GYNI jective measurements on their subsystem dependent on than using classical resources alone. We confirm this in- their inputs xi. They then output their measurement results xi tuition and prove that, indeed, quantum correlations pro- ai. Denoting Mai the projection operator associated with vide no advantage over classical correlations. Surprisingly, the output ai for the input xi, the probability that the N however, the no-signaling principle is not at the origin of players produce the correct output is thus given by the quantum limitation: for N 3, there exist input dis- ...... x1 ... xn tributions q for which no-signaling correlations provide an Pða1 ¼ x2; ;aN ¼ x1jx1; ;xNÞ¼hMx2 Mx1 i; advantage over the best classical and quantum strategies. and the average winning probability is This suggests the possibility that in a multipartite scenario, X quantum correlations obey a qualitatively stronger version ! ¼ qðxÞhMxi; (5) of the usual no-signaling conditions. x Each of the input distributions q associated with a non- ¼ x1 xn trivial no-signaling strategy defines a Bell inequality where we have written Mx Mx2 Mx1 for short. whose maximal classical and quantum values coincide, The operators Mx satisfy the following properties: but whose no-signaling value is strictly larger. Inter- 2 Mx ¼ Mx; (6) estingly, some of these inequalities define facets of the polytope of local correlations. We thus prove the existence and of nontrivial facet Bell inequalities with no quantum vio- 0 x y y lation, answering a question raised by Gill [11]. Moreover, MxMy ¼ if ; : (7) since these Bell inequalities are facets, the GYNI game The first property follows from the fact that the Mx are identifies a portion of the boundary of the set of quantum projection operators. The second property follows from the correlations of nonzero measure, in contrast with previous xi xi orthogonality relations Ma M ¼ 0 and observation (4). information-theoretic or physical limitations on nonlocal- i ai ity [8–10,12–14]. Note that protocols involving mixed states or general mea- GYNI with classical and quantum resources.—We start surements can all be represented in the above form by expanding the dimensionality of the initial state.P by showing that the optimal classical and quantum winning x strategies are identical for any prior distribution q of the We now show, using (6) and (7), that ! ¼ xqð ÞMx inputs. Let us first show that there is a simple classical !c, where should be understood as an operator inequality; i.e.,PA B meansP that hAihBi for all jc i. First note strategy achieving a winning probability 0 0 that xqðxÞMx xq ðxÞMx, where q ðxÞ¼qðxÞþ x x 2 x !c ¼ max½qðxÞþqðxÞ; (3) ½!c qð Þqð Þ= , since by definition !c qð Þ x qðxÞ0. It is thus sufficient to consider weights q such x x x where x denotes the ‘‘negation’’ of the input string x, x ¼ that qð Þþqð Þ¼!c for all . We can then write ... 1 ðx1; ; xNÞ with xi ¼ xi , and denotes addition mod- X pffiffiffiffiffiffi X 2 ulo 2. This strategy is based on the following simple !c qðxÞMx ¼ !c xMx observation. x x X 1 2 Let y be an arbitrary string: If x y; y; þ ½xMx x Mx ; (8) 2 x there exists an i such that x ¼ y and x 1 y 1: (4) ffiffiffiffiffiffi ffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i iþ iþ p p where x ¼ !c qðxÞ= !c and x ¼ qðxÞqðxÞ=!c. Indeed, if this was not the case, we would have that for any To verify this identity we only need to use (6) and (7) and i, either xi yi or xiþ1 ¼ yiþ1. But this would in turn the fact that qðxÞþqðxÞ¼!c. Note now that the right- imply that either x ¼ y or x ¼ y, in contradiction with hand side of (8)is 0, since it is a sum of squareP involving the hypothesis. only Hermitian operators. This shows that xqðxÞMx Consider now a classical strategy specified by the string !c, as announced.P y, where each party outputs the bit ai ¼ yiþ1 if it received The inequality xqðxÞPðai ¼ xiþ1jxÞ!c can be in- the input yi, and outputs ai ¼ yiþ1 if it received yi.It terpreted as a Bell inequality whose local and quantum obviously follows that Pðai ¼ yiþ1jyÞ¼1 and Pðai ¼ bound coincide. It is well known that in order to achieve a yiþ1jyÞ¼1. On the other hand, Pðai ¼ xiþ1jxÞ¼0 for Bell violation in quantum theory one must perform mea- all x y; y. Indeed, from observation (4), there exists an i surements corresponding to noncommuting operators. The such that xi ¼ yi, but for which ai ¼ yiþ1 xiþ1. The above proof, however, does not distinguish noncommuting winning probability of this classical strategy is thus equal operators from ordinary, commuting numbers: it is based to ! ¼ qðyÞþqðyÞ, which yields (3) if we take y to be on the algebraic identity (8) which follows only from qðyÞþqðyÞ¼maxx½qðxÞþqðxÞ. Eqs. (6) and (7), regardless of whether the Mx’s commute We now prove that there is no better quantum (hence or not. This explains why the classical and quantum bounds classical) strategy. In the most general quantum protocol, are identical. 230404-2 week ending PRL 104, 230404 (2010) PHYSICAL REVIEW LETTERS 11 JUNE 2010

GYNI with no-signaling resources.—At first sight, it may By normalization of probabilities, the sum of the right- seem that the quantum limitation on the GYNI game arises hand sides of Eqs. (11) is upper bounded by one, and thus from the no-signaling principle: if the players were able to Pð000j000ÞþPð110j011ÞþPð011j101Þ1. Similar win with high probability, their output would somehow conditions are obtained for any of the four possible combi- depend on their neighbor’s input. This motivates us to look nations of three terms in Eq. (10). Summing over these at how players constrained only by the no-signaling prin- possibilities, we find 3½Pð000j000ÞþPð110j011Þþ ciple perform at GYNI. Pð011j101ÞþPð101j110Þ 4, or in other words !ns Formally, the no-signaling principle states that the mar- 4=3 1=4 ¼ 4=3!c. Furthermore, the inequality is satu- ...... ginal distribution Pðai1 ; ;ai jxi1 ; ;xi Þ for any subset rated only if the four probabilities appearing in (10) are all k k 1 3 fi1; ...;ikg of the n parties should be independent of the equal to = . It turns out that the remaining entries of the measurement settings of the remaining parties [7], i.e., that probability table PðajxÞ¼Pða1a2a3jx1x2x3Þ can be completed in a way that is compatible with the no-signaling ...... Pðai1 ; ;ai jx1; ;xNÞ¼Pðai1 ; ;ai jxi1 ; ;xi Þ: k k k principle; i.e., the bound !ns 4=3!c is achievable. Up to This guarantees that any subset of the parties is unable to relabeling of inputs and outputs, there exist two inequiva- signal to the other by their choice of inputs. lent classes of extremal no-signaling correlations achieving We show in Appendix A [15] that players constrained this winning probability (see Appendix B in [15]). One of a x 2 3 a x 1 3 0 a x only by no-signaling have a bounded winning probability them takes the form Pð j Þ¼ = gð ; Þþ = g ð ; Þ, where g and g0 are the following Boolean functions !ns 2!c. They thus cannot win in general with unit probability at GYNI. Furthermore, for certain input distri- a x 1 1 1 butions, such as the one where all input strings are chosen gð ; Þ¼a1a2a3ð x1Þð x2Þð x3Þ; x 1 2N 0 with equal weight qð Þ¼ = , we show as expected that g ða; xÞ¼ð1 a1Þð1 a2Þð1 a3Þx1a2a3 !ns ¼ !c. That is, for uniform and completely uncorrelated inputs, any resource performing better than a classical a1x2a3 a1a2x3 x1x2x3: (12) strategy is necessarily signaling. Surprisingly, this property is not general. There exist From this definition, it is easy to verify that distributions qðxÞ for which no-signaling strategies outper- Pða1a2a3jx1x2x3Þ satisfies the no-signaling conditions 1 3 4 3 form classical and quantum strategies. Consider for in- and achieves winning probability !ns ¼ = ¼ = !c. stance the following input distribution The existence of no-signaling correlations achieving !ns ¼ 4=3!c in the case N ¼ 3 is enough to show that N1 1=2 if x1 x ^ ¼ 0 4 3 3 qðxÞ¼ N (9) !ns = !c for any N . This can be seen as follows. 0 otherwise; Consider the situation in which the first three parties use the optimal strategy for N ¼ 3 while the remaining parties ^ ^ 1 where N ¼ N if N is odd and N ¼ N if N is even. It simply output their input. In this case, all the terms in ! easily follows from the previous analysis that for classical vanish, except the four terms Pð000; 0...0j000; 0...0Þ, 1 2N1 and quantum resources, !c ¼ = . We now prove, Pð110; 0...0j011; 0...0Þ, Pð011; 1...1j101; 1...1Þ, and however, that no-signaling resources can achieve !ns ¼ Pð101; 1...1j110; 1...1Þ, which are all equal to 1=3. 4 3 = !c. Note that the distribution (9) can be interpreted as a Beyond these analytical results, we obtained the maxi- promise that the sum of the inputs (modulo 2) is equal to mal no-signaling values of !ns up to N ¼ 7 players using zero. This prior knowledge does not yield any information linear programming. The ratios !ns=!c of no-signaling to to the parties about the value of their neighbor’s input, yet it classical winning probabilities are 4=3 for N ¼ 3; 4, 16=11 can be exploited by no-signaling correlations to outper- for N ¼ 5; 6, and 64=42 for N ¼ 7, showing that for more form classical strategies. parties there exist no-signaling correlations that can out- 3 We start by considering the case N ¼ , for which perform the optimal no-signaling strategy for N ¼ 3. (Note 1 that it can be shown that the winning probability for an odd ! ¼ 4½Pð000j000ÞþPð110j011ÞþPð011j101Þ number N of parties is always equal to the winning proba- þ Pð101j110Þ; (10) bility for N þ 1 players; see Appendix C in [15].) P GYNI Bell inequalities.—The GYNI Bell inequalities where Pð000j000Þ¼Pða1 ¼ 0;a2 ¼ 0;a3 ¼ 0jx1 ¼ 0; xqðxÞPðai ¼ xiþ1jxiÞ!c are not violated by quantum x2 ¼ 0;x3 ¼ 0Þ, and so on. Consider the first three terms in theory, but can be violated by more general no-signaling (10). The no-signaling principle implies that X X theories. In [11], Gill raised the question of whether there Pð000j000Þ Pð00a3j000Þ¼ pð00a3j001Þ; exist Bell inequalities which (i) feature this ‘‘no quantum a3 a3 advantage’’ property and (ii) define facets of the polytope X X of local correlations. Here we give a positive answer to this Pð110j011Þ Pð1a20j011Þ¼ pð1a20j001Þ; (11) question. We have checked that the GYNI inequalities a2 a2 X X defined by the distribution (9) are facet defining for N Pð011j101Þ Pða111j101Þ¼ pða111j001Þ: 7 players. More generally, we verified that the inequalities a1 a1 defined by the distribution qðxÞ having uniform support on 230404-3 week ending PRL 104, 230404 (2010) PHYSICAL REVIEW LETTERS 11 JUNE 2010

N^ 0 7 (10). In particular, it would be interesting to understand if i¼1xi ¼ are facet defining for all N . We conjecture that they are facet defining for any number of parties. Note they can be exploited for other information tasks. Finally, also that the polytope of local correlations for the case N ¼ our results suggest that the quantum limitation on the 3 (with binary inputs and outputs) was completely charac- GYNI game might originate from a generalization of the terized in [16]; the inequality corresponding to (10) be- no-signaling principle in a multipartite setting. Can this longs to the class 10 of [16]. Geometrically, our result intuition be made concrete? Are there more general infor- shows that the polytope of local correlations and the set mation tasks with no quantum advantage? of quantum correlations have in common faces of maximal We thank C. Branciard and D. Perez-Garcia for discus- dimension [we recall that a facet corresponds to a ðd sions, and the QAP Partner Exchange Programme. This 1Þ-dimensional face of a d-dimensional polytope]. work is financially supported by the Fundaçaõ para a This also implies that GYNI is an information-theoretic Cieˆncia e a Tecnologia (Portugal) through Grant game that identifies a portion of the boundary of quantum No. SFRH/BD/21915/2005, the European ERC-AG Qore correlations which is of nonzero measure. To the best of and PERCENT projects, the Spanish MEC FIS2007-60182 our knowledge, all previously introduced information- and Consolider-Ingenio QOIT projects, Generalitat de theoretic or physical principles recovering part of the Catalunya and Caixa Manresa, the UK EPSRC, the quantum boundary—including nonlocal computation [9], Interuniversity Attraction Poles (Belgian Science Policy) nonlocality swapping [12], information causality [10,13], project IAP6-10 Photonics@be, the EU project QAP and macroscopic locality [14]—only single out a portion of Contract No. 015848, and the Brussels-Capital region zero measure [17]. through a BB2B grant. Discussion and open questions.—Our work raises plenty of new questions. First, it would be interesting to understand the structure of those input distributions q leading to a gap between no-signaling and classical or quantum cor- [1] J. S. Bell, Physics 1, 195 (1964). relations (see Appendix A in [15] for a class of distribu- [2] H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, tions for which there is no gap). For instance, in the case of arXiv:0907.3584 [Rev. Mod. Phys. (to be published)]. four parties, the distribution q having uniform support on [3] A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Phys. Rev. Lett. 98, 230501 (2007). x1 x2 x3 x1x2x3 ¼ 0 leads to !ns ¼ 4=3!c.How- ever, the corresponding Bell inequality is not a facet. [4] C.-E. Bardyn, T. C. H. Liew, S. Massar, M. McKague, and V. Scarani, Phys. Rev. A 80, 062327 (2009). Another question is thus to single out, among all relevant [5] S. Pironio et al., Nature (London) 464, 1021 (2010). input distributions, those corresponding to facet Bell in- [6] S. Popescu and R. Rohrlich, Found. Phys. 24, 379 (1994). equalities. For three parties, it follows from [16] that the [7] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, distribution (9) is the unique possibility. and D. Roberts, Phys. Rev. A 71, 022101 (2005). A further interesting problem is whether there exist facet [8] W. van Dam, arXiv:quant-ph/0501159v1; G. Brassard, H. Bell inequalities with no quantum advantage in the bipar- Buhrman, N. Linden, A. A. Me´thot, A. Tapp, and F. Unger, tite case. Note that our GYNI inequalities are nontrivial Phys. Rev. Lett. 96, 250401 (2006); N. Brunner and P. only for N 3; for the case N ¼ 2, the classical and no- Skrzypczyk, Phys. Rev. Lett. 102, 160403 (2009). signaling bounds are always equal. In Ref. [9], examples of [9] N. Linden, S. Popescu, A. J. Short, and A. Winter, Phys. bipartite Bell inequalities with no quantum advantage have Rev. Lett. 99, 180502 (2007). been presented in the context of nonlocal computation. [10] M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Zukowski, Nature (London) 461, 1101 However, as mentioned earlier, none of the Bell inequal- (2009). ities associated with nonlocal computation have been [11] See Problem 26.b at http://www.imaph.tu-bs.de/qi/ proven to be facet defining. We studied this question here problems/26.html. and could prove that none of the simplest inequalities from [12] P. Skrzypczyk, N. Brunner, and S. Popescu, Phys. Rev. [9] (corresponding to the family of inequalities specified by Lett. 102, 110402 (2009). the parameters n ¼ 2; 3 in [9]) are facet inequalities. The [13] J. Allcock, N. Brunner, M. Pawlowski, and V. Scarani, proof uses a mapping from these inequalities to the space Phys. Rev. A 80, 040103(R) (2009). of correlation inequalities for n parties, two settings and [14] M. Navascues and H. Wunderlich, Proc. R. Soc. A 466, two outcomes, which was fully characterized in Ref. [18]; 881 (2010). see Appendix D in [15] for a detailed proof. We conjecture [15] See supplementary material at http://link.aps.org/ that none of the Bell inequalities introduced in [9] are facet supplemental/10.1103/PhysRevLett.104.230404 for detailed proofs. defining. [16] C. Sliwa, Phys. Lett. A 317, 165 (2003). Coming back to our original motivation, it would be [17] Note that Ref. [14] can actually recover the entire quantum interesting to get a deeper understanding of the structure boundary in the correlator space. and information-theoretic properties of the no-signaling [18] R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 correlations giving an advantage over classical or quantum (2001); M. Zukowski and C. Brukner, Phys. Rev. Lett. 88, correlations, for instance, those associated with inequality 210401 (2002).

230404-4 6

AUXILIARY ONLINE MATERIAL APPENDIX B

APPENDIX A Here we describe two inequivalent no-signaling correlations which attain ωns = 4 /3 ωc for the tripartite inequal- Here we derive the upper bound ω 2ω for the ity (10). These correlations are extremal non-local boxes ns ≤ c winning probability ωns of no-signalling strategies. We in the sense of being vertices of the no-signaling polytope then show that ωns = ωc for all input distributions q(x) for three parties and binary inputs and outputs [7]. such that q(x) q(y) = q(¯y ) for some input string y. Writing ( a, b, c ) for ( a1, a 2, a 3) and ( x, y, z ) for ≤ Such distributions include in particular the uniform dis- (x1, x 2, x 3), we can write the first box as tribution where all input strings are chosen with equal N 1 weight q(x) = 1 /2 . P1(a, b, c x, y, z ) = f(a, b, c, x, y, z ) (16) To start we derive the upper bound ω 2ω , valid | 3 ns ≤ c for any distribution q(x). From the definition (3), we where f(a, b, c, x, y, z ) is the boolean function have that q(x) ω for every input string x. This triv- ≤ c ially leads to the upper-bound f(a, b, c, x, y, z ) =(1 b x y xy )(1 c z) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ (17) ω ω P (a = x x) . (13) a(1 y cy b(c z)) . ns ≤ c i i+1 | ⊕ ⊕ ⊕ ⊕ ⊕ x X Similarly, we can write the second box as Notice that this bound is only meaningful when the right- hand side is smaller than 1, since obviously ωns 1. 2 1 ≤ P2(a, b, c x, y, z ) = g(a, b, c, x, y, z ) + g′(a, b, c, x, y, z ) We now show that for all no-signalling distributions | 3 3 (18) x P (ai = xi+1 x) 2, from which the bound ωns | ≤ ≤ with g and g the two boolean functions 2ωc immediately follows. ′ PFirst note that from the no-signaling condition, g(a, b, c, x, y, z ) = abc (1 x)(1 y)(1 z) ⊕ ⊕ ⊕ P (a1,...,a k 1 x1,...,x k 1) P (a1,...,a k x1,...,x k) . g′(a, b, c, x, y, z ) =(1 a)(1 b)(1 c) (19) − | − ≤ | (14) ⊕ ⊕ ⊕ xbc ayc abz xyz . We now write ⊕ ⊕ ⊕ ⊕ P (a = x x) Among the boxes that are equivalent to P1 under relabel- i i+1 | x ing of parties, inputs, and outputs, a total of 24 of them X violate maximally the Bell inequality (10), and similarly = P (a1 = x2,...,a N = x1 x1,...,x N ) | for 8 of those that are equivalent to P2. Even though x1,...,x N X other tripartite no-signaling boxes (inequivalent to P1 or P (a1 = x2,...,a N 1 = xN x1,...,x N 1) P2 under relabeling of parties, inputs, or outputs) violate ≤ − | − x ,...,x 1X N the Bell inequality (10), those 32 boxes obtained from P1 and P are the unique ones that violate it maximally. = P (a1 = x2,...,a N 2 = xN 1 x1,...,x N 2) 2 − − | − x1,...,x N 1 X − where the inequality follows from (14) and in the last APPENDIX C equality we used the no-signaling condition after summing over xN . Iteratively performing this last step, we Here we show that for the input distribution (9), the finally obtain no-signaling bound for an even number of parties N + 1 is always equal to the no-signaling bound for N parties. P (a = x x) P (a = x x ) 2 . (15) i i+1 | ≤ 1 2| 1 ≤ Start by considering N +1-GYNI game, where the first N x x ,x X X1 2 players use the optimal strategy for the N-player case and We now analyze the no-signalling winning probability player N + 1 outputs its input. They then achieve a no- for distributions satisfying q(x) q(y) = q(¯y ) for some signaling violation equal to the N case, which imposes the ≤ input string y. Note that for such weights ωc = q(y) + lower bound ωns (N + 1) ωns (N). But this is actually q(¯y ) = 2 q(y), as easily follows from (3). We thus have the best average success these≥ N + 1 players can obtain. ω To see that, consider the game for N +1 parties. Allowing ω = q(x)P (a = x x) c P (a = x x) . ns i i+1 | ≤ 2 i i+1 | players N and N+1 to communicate can only increase the x x X X achievable value of ωns (N + 1). Indeed, in this situation But, as we have shown above, P (a = x x) 2 for the best strategy that player N can adopt is to output x i i+1 | ≤ all no-signalling distributions , and thus ωns ωc. Since xN+1 , which was communicated to him by player N + P ≤ any classical strategy is also a no-signalling strategy, it 1, while player N + 1 needs to guess x1 given xN and actually holds that ωns = ωc. xN+1 . Clearly, the knowledge of xN+1 is of no use for 7 him since this bit is completely uncorrelated with the space for which the complete set of tight Bell inequalities rest of the input string. Consequently, the situation is has been provided in Ref. [17]. analogous to having players 1 , . . . , N 1, N + 1 (i.e. all To any inequality of the form (20) defined by the triple players except player N) play the game− for N parties. (n, f, p˜), we associate the following Bell inequality in the Therefore ω (N + 1) ω (N) and we have finally that (n, 2, 2) full-correlation space: ns ≤ ns ωns (N + 1) = ωns (N) for odd N. n I (n, f, p˜) = c(z) C . . . C 2− k(n, f, p˜) n22 h z1 zn i ≤ z X (22) APPENDIX D where c(z) = ( 1) f(z)p˜(z), and where we view z 0, 1 − i ∈ { } as denoting one of two possible observables Czi of party i In what follows, we derive a criterion that is necessarily taking values 1, 1 (with i = 1 ,...,n ). satisfied by any facet-defining Bell inequality associated {− } to the task of nonlocal computation (NLC) [9], and show Lemma. If the NLC inequality I(n, f, p˜) for n that none of the NLC Bell inequalities for boolean func- bits is facet-defining, then the corresponding inequal- tions of two and three input bits are facet-defining. ity In22 (n, f, p˜) is facet-defining in the (n, 2, 2) full- Nonlocal computation is a distributed task of two par- correlation space. ties, where the goal is to compute a given boolean func- Proof. The deterministic extremal points of the tion f(z) of an n-bit string z. The input bit string is de- (n, 2, 2) full-correlation polytope are of the form [17] composed into two strings x and y, such that x y = z. ⊕ u z u z δ u z δ The bit string x is sent to party A while the bit string y C . . . C = ( 1) 1 1 . . . ( 1) n n ( 1) = ( 1) · ⊕ h z1 zn i − − − − is sent to party B. Upon receiving their input bit strings, (23) A and B each output a single bit, a and b respectively, where u 0, 1 specifies the local strategy of each party i ∈ { } such that the following relation holds: a b = f(z). Im- and δ 0, 1 represents an additional global sign flip, ∈ { } portantly, each party has locally no information⊕ about which we can think of as being carried out by the last the input bit string z, that is P (xi = zi) = 1 /2 for all party. These deterministic points are thus specified by i = 1 , ..., n . For each n, f(z), and distribution of inputs the single bit δ and the n-bit string u, and are therefore in p˜(z), we obtain a Bell expression whose value is associ- one-to-one correspondence with the extremal points (21) ated to the probability of success at the task. These NLC saturating the inequalities (20). For any such strategy inequalities have the form specified by δ and u, we have that

u (x+y) δ f(z) AxBy = ( 1) · ⊕ I(n, f, p˜) = ( 1) p˜(z) AxBy k(n, f, p˜) h i − − h i ≤ x y=z x y=z (24) z x y=z ⊕X ⊕X X ⊕X n u z δ n (20) = 2 ( 1) · ⊕ = 2 C . . . C . − h z1 zn i where Ax and By are observables which take values 1, 1 . Notice that each party measures 2 n observables. It immediately follows from the above identity that the {− } In Ref. [9] it is proven that the best classical strategy inequalities (22) are valid for the ( n, 2, 2) full-correlation ax by polytope. is given by Ax = ( 1) and By = ( 1) with − − Let us now suppose that the Bell inequality n a = u x, b = u y δ , (21) In22 (n, f, p˜) 2− k(n, f, p˜) is not facet-defining. Then x y ≤ 1 2 · · ⊕ we can write In22 (n, f, p˜) = In22 (n, f, p˜) + In22 (n, f, p˜) 1 2 where δ denotes a single bit and u an n-bit string shared and k(n, f, p˜) = k (n, f, p˜) + k (n, f, p˜) for some 1 2 1 2 by the parties. This classical strategy, which is a lin- In22 (n, f, p˜), In22 (n, f, p˜), k (n, f, p˜), and k (n, f, p˜) such ear approximation of the function f, achieves a winning that probability as high as any quantum resource. Thus the 1 n 1 I (n, f, p˜) 2− k (n, f, p˜) (25) local and quantum bounds of inequalities (20) coincide. n22 ≤ There exist however no-signaling correlations which can and perform with winning probability one at this game. 2 n 2 I (n, f, p˜) 2− k (n, f, p˜) (26) Checking whether the NLC inequalities (20) are facet- n22 ≤ defining is in general a hard problem since one should are valid inequalities for the ( n, 2, 2) full-correlation poly- consider any input size n, boolean function f, and dis- tope, i.e., they are satisfied by all deterministic points of tributionp ˜ (z). Below we give a first simplification to the form (23). But then it follows from the above cor- this problem by deriving a necessary criterion satisfied respondence between deterministic point of the ( n, 2, 2) by facet NLC inequalities. Our method is based on a n n polytope and the (2 , 2 , 2) polytope that I(n, f, p˜) = mapping from the (2 , 2 , 2) correlation space – i.e. (2 I1(n, f, p˜) + I2(n, f, p˜), where parties, 2 n settings, 2 outcomes) – in which the NLC inequalities are defined, into the ( n, 2, 2) full-correlation I1(n, f, p˜) k1(n, f, p˜) (27) ≤ 8 and Ref. [17] a construction for the coefficients c(z) of all facet (n, 2, 2) correlation Bell inequalities has been given. For 2 2 I (n, f, p˜) k (n, f, p˜) (28) small number of inputs, i.e. n = 2 , 3, we have explicitly ≤ verified that none of the corresponding NLC inequalities are valid NLC inequalities. This implies that I(n, f, p˜) are facet-defining; all these inequalities can actually be n ≤ k(n, f, p˜) is not facet-defining for the (2 , 2 , 2) polytope, expressed as sums of CHSH inequalities. For larger n from which the statement of the Lemma follows. however, a similar analysis becomes difficult due to the The above Lemma implies that it is sufficient to re- large number of facet ( n, 2, 2) Bell inequalities and the strict our analysis on NLC inequalities associated with high dimensionality of the (2 , 2n, 2) correlation space. facet inequalities in the ( n, 2, 2)-full correlation space. In

Paper E

Looking for symmetric Bell inequalities

J.-D. Bancal, N. Gisin and S. Pironio

Journal of Physics A: Mathematical and Theoretical 43, 385303 (2010)

115

IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 43 (2010) 385303 (16pp) doi:10.1088/1751-8113/43/38/385303

Looking for symmetric Bell inequalities

Jean-Daniel Bancal1, Nicolas Gisin1 and Stefano Pironio2

1 Group of Applied Physics, University of Geneva, 20 rue de l’Ecole-de Medecine,´ CH-1211 Geneva 4, Switzerland 2 Laboratoire d’Information Quantique, Universite´ Libre de Bruxelles, Belgium

E-mail: [email protected]

Received 23 April 2010, in ﬁnal form 11 July 2010 Published 6 August 2010 Online at stacks.iop.org/JPhysA/43/385303

Abstract Finding all Bell inequalities for a given number of parties, measurement settings and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238 885 new Bell inequalities and 1085 new Svetlichny inequalities. We ﬁnd, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins–Gisin–Linden–Massar–Popescu type.

PACS numbers: 03.65.Ud, 03.67.−a

1. Introduction

Already discovered by Boole in the theory of logic and probabilities as ‘conditions of possible experience’ [1], Bell inequalities found a new dimension with the work of John Bell who showed that quantum physics could violate these conditions in some situations, highlighting what is now known as quantum nonlocality [2]. Complete set of Bell inequalities are known only for setups involving small numbers of parties, measurement settings and measurement outcomes. This may already be sufficient for various applications, such as exhibiting the nonlocality of a noisy quantum state in a real experiment [3] or establishing the security of a device-independent quantum key distribution protocol [4, 5]. But the simplest inequalities are not always optimal. For instance, certain inequalities with a large number of measurement settings are much more resistant to the detection inefficiencies than the CHSH inequality [6, 7] or are violated by quantum states that do not violate the CHSH inequality [8, 9]. This motivates a search for Bell inequalities involving more parties, measurements or outcomes. But finding all Bell inequalities pertaining to a given experimental setup is in general a hard task [1, 10], and a complete search with current-day techniques is not feasible in most instances. It is instructive, however, to realize that many useful Bell inequalities, like CHSH [11], Mermin [12]orCGLMP[13] to cite just a few, can be written in a form that is invariant under

1751-8113/10/385303+16$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1 J. Phys. A: Math. Theor. 43 (2010) 385303 J-D Bancal et al any permutation of the parties. Symmetric inequalities are also attractive because they are likely to be easier to handle. Motivated by this observation, we show here how to exploit a symmetric version of the full Bell polytope to generate all symmetric Bell inequalities. This symmetrized polytope is much easier to handle than the full Bell polytope. In particular, for the Bell experiment with binary settings and outcomes, the number of extremal points of the symmetrized polytope only grows polynomially with the number of parties, which is an exponential gain compared to the general situation. Our method for ﬁnding symmetric inequalities is not restricted to Bell inequalities, but can be applied to any set of inequalities characterizing a correlation polytope, like for instance the Svetlichny inequalities, which allow to test for genuinely multipartite nonlocality [14]. We present in the next section our approach from this general perspective. We then apply in section 3 our method to several examples for which listing all (non-symmetric) inequalities is computationally intractable with present-day techniques. In the Bell scenario with two parties, two settings and four outcomes, we ﬁnd in particular facet inequalities that are not of the CGLMP form, answering an open question raised by Gill3.

2. General setting

Let (n, m, k) denote a Bell experiment where n parties can choose one out of m possible measurement settings that each yield one out of k possible outcomes4. The statistics of the observed results are described by the joint conditional probability distributions (also called correlations5)

p(r1,...,rn|s1,...,sn), (1) where si ∈{1,...,m} denotes the measurement setting of party i and ri ∈{1,...,k} denotes the corresponding measurement result. Note that in general the N = mnkn probabilities (1) are not all independent but satisfy linear constraints, such as the normalization conditions or the no-signalling conditions. We are N thus actually interested in an afﬁne subspace of R of dimension d, where d = mn(kn − 1) for normalized correlations, and d = (m(k − 1) +1)n − 1 for correlations that satisfy in addition the no-signalling conditions [9, 15]. We suppose in the following that a proper parametrization d has been introduced so that the joint distributions (1) can be identiﬁed with points p in R . d We are interested in whether a given p belongs to some special subset P ⊆ R .Inthis work, we consider sets P which are polytopes6. A polytope can be described by the list V of its vertices (or extremal points) v ∈ V , and any point p ∈ P can be written as p = ρvv, (2) v∈V = where ρv are positive and normalized weights: ρv 0 for all v and v∈V ρv 1. In the case of the Bell polytope, for instance, the extremal points are the deterministic local strategies, corresponding to the prior assignment of an outcome ri,si ,v to each measurement setting si.

3 Problem 27.A on http://www.imaph.tu-bs.de/qi/problems/27.html. 4 More general Bell scenarios with different numbers of measurement settings mi and outcomes ki for each party i are also possible, but since we will consider situations that are symmetric under permutations of the parties, we choose mi = m, ki = k for all i. 5 Throughout this paper, the term ‘correlations’ refers to probability distributions of the form (1). It should not to be confused with ‘correlator’ or ‘correlation functions’, such as E(s1,s2) = p(r1 = r2|s1,s2) − p(r1 = r2|s1,s2). 6 In general, it might also be interesting to consider sets that are not polytopes [16, 17], nor even convex [18].

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There are thus knm different vertices v, corresponding to joint probability distributions of the form 1ifr = r for all i = 1,...,n p(r ,...,r |s ,...,s ) = i i,si ,v (3) 1 n 1 n 0 otherwise. Alternatively to its representation in terms of vertices (the V-representation), a polytope P can also be described, by the Farkas–Minkowski–Weyl theorem, as the intersection of finitely many half-spaces (the H-representation). A half-space is defined by an inequality · = d ∈ d+1 h p i=1 hi pi h0 specified by the couple (h, h0) R . We say that an inequality · (h, h0) is valid for the polytope P,ifh p h0 for all points p in P. The Farkas–Minkowski– Weyl theorem states that a polytope can be characterized by a finite set of valid inequalities. That is, a point p belongs to P if and only if · ∈ h p h0 for all (h, h0) H, (4) d+1 where H is some finite set in R . This description is particularly appropriate when willing to show that a point does not belong to the polytope, since it is sufficient to exhibit the violation of a single one of the inequalities (4). In the case of the Bell polytope, these inequalities are known as Bell inequalities. A complete and minimal representation of a polytope in the form (4) is provided by the set of facets of the polytope. An inequality (f, f0) ∈ H defines a facet if its associated hyper-plane f ·p = f0 intersects the boundary of the polytope in a set of dimension d −1, i.e. if there exists d affinely independent points of P satisfying f ·p = f0.Thesetfˆ ={p | p ∈ P, f ·p = p0} then corresponds to the facet defined by (f, f0). Finding all the Bell inequalities corresponding to an experimental configuration (n,m,k) thus amounts to determining the facets (minimal H-representation) of a polytope when given its extremal points (V-representation). This conversion problem is well known and there exists several available algorithms to solve it [19]. However, when n, m or k are too large, the associated polytope becomes too complex to be handled by these algorithms. It might then be appropriate to focus the search on a subclass of all facet inequalities. We explain how this can be done for symmetric inequalities in the next section.

2.1. Focusing on symmetric facets

Some facet inequalities (f, f0) of a correlation polytope P can be written in a way that is invariant under permutations of the parties. This is the case for instance for the CHSH inequality, which can be written in the CH form as − − − p(a1) + p(b1) p(a1b1) p(a1b2) p(a2b1) + p(a2b2) 0, (5) 7 wherewewritep(a, b|x,y) for p(r1,r2|s1,s2) and deﬁne p(ax ) = p(a = 1|x), p(by ) = p(b = 1|y), p(ax by ) = p(a = 1,b = 1|x,y). In the following we will call such facets symmetric facets8. If one is interested in ﬁnding only the symmetric Bell inequalities relevant to a given scenario (n,m,k), then it is possible to restrict the space in which to search for them. This is what we show now. Let G be the set of permutations of {1,...,n}. Given a permutation π ∈ G,let

p(a1,...,an|x1,...,xn) → p(aπ(1),...,aπ(n)|xπ(1),...,xπ(n)) (6)

7 From now on, we write p(a,b,c...|x,y,z,...) for p(r1,r2,r3,...|s1,s2,s3,...). 8 Note that a symmetric inequality need not appear in a symmetric form when written in any of its equivalent forms under relabelling of measurement settings and outcomes. For instance p(a1)+p(b2)−p(a1b1)−p(a1b2)+p(a2b1)− p(a2b2) 0 is equivalent to the CH inequality (5) up to relabelling of the settings and outcomes, but is not invariant under the exchange of parties 1 and 2.

3 J. Phys. A: Math. Theor. 43 (2010) 385303 J-D Bancal et al be its action on the joint probability distributions. This permutation induces a transformation d d d π : R → R : p → πp in the vector space R in which the correlations p are represented, which by abuse of language we denote by the same symbol π. Note that the correlation polytopes that we consider here are evidently invariant under such permutations, i.e. πP = P , and any vertex v ∈ V is mapped into another vertex πv ∈ V . Given a facet-inequality (f, f0) of P,let(πf, f0) be its image under π.Note that with this definition the facet fˆ ={p | p ∈ P, f · p = f0} is mapped onto πfˆ ={πp | p ∈ P, f ·p = f0} since {p | p ∈ P,(πf)·p = f0}={p | p ∈ P,f ·(π †p) = f0}={πp | p ∈ P, f · p = f0}, where we used the fact that the transformations π are −1 unitary, i.e. π † = π . We say that a facet (f, f0) is symmetric if πf = f for all π ∈ G. Consider now the symmetrizing map 1 = π (7) |G| π∈G d and let ˜ denote the projection on the symmetric affine subspace S of R of dimension = d ds dim(S). We suppose to simplify the presentation that the origin of R is contained in S (this can always be achieved by a proper translation of the correlation vectors p), so that S is d ∈ d = actually a linear subspace of R . An arbitrary vector p R can then be written as p ps +pt , where ps = p˜ is the projection of p on the symmetric subspace S and pt = (1 − )p˜ on the = − ˜ d complementary space T (1 )R . Similarly an arbitrary inequality (f, f0) can be written as (fs ⊕ ft ,f0). A symmetric inequality then takes the form (fs ⊕ 0,f0) where 0 denotes the − d ds ∈ ds +1 null vector in R . Given an inequality (fs ,f0) R defined in the symmetric subspace S, we denote its symmetric extension as the inequality (f, f0) = (fs ⊕ 0,f0) defined in the d full space R . Let Ps = P˜ ={ps | p ∈ P } be the projection of the polytope P on the symmetric ∈ = ∈ subspace. Any vertex w Ps is necessarily the projection w vsof some vertex v P . = ∈ = Indeed, suppose that w is the projection of a non-extremal point p i qi vi P , i qi 1, = ˜ = ˜ = 0

∈ ds Theorem. Let (fs ,f0) be a facet inequality for the polytope Ps R . Then its symmetric ⊕ ∈ d+1 ∈ d extension (fs 0,f0) R defines a valid inequality for the full polytope P R . Moreover all symmetric facet inequalities of the full polytope P are the symmetric extension of some facets of the symmetrized polytope Ps. = ⊕ · Proof. The symmetric extension (f, f0) (fs 0,f0) is valid for P if f p f0 is satisfied by all points p ∈ P . But this immediately follows from the fact that f · p = ⊕ · ⊕ = · · ∈ (fs 0) (ps pt ) fs ps f0 and the fact that fs ps f0 is valid for all ps Ps . Now, let (g, g0) = (gs ⊕ 0,g0) be a symmetric facet of P. Clearly, it is the symmetric ∈ ds extension of the inequality (gs ,g0) R , which is valid for Ps. Moreover, (gs ,g0) defines a facet of Ps as there exist ds affinely independent points in Ps that saturate it. Indeed, since (g, g0) defines a facet of P, there exist d affinely independent points p in P that satisfy (gs ⊕ 0) · p = g0. These points are of the form p = ps ⊕ pt , where pt can clearly be arbitrary. Since the complementary space T is of dimension d − ds , there must therefore be at least

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(a)(b)(c)

3 Figure 1. (a) Example of a polytope P in the vector space R .(b) Subspace S symmetric under the exchange of coordinates e1 and e2. Ps (grey) is the projection of the polytope onto this subspace. (c) fs and gs are two facets of Ps,andf and g are their symmetric extensions to the whole space 3 R . f is a symmetric facet of the original polytope P, whereas g is just a valid inequality for P.

d − (d − ds ) = ds affinely independent points of the form ps ⊕ 0 that saturate the inequality (gs ⊕ 0,g0). These points obviously define ds affinely independent points in Ps that saturate the inequality (gs ,g0).

Note however that the converse of the theorem is not true, as illustrated in ﬁgure 1: facets d of Ps do not necessarily extend to facets of the polytope P in the general space R .Weshow in section 2.3 how it is nevertheless possible to take advantage of such inequalities to generate new (not necessarily symmetric) facet inequalities for the original polytope P.

2.2. Illustration on the (2, 2, 2) bell scenario We now illustrate in detail the above approach on the (2, 2, 2) Bell scenario. This scenario is characterized by 16 probabilities p(ab|xy), where x ∈{1, 2} denote the measurement setting of Alice and a ∈{1, 2} the corresponding outcome, and where similarly y ∈{1, 2} and b ∈{1, 2} denote Bob’s measurement setting and outcome. These probabilities satisfy normalization p(ab|xy) = 1 for all x,y = 1, 2, (8) a,b=1,2 and no-signalling p(a|x) ≡ p(ab|xy) for all a,x,y = 1, 2 b=1,2 (9) p(b|y) ≡ p(ab|xy) for all b, x, y = 1, 2. a=1,2 In total, only 8 of the 16 probabilities p(ab|xy) are therefore independent and we can represent 8 the correlations p as elements of R . For speciﬁcity, we choose the following parametrization

p = [p(a1), p(a2), p(b1), p(b2), p(a1b1), p(a1b2), p(a2b1), p(a2b2)], (10) where p(ax ) = p(1|x), p(by ) = p(1|y) and p(ax by ) = p(11|xy).

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The Bell-local polytope is described by 16 vertices = va1a2b1b2 [δ(a1, 1), δ(a2, 1), δ(b1, 1), δ(b2, 1), δ(a1b1, 1), δ(a1b2, 1), δ(a2b1, 1), δ(a2b2, 1)], (11) where a1,a2,b1,b2 ∈{1, 2} speciﬁes the deterministic assignment of an outcome to each measurement setting, and where δ is the Kronceker delta. The group G of permutation of two parties contains two elements: the identity 11 and the permutation π acting as follows on a vector p:

πp = [p(b1), p(b2), p(a1), p(a2), p(a1b1), p(a2b1), p(a1b2), p(a2b2)]. (12) The symmetrizing map projecting on the space S of symmetric correlations is thus equal to = 1 − = 1 − 2 (11+π), while the map projecting on the complementary space T is 11 2 (11 π). Arbitrary correlations p can thus be decomposed into a symmetric and an asymmetric part p = ps ⊕ pt , where p + πp ps = 2 p(a ) + p(b ) p(a ) + p(b ) p(a ) + p(b ) p(a ) + p(b ) = 1 1 , 2 2 , 1 1 , 2 2 , 2 2 2 2 p(a b ) + p(a b ) p(a b ) + p(a b ) p(a b ), 1 2 2 1 , 1 2 2 1 ,p(a b ) , (13) 1 1 2 2 2 2 and p − πp pt = 2 p(a ) − p(b ) p(a ) − p(b ) −p(a ) + p(b ) −p(a ) + p(b ) = 1 1 , 2 2 , 1 1 , 2 2 , 2 2 2 2 p(a b ) − p(a b ) −p(a b ) + p(a b ) 0, 1 2 2 1 , 1 2 2 1 , 0 . (14) 2 2 8 Note that the symmetric part ps lives in a five-dimensional subspace of R and can thus be expressed in an appropriate basis as p(a ) + p(b ) p(a ) + p(b ) p(a b ) + p(a b ) p = 1 1 , 2 2 ,p(a b ), 1 2 2 1 ,p(a b ) . (15) s 2 2 1 1 2 2 2 8 Similarly, pt lives in a three-dimensional space of R and can be decomposed in a proper basis as p(a ) − p(b ) p(a ) − p(b ) p(a b ) − p(a b ) p = 1 1 , 2 2 , 1 2 2 1 . (16) t 2 2 2 The projection of the 16 deterministic points (11) on the symmetric subspace defined by (15)aregivenby = δ(a1, 1) + δ(b1, 1) δ(a2, 1) + δ(b2, 1) vs;a1a2b1b2 , , 2 2 δ(a b , 1) + δ(a b , 1) δ(a b , 1), 1 2 2 1 ,δ(a b , 1) . (17) 1 1 2 2 2 Note that some vertices of the original polytope are projected onto the same point of the symmetric polytope. For instance, vs;1112 = vs;1211. In total, it can be verified that the set defined by (17) contains only ten extremal points.

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We thus have reduced the original eight-dimensional polytope defined by 16 vertices to a five-dimensional polytope with ten vertices. Applying to this symmetrized polytope a standard algorithm performing the transformation from the V-representation to the H-representation, we find four different types (up to relabelling of settings and outcomes) of facets of the symmetric polytope:

P(a1b1) 0

P(a1b2) + P(a2b1) 0 (18) − P(a1) + P(b1) 2P(a1b1) 0 − − − P(a1) + P(b1) P(a1b1) P(a1b2) P(a2b1) + P(a2b2) 0. We recognize the first inequality as the positivity condition for the joint probabilities and the last one as the CHSH inequality (written in the CH form as in equation (5)). These two classes of inequalities define symmetric facets of the full polytope. The two other inequalities are valid inequalities for the full local polytope, but do not correspond to facets (although they are facets of the symmetrized polytope, as the inequality g in figure 1). Note that in this (2, 2, 2) Bell scenario, the only two types of facet inequalities of the full polytope (the positivity condition and the CHSH inequality) can be written in a symmetric way. Hence in this simple case finding the facets of the symmetrized polytope is sufficient to generate all Bell inequalities.

2.3. Generating facet inequalities from valid inequalities

As we mentioned earlier, and was illustrated above, facets of the symmetrized polytope Ps can correspond to inequalities which are not facets of the original polytope P. These inequalities are nonetheless valid inequalities which are satisfied by all points in P and which might be violated by points that do not violate any of the symmetric facets of P. These inequalities may be used to generate new (non-symmetric) facets of P. There exist various deterministic or heuristic algorithms which may generate new facet inequalities starting from a valid (not facet-defining) inequality, see for instance [23]. Here, we describe a procedure that can be used whenever the starting valid inequality corresponds to a high-dimensional face of P, i.e. when the number of affinely independent vertices that saturate the inequality is large. In this case, it is possible to find all the facets that contain this high-dimensional face by completing the list of vertices with all possible combinations of vertices that do not saturate the inequality, as detailed by the following algorithm.

(1) Let (f, f0) be a valid inequality for P and let W ={v | f · v = f0} be the set of vertices saturating this inequality. (2) Let dim(W) denote the number of afﬁnely independent points in W.

• If dim(W) = d, then (f, f0) is a facet of P. • If dim(W) < d,letU ={v | dim(W ∪ v) > dim(W)}. For every u ∈ U,let(g, g0) be the hyperplane passing through the points in U.If (g, g0) is a valid inequality for P, it now deﬁnes a face of P of dimension dim(W)+1; in this case, go back to point 1 with (g, g0) as a starting inequality.

3. Applications

We now illustrate our method in several situations for which generating the complete set of facet inequalities using standard polytope software [19] is too time-consuming to be feasible.

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Table 1. Summary of our numerical results. For each scenario we give the dimension of the space the polytope lives in as well as its number of extremal vertices, both before (d, |V |) and after projection on the symmetric subspace (ds , |Vs |). We give the number of inequivalent symmetric inequalities valid for P (but not necessarily facet-deﬁning) obtained by resolving the symmetric polytope Ps and the number of those inequalities that are facet deﬁning. # symmetric # symmetric

Bell scenario dds |V ||Vs | inequalities facets (2, 2, 4) 48 27 256 136 29 12 (2, 4, 2) 24 14 256 136 90 55 (4, 2, 2) 80 14 256 35 627 392 (5, 2, 2) 242 20 1024 56 >238 464 238 464 Correlators (3, 3, 2) 27 10 512 40 44 20 Svetlichny (3, 2, 2) 56 14 2944 132 1204 1087

Due to the large number of inequalities that we have found, we only explicitly write a few of them here. Complete lists of all the facet inequalities that we generated are posted on the website [20]. Our results are summarized in table 1. Note that we list here only inequalities that belong to different equivalence classes, where two inequalities are considered equivalent if they are related by a relabelling of parties, measurement settings or measurement outputs, or if they correspond to two different liftings of the same lower-dimensional inequality [15]. In appendix A, we introduce a parametrization of the correlation space that naturally induces several invariants for each equivalence class. These invariants are easily computed and are useful to determine quickly whether two inequalities are equivalent (two equivalent inequalities have equal invariants).

3.1. (2, 2, 4)

We ﬁrst consider bipartite experiments with two measurement settings per sites and four possible outcomes. Note that the case (2, 2, 3) was completely solved by Kaszlikowski et al [21] and Collins et al [13], who showed that all facets of the (2, 2, 3) polytope either correspond to the positivity of probabilities, the CHSH inequality or the CGLMP inequality [13]. The CGLMP inequality was introduced for any number of outcomes k 3in[13], and Gill raised the question (see footnote 3) whether all non-trivial facet inequalities of (2, 2,k) are of this form. Using our method, we found that the Bell polytope corresponding to (2, 2, 4) contains 12 inequivalent symmetric Bell inequalities. Among them, eight involve the four possible outcomes in a nontrivial way, i.e. they correspond to genuine four-outcome inequalities that cannot be seen as liftings of inequalities with lower numbers of outcomes; these inequalities are listed in appendix B. The list of these eight inequalities contains the CGLMP inequality, but surprisingly, it also contains seven inequalities that are inequivalent to it, thus answering in the negative Gill’s question (see footnote 3).

3.2. (2, 4, 2)

We now consider a bipartite scenario involving four settings with binary outcomes. The simpler case (2, 3, 2) was solved in [9, 11, 22] and contains a single new inequality besides the positivity constraints and the CHSH inequality.

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With four settings, we could use our method to ﬁnd all of the 90 inequivalent facets of the symmetrized polytope. Among these, 55 turn out to be facets of the (2, 4, 2) full local polytope; there are thus in total 55 symmetric inequalities for this scenario. Most of them were already known (see [7, 23, 24]), but we could not ﬁnd the two following ones in the literature, given here in the notation of [9]: − − − − − − − 1 2 2 2 0 2 2 2 −1−3322 0 −32−21 51 = 52 = S(2,4,2) −2 32−1 −1 0,S(2,4,2) −2 2022 0. (19) −2 2 −1 −13 −2−22 4−1 −2 2 −13 0 −2 12−11

3.3.(4,2,2)and(5,2,2) Since our method takes advantage of the symmetry between parties, we expect that it will be particularly useful for multipartite Bell scenarios. Indeed for the Bell scenario (n,2,2), corresponding to n parties with binary settings and outcomes, the full local polytope has 4n vertices and is embedded in a space of dimension 3n − 1. The symmetrized polytope, on the 1 other hand, has at most 6 (n+1)(n+2)(n+3) vertices and is embedded in a space of dimension = 1 ds 2 n(n +3). These quantities are polynomial in n and represent an exponential advantage with respect to the general, non-symmetric situation. Note that it therefore follows that it is possible using linear programming to decide in polynomial time in n if a given symmetric correlation vector p is local. The case (3, 2, 2) was already completely solved in [25]. For (4, 2, 2) we found a total of 627 inequivalent symmetric inequalities, of which 392 are facet-deﬁning. These facet inequalities correspond to the positivity conditions and to 391 genuinely four-partite inequalities. Amongst them, the following one is quite interesting, as it can be violated by a four-partite W-state with measurements lying in the x–y plane (contrary to the three-partite case where no inequality is known that can be violated by a W-state with measurements lying in the x–y plane):

IW =−p(a1b1) + p(a1b1c1) + p(a1b1c2) − p(a2b2c2) − p(a1b1c1d1) − − p(a1b1c1d2) p(a1b1c2d2) + p(a1b2c2d2) + p(a2b2c2d2) +sym 0. (20) The notation ‘sym’ stands for the symmetric terms that are missing in (20), such as p(a1c1), p(a1b2c1), etc. If we consider the W state |0001 + |0010 + |0100 + |1000 and measurements in the x–y plane at an angle φ with respect to the x axis, and set 1 1 φA = φC = 0 φB = φD = arccos − 2 arcsin 1 1 1 1 4 4 (21) = = 1 = =− 1 φA2 φC2 arccos 4 φB2 φD2 2 arcsin 4 , we find a value IW = 1/16 > 0. For (5, 2, 2), we found 238 464 inequivalent symmetric facets. Note that all of these inequalities, except the positivity of the probabilities, are truly five-partite ones (i.e. they do not correspond to lifting of inequalities involving less parties). Among these inequalities, nine of them involve only full (five-partite) correlators and were already given in [10].

3.4. Correlation inequalities for (3, 3, 2) We considered also a tripartite scenario with three binary measurements per party. Since in this case even the symmetrized polytope is quite time-consuming to solve, we made a further

9 J. Phys. A: Math. Theor. 43 (2010) 385303 J-D Bancal et al restriction by considering only ‘full-correlator’ inequalities, which can be written using only termsoftheform Ax By Cz =p(a + b + c = 0|x,y,z) − p(a + b + c = 1|x,y,z).This corresponds to performing a projection of the polytope on the subspace deﬁned by

Ai = Bi = Ci = Ai Bj = Ai Cj = Bi Cj =0 for all i, j = 1, 2, 3. (22) We obtained 40 inequalities in this way, 18 of which are facets of the full original polytope that truly involve three inputs per party. Using the method presented in section 2.3, we found 13 supplementary facet inequalities, all of which involve again full-correlators only.

3.5. Svetlichny inequalities for (3, 2, 2) To illustrate that our method is not restricted to the Bell-local polytope, but can address any correlation polytope, we consider the Svetlichny polytope for three parties [14], which characterizes true tripartite nonlocality [26–28]. In a Svetlichny model, two of the three subsystems are allowed to communicate once the measurement settings have been chosen. There are thus three types of Svetlichny vertices vAB/C , vAC/B , vBC/A, depending on which pairs of parties are allowed to communicate. A vertex of the form, e.g. vAB/C, corresponding to a deterministic strategy where outcomes α(x,y) and β(x,y) are jointly determined for parties 1 and 2, and an outcome γ(z)is assigned to party 3. This deﬁnes a joint distribution of the form 1ifa = α(x,y), b = β(x,y), c = γ(z) p(a, b, c|x,y,z) = (23) 0 otherwise. Such probability points do not satisfy the no-signalling conditions. For binary settings and outcomes, the Svetlichny polytope thus lives in a vector space of dimension d = 56. The subspace which is symmetric under the exchange of the three parties, however, has only dimension ds = 14. This great reduction in the space dimension, together with a reduction in the number of extremal points (see table 1), allowed us to ﬁnd all symmetric Svetlichny inequalities. After projection on the no-signalling space9 a total of 1087 facet symmetric Svetlichny inequalities were found. Interestingly, there are only two symmetric Svetlichny inequalities that involve only full- triparite correlation terms: the original Svetlichny inequality [14] and the following one:

ICorr =− A1B1C1 + A1B1C2 + A1B2C1 −3 A1B2C2 − − − + A2B1C1 3 A2B1C2 3 A2B2C1 3 A2B2C2 10. (24) This last inequality can be violated by quantum states, for instance by GHZ states having a visibility larger than 95.68%. The following inequality is also interesting: =− − − IGHZ 3P(a2) + P(a1b2) P(a1b1c1) P(a1b2c2) +7P(a2b2c2) +sym 0, (25) where ‘sym’ stands for the missing symmetric terms. It can be shown [29] that it is violated by every GHZ-like state of the form |GHZ =cos θ|000 +sinθ|111 . (26)

9 Indeed, we are only interested in whether these inequalities are violated by quantum correlations, which satisfy the no-signalling conditions.

10 J. Phys. A: Math. Theor. 43 (2010) 385303 J-D Bancal et al

4. Outlook

Motivated by the number of interesting Bell inequalities that are invariant under permutations of the parties, we introduced a method to list all symmetric inequalities. This method works even in cases where solving the full correlation polytope is computationally intractable. Our method can also be used as a starting point to generate more general, non-symmetric inequalities using algorithms such as the one described in section 2.3 or in [23]. Our method allowed us to find a number of new Bell inequalities. But a new problem is now at sight: so many different inequalities are generated, even for simple situations, that it is difficult to find which ones are the most interesting. Evidence of this problem was already put forward in [10] and in [24] where it was shown that the number of inequivalent Bell inequalities increases very quickly with the number of parties or measurement settings. New insights are thus necessary in order to classify these inequalities and understand which ones are the most relevant. For simple Bell scenarios, such as (2, 2, 2), (2, 2, 3) or (2, 3, 2) all facet inequalities happen to be symmetric inequalities. What is the proportion of symmetric inequalities in more complicated scenarios? Are there other useful symmetries or properties that can be exploited to generate more inequalities?

Acknowledgments

We thank K F Pal´ and T Vertesi´ for useful remarks. This work was supported by the Swiss NCCR Quantum Photonics, the European ERC-AG QORE and the Brussels-Capital region through a BB2B grant.

Appendix A. Correlators and inequality invariants

When dealing with Bell inequalities, it is useful to check quickly if two inequalities are equivalent under relabelling of parties, measurement settings or outcomes. We were confronted with this problem when classifying the inequalities that we derived here. In this appendix, we introduce a parametrization of the correlation space which naturally induces several invariants for each equivalence class. Two inequalities that are equivalent have equal invariants. Let us consider an (n,m,k)Bell scenario and let I ⊆{1,...,n} be a subset of the n parties. | | | = Deﬁne p(rI s) as the probability that these I parties obtain results rI ri1 ,...,ri|I| given that measurements s = s1,...,sn have been made on all parties. Note that with this notation, | = | = the no-signalling condition is expressed as p(rI s) p(rI sI ), where sI si1 ,...,si|I| . Deﬁne now single-party ‘correlators’ E(ri |s) by

E(ri |s) = kp(ri |s) − 1, (A.1) and deﬁne by induction multipartite correlators E(rI |s) through | = | ∗ | E(rI1 ,rI2 s) E(rI1 s) E(rI2 s), (A.2) where the ∗ operation is just the usual multiplication, except when acting on two probabilities, | ∗ | = | in which case it satisﬁes p(rI1 s) p(rI2 s) p(rI1 ,rI2 s). The no-signalling condition is then expressed as

E(rI |s) = E(rI |sI ). (A.3)

11 J. Phys. A: Math. Theor. 43 (2010) 385303 J-D Bancal et al

r1=1

The correlators E(rI |s) are in one-to-one correspondence with the probabilities p(rI |s) and thus represent an alternative parametrization of the correlation space. Note that in the case of binary outcomes (k = 2), E(rI |sI ) coincides with the usual definition of a correlation function. The definitions (A.1) and (A.2) thus represent a possible generalization of correlation function to more outcomes. With the notation that we just introduced, a generic Bell inequality in the no-signalling space takes the form | c(rI ,sI )E(rI sI ) c(0), (A.5) I,sI ,rI where c(rI ,sI ) are the coefficients of the inequality. A property of the correlators E(rI |sI ) is that the white noise yields E(rI |sI ) = 0. The resistance to noise of an inequality written in the form (A.5) is thus directly given by the ratio of the local bound to the violation. A relabelling of parties, measurement settings or measurement outcomes simply amounts to rearrange the order of the coefficients of the inequality. Note, however, that because of the normalization conditions (A.4), the basis of the correlation space that we chose is overcomplete. The coefficients c(r ,s ) are thus not uniquely defined: adding E(r |s ) = I I ri I I 0forsomei ∈ I to the inequality (A.5) does not change the inequality itself, but does change its coefficients. To compare two inequalities, we must therefore ensure first that they are written in some standard way. The freedom that we have in adding terms of the form λ(i, r \ ,s ) E(r |s ) = 0to I i I ri I I (A.5) corresponds to define new coefficients for the inequality in the following manner:

c (rI ,sI ) = c(rI ,sI ) + λ(i, rI\i ,sI ). (A.6) i∈I

We show now that requiring the inequality coefﬁcients c (rI ,sI ) to satisfy the relation k

c (rI ,sI ) = 0 for all i ∈ I (A.7)

r1=1 allows us to deﬁne them uniquely.

Proposition. There exist values of λ(i, rI\i ,sI ) such that the newly deﬁned coefﬁcients c (rI ,sI ) satisfy relation (A.7). Moreover for all such λ’s, the c (rI ,sI ) are the same: they are thus unique.

Proof. Since equations (A.6) and (A.7) apply independently on every subset I of the parties, and on every inputs sI, we omit these indices, writing for instance c(rI ) instead of c(rI ,sI ) to lighten the notation. Moreover, all sums on the outputs ai go from 1 to k and all sums on the parties i run on I, so we also omit these bounds in the proof.

12 J. Phys. A: Math. Theor. 43 (2010) 385303 J-D Bancal et al

The existence of the λ’s can be shown by directly checking that the following formula is of the form (A.6) and satisﬁes (A.7). Let

1 1 c (r ) = 11 − ◦ ...◦ 11 − c(r ), (A.8) I k k I r1 rn = where we used the notation f + a f + b f [11+ a + b]ffor any function f and the ◦ ◦ = ◦ = ◦ = ◦ = composition satisﬁes 11 11 11, 11 a a 11 a, a b a,b and distributes over addition. To get this expression from (A.6) one possible choice of lambdas is 1 λ(1,r \ ) =− c(r ) I 1 k I r 1 2 1 1 (A.9) λ(2,r \ ) =− c(r ) + c(r ) I 2 k I k I r2 r1,r2 ... and equation (A.7) is satisﬁed:

1 1 c (r ) = ◦ 11 − ◦ ...◦ 11 − c(r ) I k k I ri ri r1 rn = ...◦ − ◦ ... c(rI ) = 0. (A.10)

ri ri Now to show the unicity of the c coefﬁcients, we note that equation (A.7) is a non- homogeneous linear system of equations in the λ variables, which we can write as λ(i, rI\i ) =− c(rI ). (A.11)

rj i rj

Every solution of this system can thus be written as λ = λp + λv where λp is a particular solution of the equation (as given by equation (A.9) for instance) and λv is a solution of the homogeneous system, where the right-hand side of equation (A.11) is replaced by zero. Thus every c (rI ) that satisﬁes equation (A.7) can be written as

c (rI ) = c(rI ) + λp(i, rI\i ) + λv(i, rI\i ). (A.12) i i

Now we show that the last term of equation (A.12) is zero for every solution λv of the homogeneous counterpart of system (A.11), which implies that the coefﬁcients c are uniquely deﬁned. For this, consider the following expression:

1 1 Z = 11 − ◦ ...◦ 11 − λ (i, r \ ). (A.13) k k v I i r1 rn It is clearly zero, since it contains the term

1 1 11 − λ (i, r \ ) = λ (i, r \ ) − kλ (i, r \ ) = 0. (A.14) k v I i v I i k v I i ri

13 J. Phys. A: Math. Theor. 43 (2010) 385303 J-D Bancal et al

On the other hand, we have that

1 1 Z = 11 − ◦ ...◦ 11 − λ (i, r \ ) (A.15) k k v I i i r1 rn i

1 1 = 11 − ◦ ...◦ 11 − λ (i, r \ ) k k v I i r1 rn−1 i 1 1 1 − 11 − ◦ ...◦ 11 − λ (i, r \ ) (A.16) k k k v I i r1 rn−1 rn i

1 1 = 11 − ◦ ...◦ 11 − λ (i, r \ ), (A.17) k k v I i r1 rn−1 i where the second term in (A.16) vanishes by deﬁnition of λv(i, rI\i ). Repeating iteratively the above step, we ﬁnd eventually that Z = λv(i, rI\i ). (A.18) i i

This, combined with the fact that Z = 0, implies the desired result.

We showed that the coefficients of an inequality can be defined in a standard and unique way by requiring them to satisfy the constraints (A.7). Now, since a relabelling of parties, measurement settings or outcomes can only rearrange the coefficients c (rI ,sI ) without changing their value, the ordered lists of coefficients c (rI ,sI ) for |I|=0, 1,...,n provide n + 1 invariants for each equivalence class. For instance the local bound c(0) is an (easily checkable) invariant. In general requiring that two inequalities have their ordered list of coefficients identical does not guarantee that they are equivalent, but in the symmetric (4, 2, 2) scenario for instance, all the 391 classes of inequalities that we generated had different lists.

Appendix B. List of symmetric inequalities with two inputs and four outcomes

The following inequalities are given in the notation of [9]:

B.1. Case with 3 outcomes for the ﬁrst setting and 4 outcomes for the second one − − 1 1 000 −1 01011 2 = −1 10101 S(2,2,(3,4)) 0(B.1) 0 010 −1 −1 0 10−10−1 0 11−1 −1 −1

14 J. Phys. A: Math. Theor. 43 (2010) 385303 J-D Bancal et al

B.2. Case with 4 outcomes for both settings − − − − − − 1 1 1 000 1 10 10 0 −1 001 111 −1 010 101 −1 011 110 −1 100 110 1 = 2 = S(2,2,4) −1 111 100 0,S(2,2,4) 0 00−1 100 0 0 111−1 −1 −1 −1 111 100 0 110−1 −10 0 010 00−1 0 100−10 0 0 100 0 −10 − − − − − − − − 1 10 1 10 1 10 1 10 −1 001 110 −1−11−1 121 −1 010 101 −1 110 101 3 = 4 = S(2,2,4) 0 100−1 −10 0,S(2,2,4) 0 −10−1 110 0 −1 11−1 120 −1 111 100 −1 10−1 220 −1 201 01−1 0 010 00−1 0 110 0 −1 −1 − − − − − − − 1 1 1 000 1 1 1 10 0 −1 011 001 −1 011 010 −1 101 010 −1 110 101 5 = 6 = S(2,2,4) −1 110 100 0,S(2,2,4) −1 10−1 201 0 0 001 0 −1 −1 −1 012 1 −10 0 010−10−1 0 100−10−1 0 100−1 −10 0 011 0 −1 −1 − − − − − − 1 1 1 000 2 1 1 000 −1 011 001 −2 201 012 −1 100 110 −1 011 100 7 = 8 = S(2,2,4) −1 101 100 0,S(2,2,4) −1 110 010 0. 0 011−10−1 0 010−1 −10 0 010 00−1 0 101−10−1 0 100−1 −10 0 200 0 −1 −2 (B.2)

References

[1] Pitowsky I 1989 Quantum Probability—Quantum Logic (Berlin: Springer) [2] Bell J S 1987 Speakable and Unspeakable in Quantum Mechanics (Cambridge: Cambridge University Press) [3] Aspect A 1999 Nature 398 189 [4] Pironio S, Acin A, Brunner N, Gisin N, Massar S and Scarani V 2009 New J. Phys. 11 045021 [5] Masanes L 2009 Phys. Rev. Lett. 102 140501 [6] Vertesi´ T, Pironio S and Brunner N 2010 Phys. Rev. Lett. 104 060401 [7] Brunner N and Gisin N 2008 Phys. Lett. A 372 3162–7 [8] Vertesi´ T 2008 Phys. Rev. A 78 032112 [9] Collins D and Gisin N 2004 J. Phys. A: Math. Gen. 37 1775–87 [10] Werner R F and Wolf M M 2001 Phys. Rev. A 64 032112 [11] Clauser J F, Horne M A, Shimony A and Holt R A 1969 Phys. Rev. Lett. 23 880 [12] Mermin N D 1990 Phys. Rev. Lett. 65 1838 [13] Collins D, Gisin N, Linden N, Massar S and Popescu S 2002 Phys. Rev. Lett. 88 040404 [14] Svetlichny G 1987 Phys. Rev. D 35 3066 [15] Pironio S 2005 J. Math. Phys. 46 062112 [16] Navascues M, Pironio S and Acin A 2007 Phys. Rev. Lett. 98 010401

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[17] Navacues M, Pironio S and Acin A 2008 New J. Phys. 10 073013 [18] Branciard C, Pironio S and Gisin N 2010 Phys. Rev. Lett. 104 170401 [19] http://cgm.cs.mcgill.ca/∼avis/C/lrs.html and http://www.zib.de/Optimization/Software/Porta/ [20] http://www.gapoptic.unige.ch/Publications/bellinequalities [21] Kaszlikowski D, Kwek L C, Chen J-L, Zukowski M and Oh C H 2002 Phys. Rev. A 65 032118 [22] Froissart M 1981 Nuovo Cimento B 64 241 [23] Pal´ K and Vertesi´ T 2009 Phys. Rev. A 79 022120 [24] Avis D, Imai H, Ito T and Sasaki Y 2005 J. Phys. A: Math. Gen. 38 10971–87 [25] Sliwa´ Cezary 2003 Phys. Lett. A 317 165–8 [26] Seevinck M and Svetlichny G 2002 Phys. Rev. Lett. 89 060401 [27] Collins D, Gisin N, Popescu S, Roberts D and Scarani V 2002 Phys. Rev. Lett. 88 170405 [28] Bancal J-D, Branciard C, Gisin N and Pironio S 2009 Phys. Rev. Lett. 103 090503 [29] Bancal J-D et al in preparation

16 Paper F

Detecting Genuine Multipartite Quantum Nonlocality: A Simple Approach and Generalization to Arbitrary Dimensions

J.-D. Bancal, N. Brunner, N. Gisin and Y.-C. Liang

Physical Review Letters 106, 020405 (2011)

133

week ending PRL 106, 020405 (2011) PHYSICAL REVIEW LETTERS 14 JANUARY 2011

Detecting Genuine Multipartite Quantum Nonlocality: A Simple Approach and Generalization to Arbitrary Dimensions

Jean-Daniel Bancal,1 Nicolas Brunner,2 Nicolas Gisin,1 and Yeong-Cherng Liang1 1Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland 2H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom (Received 4 November 2010; published 11 January 2011) The structure of Bell-type inequalities detecting genuine multipartite nonlocality, and hence detecting genuine multipartite entanglement, is investigated. We ﬁrst present a simple and intuitive approach to Svetlichny’s original inequality, which provides a clear understanding of its structure and of its violation in quantum mechanics. Based on this approach, we then derive a family of Bell-type inequalities for detecting genuine multipartite nonlocality in scenarios involving an arbitrary number of parties and systems of arbitrary dimension. Finally, we discuss the tightness and quantum mechanical violations of these inequalities.

DOI: 10.1103/PhysRevLett.106.020405 PACS numbers: 03.65.Ud, 03.67.Mn

Nonlocality is a fundamental feature of quantum me- plying the presence of genuine tripartite entanglement. chanics. On top of being a fascinating phenomenon— Svetlichny’s original inequality was later generalized to defying intuition about space and time in a dramatic the case of an arbitrary number of parties [6], inspiring way—nonlocality is also a key resource for information further studies on multipartite nonlocality in [7]. More processing [1], and has thus been the subject of intense reﬁned concepts and measures of multipartite nonlocality research in recent years. have also been investigated [8]. It is fair to say that, while our comprehension of bipartite In this Letter, we start by providing a simple and intuitive nonlocality has reached a reasonable level, multipartite approach to Svetlichny’s original inequality. Our approach, nonlocality is still poorly understood. This is partly be- which naturally extends to the case of an arbitrary number cause the phenomenon becomes much more complex when of parties, makes it clear why these inequalities detect moving from the bipartite case to the multipartite case. genuine multipartite nonlocality. It also provides an intui- Indeed, this is somehow similar to the case of entanglement tive understanding of their violations in quantum mechan- theory, where the structure of multipartite entanglement is ics, via the concept of steering [9]. Based on this approach, much richer than that of bipartite entanglement [2]. we derive Bell inequalities detecting genuine multipartite A natural issue to investigate is genuine multipartite nonlocality for an arbitrary number of systems of arbitrary nonlocality [3], which represents the strongest form of dimension. Finally, we show that the simplest of our in- multipartite nonlocality. More precisely, when considering equalities deﬁnes facets of the relevant polytopes of corre- a system composed of m spatially separated parts, it is lations and study their quantum mechanical violations. natural to ask whether all m parts of the system are non- Simple approach to Svetlichny’s inequality.—To make locally correlated, or whether it is only a subset of k

0031-9007=11=106(2)=020405(4) 020405-1 Ó 2011 The American Physical Society week ending PRL 106, 020405 (2011) PHYSICAL REVIEW LETTERS 14 JANUARY 2011

However, this notion of nonlocality does not capture the two simple arguments show immediately that S3 4 holds idea of genuine multipartite nonlocality. For instance, in for any bipartition model of the form (2). the case where Alice and Bob are nonlocally correlated, but Argument 1. Consider the bipartition AB=C. Although uncorrelated from Charlie, it would still follow that P AB are together, and could thus produce any (bipartite) cannot be written in the form (1), although the system nonlocal probability distribution, they do not know which features no genuine tripartite nonlocality. CHSH game they are supposed to play, as C is separated. To detect genuine multipartite nonlocality, one needs to Thus they are effectively playing the average game ensure that the probability distributions cannot be repro- CHSH CHSH0 (the signs specifying which game is duced by local means even if (any) two of the three parties played depend on the outputs of C). It can be checked would come together and act jointly—and consequently that the algebraic maximum of any of these average games could reproduce any bipartite nonlocal probability distri- is 4 [12]. Hence, S3 4 for the bipartition AB=C. bution. Formally, this corresponds to ensuring that P can- Argument 2. For the bipartition A=BC, B knows which not be written in the form version of the CHSH game he is supposed to play with A, X3 Z since he is together with C. However, CHSH being a non- PBða1a2a3Þ¼ pk dijðÞPijðaiajjÞPkðakjÞ; (2) local game, AB cannot achieve better than the local bound k¼1 (i.e., CHSH ¼ 2 or CHSH0 ¼ 2), as they are separated S S 4 where fi; jg fkg¼f1; 2; 3g, and the sum takes care of [13]. Thus it follows that 3 . Note that the same different bipartitions of the parties. In the following we reasoning holds for the bipartition B=AC. shall refer to such models as ‘‘bipartition models.’’ A prob- From these arguments, it follows that inequality (4) ability distribution P which cannot be expressed in the holds for any correlation of the form (2). Note that since above form features genuine tripartite nonlocality; to be the polynomial S3 is invariant under permutation of parties, reproduced classically, all three parties must come to- the proof already follows by applying either one of the two gether. Clearly, standard Bell inequalities can, in general, arguments given above. However, using both arguments not be used to test for genuine multipartite nonlocality, and above allows one in principle to deal with polynomials one needs better adapted tools. which are not invariant under permutation of parties. From now on, we shall focus on the case where each Furthermore, expressing Svetlichny’s inequality under party performs one out of two possible measurements. We the form (4) allows one to understand its optimal quantum 0 ABC denote the measurements of party j by Xj and Xj, and their mechanical violation. Suppose share a three qubit a a0 a ;a0 c 000 results by j and j. Considering the case where j j 2 Greenberger-Horne-Zeilingerpffiffiffi (GHZ) state j i¼ðj iþ f1; 1g, Svetlichny [3] proved that the inequality j111iÞ= 2. From (4) it is clear that C should choose his measurement settings in order to prepare for AB the state S a a a0 a a0 a a0 a a a0 a0 a0 a0 a0 a 3 ¼ 1 2 3 þ 1 2 3 þ 1 2 3 1 2 3 þ 1 2 3 that is optimal for the corresponding CHSH game, i.e., a 0 0 0 0 þ a1a2a3 þ a1a2a3 a1a2a3 4 (3) maximally entangled state of two qubits. Let Alice and Bob choose measurements which are optimal for CHSH—pffiffiffi holds for any probability distribution of the form (2). Thus X ¼ X0 ¼ A X ¼ð Þ= 2 1 x and p1 ffiffiffi y for ; 2 x y and a violation of inequality (3) implies the presence of genu- 0 X2 ¼ðx þ yÞ 2 for B. It is then straightforward to ine tripartite nonlocality, and hence of genuine tripartite check that the measurements of C must be X3 ¼ x and entanglement (regardless of the Hilbert space dimension 0 X ¼y. For instance, when C measures x and gets [10]). The above polynomial should be understood as a 3 0 outcome 1, he prepares the state ji¼ðj00i sum of expectation values; for instance, a1a2a3 means pffiffiffi 0 j11iÞ= 2 for AB which is optimal for the CHSH game. Eða1a2a3Þ, the expectation value of the product of the 0 Note that, given the measurement of A and B, the state outcomes when the measurements are X1, X2, and X . pffiffiffi 3 CHSH 2 2 C We now start by rewriting inequality (3)as j i gives ¼ ; thus the output of ensures that the overall sign is positive. Similarly, when C mea- 0 0 S3 ¼ CHSHa þ CHSH a3 4; (4) 1 AB 3 sures y and gets outcome ffiffiffi , he prepares for the ~ p 0 0 0 0 state j i¼ðj00iij11iÞ= 2. Given the measurements where CHSH ¼ a1a2 þ a1a2 þ a1a2 a1a2 is the usual pffiffiffi A B ~ CHSH0 2 2 Clauser-Horne-Shimony-Holt polynomial [11] and of and , the state j i givespffiffiffi ¼ . Thus 0 0 0 0 0 CHSH ¼ a1a2 þ a1a2 þ a1a2 a1a2 is one of its ABC achieve the score of S3 ¼ 4 2, which is the optimal equivalent forms, obtained by inverting the primed and quantum violation as can be checked using the techniques nonprimed measurements; equivalently one could apply of Ref. [14]. Moreover, the idea of steering also allows one 0 0 the mapping a2 ! a2 and a2 !a2. to understand the resistance to (white) noise of this quan- The main point of our observation is now the following: tum violation. Basically, Svetlichny’s inequality should be It is the input setting of Charlie that defines which version violated if and only if the state of AB (prepared by a of the CHSH game Alice and Bob are playing. When C gets measurement of C) violates CHSH. Thus we expect the 0 the input X3, then AB play the standard CHSH game; when resistance to noise of the GHZ state for Svetlichny’s in- 0 C gets the input X3, AB play CHSH . From this observation, equality to coincide with the resistance to noise of a 020405-2 week ending PRL 106, 020405 (2011) PHYSICAL REVIEW LETTERS 14 JANUARY 2011

S0 S maximally entangled two qubit state for CHSH.p Indeed,ffiffiffi in where ðm1Þ;d is obtained from ðm1Þ;d using the rule (7). both cases we get the critical visibility w ¼ 1= 2. For instance, for the case of m ¼ 4 parties we obtain The form of inequality (4) also suggests a straightfor- S a a a a 1 a a a a0 1 ward generalization to an arbitrary number of parties m: 4;d ¼½ 1 þ 2 þ 3 þ 4 þ þ½ 1 þ 2 þ 3 þ 4 þ 0 0 0 0 0 S S a0 S0 a 2m1; þ½a1 þ a2 þ a3 þ a4þ½a1 þ a2 þ a3 þ a4 m ¼ m1 m þ m1 m (5) 0 0 0 0 S0 S þ½a1 þ a2 þ a3 þ a4 1þ4ðd 1Þ; (10) where m1 is obtained from m1 by interverting primed and nonprimed settings. From argument 2 above it is clear where terms obtained by permuting the players are S 2m2 that if inequality m1 holds for any bipartition of omitted. m 1 the parties, then inequality (5) holds for any bipar- Proof of inequality (9).—The proof that (9) holds for m tition where party is not alone. The fact that (5) holds for any bipartition of the m players is again based on argu- this partition as well follows from the fact that the poly- ment 2 and goes by induction. Let us suppose that nomial Sm is symmetric under permutation of the parties m3 (i) S m 1 ;d 2 ðd 1Þ holds for any bipartition of (see below). Inequalities (5) are the generalizations of ð Þ the m 1 parties and that (ii) Sðm1Þ;d is invariant under Svetlichny’s inequality presented in Ref. [6]. m 1 any permutation of parties and contains all possible 2 Detecting genuine multipartite nonlocality in systems of terms. Then, it follows from (i) that Sm;d holds for all arbitrary dimension.—The form (4) suggests further gen- bipartitions, except for the one in which party m is alone. eralizations. We now present a family of inequalities detect- To deal with this last bipartition, we need to show that ing genuine multipartite nonlocality for scenarios involving the polynomial Sm;d is invariant under any permutation of an arbitrary number of parties and systems of arbitrary parties. This is done in two steps. First, note that by dimension. The main idea here consists of replacing the m construction Sm;d contains all 2 possible terms. So it CHSH expression in (4) with the Collins-Gisin-Linden- Massar-Popescu (CGLMP) expression [15], which gives remains to be shown that all terms featuring a given bipartite Bell inequalities for systems of arbitrary dimen- number of unprimed inputs appear with the same type of sion. Here we use the form of CGLMP of Ref. [16]; that is, brackets. To see this, notice that the brackets associated with terms with an increasing number of unprimed mea- 0 0 S2;d ¼½a1 þ a2þ½a1 þ a2 þ½a1 þ a2 surements follow a regular pattern; terms featuring only primed measurements have ½... 1; terms with one un- þ½a0 þ a0 1d 1; 1 2 (6) primed measurement have ½...; terms with two unprimed P X d1 jP X j mod d X X measurements have ½..., etc. In order to determine the where ½ ¼ j¼0 ð ¼ Þ and ½ ¼½ . Note that for convenience the measurement outcomes bracket of the following terms, one simply iterates the rule a 0; 1; ...;d 1 (7). So, the bracket of terms featuring k unprimed mea- are now denoted j 2f g. Note also that for ... 1 d 2 surements is obtained by starting from the bracket ½ ¼ , the CGLMP inequality reduces to CHSH. k S and iterating times the rule (7). Now, note that terms in To construct 3;d we use the idea of Eq. (4). First we S k 0 m;d featuring a fixed number of unprimed measurements define S , an equivalent form of S2;d [17] obtained using 2;d can come from two possible terms: first, from terms in S k ½ ! ½ þ 1 and ½ !½: (7) ðm1Þ;d featuring unprimed measurements; second from 0 terms in S m 1 ;d featuring k 1 unprimed terms. From the S ¼ S a0 þ S0 a ð Þ Next we construct 3;d 2;d 3 2;d 3 and obtain pattern described above, it follows that both of these terms 0 0 S3;d ¼½a1 þa2 þa3 þ1 þ½a1 þa2 þa3þ½a1 þa2 þa3 appear within exactly the same type of bracket. Thus we have that Sm;d is symmetric under permutation of the þ½a0 þa þa þ½a þa0 þa0 þ½a0 þa þa0 1 2 3 1 2 3 1 2 3 parties, which completes the proof. j 0 0 0 0 0 þ½a1 þa2 þa3 þ½a1 þa2 þa3 12ðd1Þ; Note that the arguments presented above also allow us to construct Sm;d directly using rule (7) starting from the (8) bracket that contains only primed terms. Moreover, it can where the rule to include the third party works by simply be shown that Sm;2 is equivalent to the generalizations of 0 inserting its outcomes (a3 or a3) into the brackets. In the Svetlichny’s inequalities given in Ref. [6]. case d ¼ 2, this rule reduces to Eq. (4). Tightness.—Among Bell inequalities, those which define From the fact that S2;d is a Bell inequality and from facets of the polytope of local correlations are of particular argument 2, it follows that the inequality (8) holds for interest, since they form a minimal set of inequalities to the bipartitions A=BC and B=AC. Moreover, since the characterize local correlations [18]. These inequalities are polynomial S3:d is symmetric under permutation of the referred to as ‘‘tight’’ Bell inequalities. In this Letter, we parties, the inequality (8) holds for any bipartition. focus on Bell-type inequalities detecting genuine multi- This construction can be generalized to an arbitrary partite nonlocality. These inequalities are thus satisfied by number of parties m. Specifically, we take any bipartition model of the form (2). Indeed the set of bipartition correlations also forms a polytope—which is S ¼ S a0 þ S0 a 2m2ðd 1Þ; m;d ðm1Þ;d m ðm1Þ;d m (9) strictly larger than the local polytope [19]. Here we have 020405-3 week ending PRL 106, 020405 (2011) PHYSICAL REVIEW LETTERS 14 JANUARY 2011 checked that inequalities (8) and (10) are facets of the other possible generalizations. For instance it would be respective polytope for d ¼ 2; 3. We conjecture that all interesting to investigate the case where the parties can inequalities (9) correspond to facets. perform more than two measurements. Quantum violations.—Finally we discuss the quantum We thank D. Cavalcanti, S. Popescu, S. Pironio, O. violation of our inequalities. In the case of Svetlichny’s Gu¨hne, and P. Skrzypczyk for insightful discussions. We original inequality, it turned out that writing the inequality acknowledge financial support from the UK EPSRC, the in the form (4) naturally leads us to consider steering in ERC-AG QORE, and the Swiss NCCR Quantum Photonics. order to find the optimal quantum violation. Indeed, since Note added.—Recently, we became aware of the work of the structure of our inequalities (9) is based on (4), we Ref. [22], which presented an inequality sharing similar follow a similar approach here, which will lead us to the properties with our inequality Eq. (9). optimal quantum violations as well. First we recall that, in the bipartite case and for d ¼ 3, the maximal violation of the CGLMP inequality (6)is obtained by performing measurements on a partially en- [1] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991); A. Acıń et al., c 00 ibid. 98, 230501 (2007); H. Buhrman et al., Rev. Mod. tangled statep offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffi two qutrits given by j 2i¼ðj iþ pffiffiffiffiffiffi pffiffiffi Phys. 82, 665 (2010); S. Pironio et al., Nature (London) j11iþj22iÞ= 2 þ 2 ¼ð 11 3Þ=2 , where [20]. 464, 1021 (2010). The optimal measurements are so-called Fourier transform [2] W. Du¨r, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 measurements [15,21]; the basis is defined by the nonde- 1 P 2i (2000); R. Horodecki et al., Rev. Mod. Phys. 81, 865 u pffiffi 2 exp v u v generate eigenvectors j i¼ 3 v¼0 ½ 3 ð m þ Þj i (2009). 0 for party m, where 1 ¼ 0, 1 ¼1=2, and 2 ¼ 1=4, [3] G. Svetlichny, Phys. Rev. D 35, 3066 (1987). 0 [4] S. Campbell and M. Paternostro, Phys. Rev. A 82, 042324 2 ¼1=4. This gives S2;3 ¼ 1:0851, corresponding to a resistance to (white) noise of w ¼ 0:6861. (2010); D.-L. Deng, S.-J. Gu, and J.-L. Chen, Ann. Phys. (N.Y.) 325, 367 (2010). Now moving to the case of three parties, it appears natural [5] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 to choose the measurements of Alice and Bob to be the ones (2001). which are optimal for CGLMP (i.e., as above). Next we [6] D. Collins et al., Phys. Rev. Lett. 88, 170405 (2002);M. choose the tripartite state and Charlie’s measurements to be Seevinck and G. Svetlichny, ibid. 89, 060401 (2002). such that, by measuring his system, C prepares the desired [7] J. L. Cereceda, Phys.Rev.A66, 024102 (2002);P.Mitchell, state for A and B. For instance, we can take simply jc 3i¼ S. Popescu, and D. Roberts, ibid. 70, 060101 (2004);S. Ghose et al., Phys. Rev. Lett. 102, 250404 (2009);M.L. pffiffiffiffiffiffiffiffiffi1 ðj000iþj111iþj222iÞ and fix Charlie’s measure- 2þ2 Almeida et al., Phys. Rev. A 81, 052111 (2010). ments to be Fourier transform as well—we take 3 ¼ 1=2 [8] N. S. Jones, N. Linden, and S. Massar, Phys. Rev. A 71, 0 042329 (2005); J.-D. Bancal et al., Phys. Rev. Lett. 103, and 3 ¼ 0. With these parameters we obtain the violation 090503 (2009). S3;3 ¼ 2:1703, which we have checked to be the optimal qua- [9] E. Schro¨dinger, Math. Proc. Cambridge Philos. Soc. 31, ntum violation using the techniques of Ref. [14]. Note also c w 0:6861 555 (1935). that the resistance to noise of j 3i here is ¼ ,which [10] J.-D. Bancal et al. (to be published). corresponds exactly to that obtained for CGLMP with jc 2i. [11] J. F. Clauser et al., Phys. Rev. Lett. 23, 880 (1969). From the structure of our inequalities (9), we conjecture [12] This can be achieved by a local strategy between AB. that this idea of steering always provides the optimal [13] Charlie’s output could in principle depend on Bob’s input. However, for inequalities involving only full correlations, quantum violation, obtained from the state jc mi¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi such flip performed by C can also be performed by B. 0 m 1 m 2 m = 2 2 ðj i þ j i þj i Þ þ and Fourier transform [14] M. Navascue´s, S. Pironio, and A. Acıń, Phys. Rev. Lett. measurement. From this we expect the resistance to noise 98, 010401 (2007). to be independent of the number of parties m and given by [15] D. Collins et al., Phys. Rev. Lett. 88, 040404 (2002). w ¼ 0:6861. We could check numerically that this is in- [16] A. Acıń, R. Gill, and N. Gisin, Phys. Rev. Lett. 95, 210402 deed the case for S4;3. Also, we expect a similar behavior (2005). S0 S for higher dimensions d. [17] Instead of rule (7), m;d can also be obtained from m;d via a a0 a0 a 1 S0 Conclusion.—The main focus of this Letter is to provide the relabeling 1 ! 1 and 1 ! 1 þ . Thus m;d is an S an intuitive approach to Bell-type inequalities detecting equivalent form of m;d. genuine multipartite nonlocality. First, we provided a [18] I. Pitowski, Quantum Probability, Quantum Logic (Springer, Heidelberg, 1989). natural form for Svetlichny’s inequality, which allows [19] J.-D. Bancal, N. Gisin, and S. Pironio, J. Phys. A 43, one to better understand its structure as well as its quantum 385303 (2010). violation. Based on this understanding, we then derived a [20] A. Acıń et al., Phys. Rev. A 65, 052325 (2002). family of Bell-type inequalities detecting genuine multi- [21] D. Kaszlikowski et al., Phys. Rev. Lett. 85, 4418 partite nonlocality for an arbitrary number of systems of (2000). arbitrary dimensionality. Finally, our approach suggests [22] J. L. Chen et al., arXiv:1010.3762.

020405-4 Paper G

Extremal correlations of the tripartite no-signaling polytope

S. Pironio, J.-D. Bancal and V. Scarani

Journal of Physics A: Mathematical and Theoretical 44, 065303 (2011)

139

IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 44 (2011) 065303 (19pp) doi:10.1088/1751-8113/44/6/065303

Extremal correlations of the tripartite no-signaling polytope

Stefano Pironio1, Jean-Daniel Bancal2 and Valerio Scarani3

1 Laboratoire d’Information Quantique, Universite´ Libre de Bruxelles, 1050 Brussels, Belgium 2 Group of Applied Physics, University of Geneva, Switzerland 3 Centre for Quantum Technologies and Department of Physics, National University of Singapore, Singapore 117543

E-mail: [email protected]

Received 19 November 2010, in ﬁnal form 20 December 2010 Published 12 January 2011 Online at stacks.iop.org/JPhysA/44/065303

Abstract The no-signaling polytope associated with a Bell scenario with three parties, two inputs, and two outputs, is found to have 53 856 extremal points, belonging to 46 inequivalent classes. We provide a classification of these points according to various definitions of multipartite nonlocality and briefly discuss other issues such as the interconversion between extremal points seen as a resource and the relation of the extremal points to Bell-type inequalities.

PACS number: 03.65.Ud

1. Introduction

Quantum correlations, i.e. probability distributions characterizing the outcomes of measurements performed on entangled quantum states, belong to the set of no-signaling probability distributions. No-signaling captures one of the essential properties of quantum correlations: the impossibility of using them to send a message. Popescu and Rohrlich surmised that this property may define quantum correlations exactly; upon studying the question, however, they realized that it is not so and showed a probability distribution that satisfies no-signaling, but cannot be obtained from quantum physics. This mathematical object is known as a PR-box [1], although other authors had discussed it earlier [2, 3]. In recent years, various authors have studied no-signaling distributions, with their motivations ranging from sheer mathematical interest to the hope of describing something that may be discovered in nature. At any rate, the set of no-signaling probability distributions provides a thinking space [4], in which one can meaningfully ask why nature is not more nonlocal [5], what the physical principles are that underlie quantum physics [6–9], or how to exploit nonlocality for information processing [10, 11]. By treating each probability distribution as a point in a high-dimensional space, one obtains a geometric characterization of the set of no-signaling probability distributions: this set is a polytope, i.e. a convex set with finitely many extremal points. The first no-signaling

1751-8113/11/065303+19$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA 1 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al polytope to be characterized is associated with the simplest meaningful scenario: two parties, each with two inputs (the measurement settings) and two outputs (the measurement outcomes) [3, 4]. This polytope lives in an eight-dimensional space. It has 24 vertices, 16 of which describe local deterministic distributions, while the eight nonlocal points are all equivalent to the PR-box under suitable relabeling of the inputs and outputs. The facets of the local polytope also belong to two classes upon relabeling: 16 of them are positivity inequalities enforcing the constraint that probabilities must lie between 0 and 1, and eight are non-trivial facets, all equivalent to the Clauser–Horne–Shimony–Holt (CHSH) Bell-type inequality [12]. The elegance of the construction is completed when one notes that each of the eight nonlocal points lies above one of the eight non-trivial facets of the local polytope. In the past few years, other no-signaling polytopes for bipartite scenarios have been studied, namely two inputs and d outputs [4], and m inputs and two outputs [13, 14]. In these examples, it was possible to give a compact description of the geometry of the polytopes, notwithstanding their growing complexity. The structure of no-signaling theories in multipartite scenarios, in contrast, has been only partially addressed in some of the initial studies [4, 13]. Recent results motivate the need for a better understanding [15]. In this paper, we present the no-signaling polytope for three parties, two inputs and two outputs, and derive its extremal points. This is the simplest multipartite scenario; nevertheless, the complexity of the geometry of the polytope is far greater than in the bipartite case. We discuss some possibilities for classifying the extremal boxes, being aware though that many questions remain open. In section 2, we deﬁne the mathematical objects, provide the list of extremal tripartite boxes, and discuss a few simple examples. In section 3, we sketch several criteria for multipartite nonlocality and classify the extremal boxes according to these. In section 4, we brieﬂy mention some known results on simulating some boxes using other ones. In section 5, we study the violation of Bell’s inequalities by the extremal boxes.

2. Tripartite no-signaling boxes

2.1. Notation and deﬁnitions We are interested in the set of tripartite no-signaling boxes, where each party has two inputs and two outputs. Let x,y,z ∈{0, 1} denote the inputs of each party and a,b,c ∈{−1, 1}, the outputs. The boxes are characterized by the joint probabilities P(abc|xyz) of obtaining the triple of outputs (a,b,c) given the triple of inputs (x,y,z). These probabilities satisfy positivity P(abc|xyz) 0, for all a, b, c, x, y, z,(1) normalization P(abc|xyz) = 1, for all x,y,z,(2) a,b,c and no-signaling P(abc|xyz) = P(abc|xyz), for all a,b,x,y,z,z,(3) c c where the last condition also holds for cyclic permutations of the parties. These no-signaling conditions guarantee that signaling among any partitions of the parties is impossible, e.g., that box C cannot signal to box A or B, or to the combined system AB, or conversely that the combined system AB cannot signal to C. Due to the equality constraints (2) and (3), only 26 out of the 64 probabilities P(abc|xyz) are independent, i.e. the set of no-signaling boxes is contained in an afﬁne space of dimension

2 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al

26. It is convenient to write the no-signaling boxes in this 26-dimensional space using the following parametrization: | = 1 P(abc xyz) 8 [1 + a Ax + b By + c Cz + abAx By + acAx Cz + bcBy Cz + abcAx By Cz], (4) where Ax =P(a = 1|x) − P(a =−1|x) is the expectation value of the outcome a for the input x, Ax By =P(ab = 1|xy) − P(ab =−1|xy) is the expectation value of the product ab for the inputs x and y, and so on. Note that the single-party and two-party expectations are well defined and do not depend on the other parties inputs (e.g., Ax =Axyz) thanks to the no-signaling conditions. In total, there are 6 single-party expectations, 12 two-party expectations, and 8 three-party expectations, adding up to a total of 26 numbers that fully specify a probability point in the no-signaling set. It is sometimes useful to consider the ‘computer scientist’ notation where outputs take values in {0, 1}, instead of the ‘physicist’ notation where they take value in {−1, 1}.We will therefore also consider the alternative labeling a,ˆ b,ˆ cˆ ∈{0, 1} for the outputs, with aˆ bˆ cˆ a = (−1) , b = (−1) , and c = (−1) . With this labeling, it is convenient to parametrize ˆ = | a no-signaling point through the ‘sum modulo 2’ expectations Ax aˆ P(aˆ x)aˆ, ˆ ˆ = ˆ| ˆ ˆ ˆ ˆ = ˆ | ˆ Ax +By a,ˆ bˆ P(aˆb xy) (aˆ +b), and Ax +By +Cz a,ˆ b,ˆ cˆ P(aˆbcˆ xyz) (aˆ +b+c)ˆ . These expectations are in one-to-one correspondence with the ‘product’ expectations defined above through Ax =1 − 2Aˆ x , Ax By =1 − 2Aˆ x + Bˆ y , and Ax By Cz=1 − 2Aˆ x + Bˆ y + Cˆ z. As an illustration, the PR-box is defined in the physicist notation by Ax =0, By =0, and xy Ax By =(−1) , and in the computer scientist notation by Aˆ x =1/2, Bˆ y =1/2, and Aˆ x + Bˆ y =xy.

2.2. Extremal boxes Since constraints (1), (2), and (3) are linear, the set of no-signaling boxes is a polytope. Boxes of particular interest are the extremal ones, which correspond to the vertices of this polytope. They fully characterize the no-signaling polytope since any box can be decomposed as a convex combination of the extremal ones. Given a polytope described in terms of linear constraints, there exist algorithms that can enumerate all its vertices, although they are efficient only for low-dimensional problems. We determined the extreme boxes of the tripartite no-signaling polytope using both the algorithms PORTA [16] and cdd [17]. It turns out that there are 53 856 extremal points. These points can be classified by equivalence classes under relabeling of the parties, inputs, and outputs4. Once the extremal points are sorted according to these symmetries, they define 46 equivalence classes. We provide a representative for each class both in the physicist and computer scientist notation (tables 1 and 2). These lists are also available in electronic format [18].

2.3. More detailed presentation of some boxes Let us start by featuring some extremal boxes of particular interest or which have already appeared in the literature. • Deterministic boxes (boxes of class 1). Boxes in this category have deterministic outputs, for instance the representative provided in tables 1 and 2 satisﬁes

aˆx = 0, bˆy = 0, cˆz = 0. (5)

4 The relabeling of the inputs and outputs must be deﬁned by a local processing. For instance, aˆ → aˆ + x (sum modulo 2) is allowed, and aˆ → aˆ + y is not.

3 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al denotes e n e n 1536 1536 1 4 1 3 196 1 384 1 384 1 1536 1 1536 11 512 512 1 1536 1 192 00 1536 0 3072 1536 0 1536 − − − − − − − − − − − nputs, and outputs. 1 2 1 2 1 0 768 1 1 4 2 3 2 3 4 5 3 1 3 1 − − − 1 2 1 2 1 0 0 128 1 0 0 384 1 1 1 0 0 192 1 4 1 3 11 1 0 0 0 1536 0 3072 − − − − − − − 1 4 1 2 z 2 3 1 3 1 2 01 001 1 1 111 − − C y B x 1 2 1 2 1 2 1 3 11 1 1 11100384 10000768 1 11 1 10 0 1 1 1 1 2 1 3 A − − − − − − − − − − − − − 1 2 1 2 1 2 1 3 2 3 1 111 − − − − 1 2 1 3 1 2 1 2 3 3 0 0111 11 11 − − 2 5 1 2 1 3 1 3 1 3 1 2 1 2 3 3 1 2 001 111 111 001 111 111 − − − − − 1 2 1 2 1 3 1 3 1 2 1 3 1 3 1 2 1 3 1 3 1111 5 3 2 2 0 − − − − − − − − − − − 1 2 2 1 3 2 1 3 3 2 5 2 3 2 3 3 1 1 1 1 1 1 1 2 00 1 1 z C y 1 5 B 2 2 3 2 2 1 2 3 3 1 1 1 1 1 1 1 00 − 2 3 2 3 1 2 1 2 5 3 1 3 1 1 1 1 1 1 00 00 00 00 2 5 1 3 1 3 2 2 1 3 1 3 4 3 3 1 1 1 1 1 0 00 − − − 2 1 1 1 1 2 1 2 3 4 1 3 1 3 1 2 5 2 3 3 z 001001 001100 00 C x A 1 1 1 1 1 1 2 2 2 5 1 2 3 3 00 000 1 2 1 2 3 1 2 5 3 1 3 2 3 1 1 1 001 00 00 00 00 1 3 1 3 1 3 4 1 3 1 3 2 1 5 1 1 0 00 − 1 1 1 3 4 1 2 2 2 3 y 00 00 00 B x A 2 2 1 2 1 2 3 3 1 1 1 2 00 00 000 1 2 1 2 1 3 3 5 3 2 3 2 3 1 3 1 010 0 00 z C 1 3 1 2 3 1 00 00 5 3 3 1 1 1 000100000 0 000 000 1 0 0 000 0 0 y B 1 2 1 3 1 3 1 3 1 2 5 1 2 3 1 3 2 1 0000 0 0 0000 4 5 3 3 1 1 1 1 0 00000 0 000000 1 000000 0 0 0 0 0 00000 0 000000 1 0 0 0 0 0 Extremal boxes of the tripartite no-signaling polytope. A representative is given for each equivalence class of boxes under relabeling of parties, i x A 1 2 1 2 1 2 1 2 1 3 1 3 1 2 1 3 1 3 1 2 1 2 2 5 1 2 1 3 2 3 1 3 the number of representatives in each class. Table 1. 1 1 1No 1 0 1 1 1 01111111111111111111 1 1 1 0211000000000000111 1 64 00 01 10 11 00 01 10 11 00 01 10 11 000 001 010 011 100 101 110 111 000100100 0 0 0 0 0 13000000110000100000 0 17 18 19 21 16 24 14 15 22 23 20 10 11 000100100 0 0 1 0 0 4000000100000100001 13 12 000101010 0 0 0 1 0 5000000100110001000 000100000 0 0 0 0 1 16000000110000000000 1 000100000 0 0 1 1 1 7000000100100000000 000101011 0 0 0 0 0 18000000110010001010 0 9

4 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al e n 1536 3072 1536 3072 3 5 3 5 116 1 384 1 128 1 1536 1 1536 1 512 1 1536 1 1536 1 1536 1 1536 1 5 1 2 1 256 0 3072 00 3072 1536 − − − − − − − − − − − − 3 5 2 3 2 3 2 3 1 11 48 1 1 1 1 384 1 3 1 1 0 − − − − − − − − − 3 5 3 4 2 3 1 1 1 768 1 1 11 1 1 1 1 2 1 1 5 5 3 1 1 0 512 − − − − − − − − − − − 3 5 3 5 2 3 1 z 3 1 3 1 3 4 2 3 2 3 111 111 1 1 − − − − C y B x 1 2 3 5 3 4 1 2 11 1 1 1 11 1 1 1 11 1 1 1 1 11 1 1 3 1 5 1 5 A − − − − − − − − − − − − − − 1 2 3 5 1 2 1 1 1 1 3 2 5 3 3 3 5 3 3 4 7 1 − − − − 1 2 3 5 1 2 1 1 2 5 1 1 3 3 3 3 3 5 2 3 3 7 11 0111 − − − 1 3 1 2 1 3 3 5 1 5 1 2 1 2 3 3 3 7 1 1 2 1 1 111 1 111 111 − − − − − − − 1 3 1 3 1 3 1 3 3 5 3 5 3 5 1 3 1 3 1 3 1 3 1 7 1 5 4 1 01 1 − − − − − − − − − − − − 1 1 3 1 3 3 5 7 1 1 1 1 1 1 1 1 1 3 3 5 5 5 1 3 2 4 3 3 z − − − − C y 1 3 1 3 3 5 1 4 1 7 B 1 1 1 1 1 1 1 1 3 3 5 5 5 4 3 3 00 − − − − − 1 3 1 7 1 1 1 1 1 1 3 1 3 5 5 5 5 3 1 4 4 3 3 1 1 1 1 00 − − 1 3 1 3 3 5 3 5 1 4 1 7 1 1 1 1 1 5 5 1 3 3 3 3 0 − − − − − − 1 3 3 5 1 7 1 1 1 1 1 5 5 5 1 3 2 4 1 3 1 3 1 000000 000000 z − − − C x 1 3 1 3 3 5 1 5 1 7 A 1 1 1 1 1 1 2 5 5 4 4 2 000 000 00 − − − − − 1 3 1 1 1 3 1 2 3 1 2 2 3 5 5 5 5 3 1 4 1 4 1 2 7 3 1 00 00 00 − 1 1 1 3 3 7 1 1 1 1 1 1 5 5 5 5 1 3 1 3 3 3 0 − − − 1 3 1 3 3 5 1 7 1 1 1 1 5 5 5 4 1 3 1 3 00 00 y − − − − B x 1 3 1 3 3 5 1 5 1 7 A 2 5 5 1 4 2 1 1 1 1 − − 000 000 − − 00 00 000 − 2 3 1 2 2 3 5 5 5 5 1 3 1 3 1 2 1 2 7 1 1 1 3 3 1 1 3 3 5 5 5 5 7 1 1 1 1 1 1 1 0 z C 3 3 5 5 5 5 1 3 7 1 1 1 1 1 1 1 00 3 3 5 5 5 5 4 7 1 1 1 1 1 1 1 1 0 000 000 0 y B 3 3 5 5 5 5 1 3 1 3 1 4 4 7 1 1 1 1 1 1 1 1 0000 0000 3 3 3 3 5 5 5 5 4 7 1 1 1 1 1 1 1 1 1 1 00000 0 0 0 00000 0 0 00000 0 0 00000 (Continued.) x A 1 3 1 3 1 2 1 3 1 3 1 5 1 5 1 5 3 5 1 3 1 3 1 4 1 4 1 3 1 3 1 2 1 7 Table 1. No 0 1 025 1 0 1 00 01 10 11 00 01 10 11 00 01 10 11 000 001 010 011 100 101 110 111 31 33 34 38 29 35 26 37 30 32 36 39 40 27 28 0 0 1 1 1 1 1 41000000100000000000 40000000001 1 1 1 1 1 1 44000000000000000000 20000000000 0 1 1 1 42000000100000000000 43 50000000001 1 1 1 1 1 1 1 45000000000000000000 46000000000000000000

5 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al xyz 4 5 xyz 2 − − xyz yz 2 3 1 5 yz xyz + 3 2 2 3 − + xz − yz xyz 6 5 xyz + xz xyz yz + yz 2 3 + 3 2 2 3 2 3 xyz xyz + 2 3 xyz xz + xyz xyz + + yz xyz − 1 2 xy − + 3 2 4 5 − − − xz yz − xz xy 1 3 − yz 1 3 + 1 3 3 2 − xy xz yz xz yz) + xyz xz z + − yz xyz + + 1 5 + + + − + yz + 2 3 xyz − z z + z xy xz xy 3 4 2 3 + − z 1 6 2 3 + 1 3 xy xy xy xz − 1 3 xy 1 2 xz y + + + + − − + + + xz 1 4 + z 7 + − xy xyz yz yz 10 1 4 ˆ y y y y y y y z 3 2 C y 1 2 xyz xyz y y 1 2 1 2 1 6 1 2 3 4 2 3 2 3 1 3 + − + xy 1 6 + xyz 1 2 1 2 − + + − + − − y + − − − − − − − − x + + − xy xy xy y ˆ xz) y B 1 4 1 3 7 x x xz x x x x x x x 1 2 3 4 10 xz x x x yz 1 2 1 2 1 2 1 2 1 6 1 2 3 4 2 3 2 3 2 3 + 1 2 1 4 1 4 1 6 − − + − − + + − x 1 1 + − − − − + + − + − − − − − ( ( x x x x x ˆ A 7 1 1 1 2 1 2 1 2 2 2 1 2 1 2 1 2 1 4 1 4 1 3 1 6 5 8 1 2 1 2 1 4 3 4 10 3 4 2 3 2 3 2 3 yz yz 1 5 1 2 − + yz z z yz yz yz 1 2 1 5 yz z 1 4 1 2 1 3 1 3 1 6 + − − + + + − z + z z y yz) ˆ C y y y y y 1 2 1 2 yz y yz yz yz y yz yz 1 1 4 1 6 1 6 10 1 4 1 3 1 2 1 6 1 2 1 3 1 2 1 4 1 3 1 3 + − + + y 1 − + + − + − + + + − − + − + ( y z y y ˆ B 1 1 1 2 1 2 1 2 2 2 1 2 1 2 1 4 1 2 1 2 1 3 1 2 1 4 1 2 1 3 1 4 1 4 2 5 1 2 1 3 1 2 1 3 yz yz xz xz xz 1 2 1 3 1 3 + − − xz xz xz z xz xz xz 1 2 1 2 2 5 z z 1 4 xz) xz) 1 3 1 4 1 4 1 6 1 3 + + + − − z + + + − + + z z x ˆ x x C x x x x x x x x 1 2 1 2 x x x x xz xz x 1 1 4 1 4 1 2 3 8 1 6 1 6 1 4 10 1 4 + 1 4 1 4 1 6 1 4 1 3 1 3 1 3 − − + + x 1 1 − − + + + − − − − − + − − + + + ( ( x x ˆ A 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 4 1 4 1 3 1 2 1 2 1 2 1 2 1 2 1 4 2 5 1 2 1 3 1 3 1 6 xy xy xy 2 5 xy xy 1 2 1 2 1 3 1 3 − + + − − xy xy xy y y y 1 2 2 3 2 3 xy xy xy xy xy 3 y y 1 4 1 4 xy) 10 1 2 1 4 1 3 1 6 1 3 − − − − − − + y + + + − − + + y y y y y ˆ y x B x x x x x x 1 2 1 2 1 2 1 2 1 2 x x x xy x x 1 2 3 8 1 2 1 4 1 4 1 4 3 + + 1 4 1 4 1 6 10 1 3 1 3 1 3 + + + + + x − + + + − − − − + − + + + x (x x x x x x x x ˆ A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 1 2 3 2 2 2 2 2 5 1 2 3 6 6 z z 1 6 1 4 z + + ˆ C 2 2 2 2 2 2 2 1 2 2 2 2 1 3 1 2 2 1 2 2 2 1 2 1 2 1 4 3 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y 1 y y y y y y 10 1 4 1 6 1 6 1 6 1 4 1 4 + y + + + + + + ˆ B 3 1 2 2 2 2 2 2 2 1 4 1 2 1 2 1 2 1 3 1 3 1 2 1 3 1 2 1 2 1 4 10 1 4 3 1 3 2 1 1 1 1 1 1 1 1 x 1 x x x x x x x x x x x x x 10 1 4 1 4 1 4 1 4 1 6 1 6 1 8 1 6 1 6 1 4 1 4 1 4 1 3 Extremal boxes in the computer scientist notation. + x + + + + + + + + + + + + + ˆ A 3 1 2 1 2 1 2 1 2 1 2 1 2 1 4 1 4 1 4 1 4 1 3 1 3 1 4 1 3 1 3 1 4 1 4 10 1 4 1 3 1 6 1 3 4 12 Table 2. No 1000020 0 0 0 22 8 10 13 5 20 19 24 3 23 9 7 14 17 6 16 11 18 15 21

6 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al xyz xyz xyz xyz 4 5 4 5 xyz 2 3 8 7 + − − − − yz yz yz yz yz xyz 1 3 6 5 6 5 1 4 yz 8 7 2 3 + + + + + + yz + 4 5 xyz xz xz xz xz xz xz yz − 1 3 1 3 6 5 2 5 5 4 − xyz xyz 2 7 xyz 2 3 4 3 4 5 + + + + + xyz + − yz xyz + xz 4 3 3 4 + 2 − 4 5 xy xy xy xy xy xyz xz xy xz + xyz 1 3 1 3 4 5 2 5 3 4 − − 2 7 − yz yz 2 + + 2 5 − + + + + + xyz + + xz yz yz − xz 4 3 xy z z z z z 3 4 + 2 3 xy xy z xyz 2 3 4 5 1 3 1 3 4 5 2 5 1 4 + xy 1 2 1 2 2 7 xy 2 + + − xz + xz − − − − − − + − + + xz 2 5 − − z + 2 3 z z xy xz ˆ yz xyz y y y y y y y y C xy y 4 5 1 6 1 4 2 3 1 3 1 3 4 5 2 5 1 6 3 4 1 2 1 2 2 − 2 3 2 7 yz + + xyz xy + + + + + − + y − + − − − − − + − − − − xy y y y xy xy ˆ B 4 5 x x x x x x x x x x 4 5 1 6 1 8 2 3 2 3 y xy x xz xz 2 3 3 4 2 3 4 5 3 5 1 6 3 4 3 4 1 2 1 2 + 2 3 2 3 4 7 + − + + + + + + x + − − − + − − − − − − − + x x x x x x ˆ A 1 3 3 1 1 3 3 2 3 4 2 3 1 3 4 5 4 5 4 5 3 5 5 6 1 6 1 8 3 4 1 6 1 6 4 2 2 7 xyz xy xy yz 4 5 − yz yz yz 2 3 yz z z 1 3 1 4 1 4 2 5 1 6 − + + z + + − z ˆ C y y 2 3 yz yz yz y yz yz y z yz yz 1 6 1 8 1 3 1 3 2 5 2 5 2 5 2 5 1 6 1 4 1 3 1 3 + + y + + + + + + + − − + + + y ˆ B 1 1 1 1 1 2 3 1 3 1 2 1 3 2 3 2 5 2 5 2 5 2 5 1 3 1 2 3 8 3 8 1 3 1 3 1 2 2 2 4 7 2 2 2 xz xz xz xz xz xz 1 3 1 3 4 5 2 5 1 3 2 7 − − − − − − xz xz xz 2 3 z z z z z z 2 3 1 4 1 3 1 3 2 5 2 5 1 6 2 7 − z + + + + + + + + z ˆ C x x x x x 2 3 x x x xz x xz x x xz x x 2 3 1 6 1 8 1 6 1 6 + 1 3 1 4 1 3 2 5 2 5 2 5 1 5 1 6 1 4 1 4 2 7 + x − + + + + + + + + − − + − − + + x ˆ A 2 2 1 1 1 2 3 1 6 1 4 1 6 2 3 5 2 5 5 1 5 1 3 1 2 3 8 3 8 1 2 1 2 1 4 1 2 1 2 2 7 2 2 2 xy xy xy xy xy xy xy xy xy 1 4 1 3 1 3 4 5 2 5 1 3 1 3 1 4 2 7 + − − − − − − − − xy xy xy xy y 2 3 2 3 1 2 1 2 y y y y y y y y 1 8 1 3 1 3 2 5 2 5 1 6 1 6 1 4 2 7 − − − − y − + + + + + + + + y y y y ˆ B x x x 2 3 2 3 1 2 1 2 x x x x x x x x x x 1 8 1 6 1 6 + 1 3 1 4 1 3 2 5 1 5 1 6 1 6 1 4 1 4 2 7 + + + + x + + + + + + + − + − − + + x x x x ˆ A 2 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 2 1 1 1 3 6 4 6 3 5 5 5 5 1 3 3 2 4 2 2 4 2 2 7 2 2 2 z 1 6 z + ˆ C 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 3 1 1 1 3 2 2 2 3 5 5 5 5 1 3 1 2 1 2 2 2 2 2 2 2 7 2 2 2 y y y 1 6 1 6 1 8 y + + + ˆ B 1 1 1 1 2 2 2 3 1 1 3 1 1 1 3 2 1 2 2 3 5 5 5 2 5 1 3 1 3 3 8 8 1 2 1 2 1 2 2 2 7 2 2 2 x x x x x x x x 1 4 1 5 1 6 1 6 1 8 1 6 1 6 1 4 (Continued.) x + + + + + + + + ˆ A 1 3 1 3 1 4 1 3 1 3 2 5 2 5 2 5 1 5 1 3 1 3 3 8 3 8 1 3 1 3 1 4 1 2 1 2 3 7 1 2 1 2 1 2 40 Table 2. No 25 46 38 37 28 41 43 39 36 45 29 35 26 27 44 31 34 33 32 30 42

7 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al

The 64 possible deterministic boxes deﬁne the extremal points of the polytope of local correlations. All the other extremal boxes are nonlocal. • PR-boxes (boxes 2). This class comprises boxes corresponding to a PR-box shared between two parties, with the third party deterministic. For instance, the representative 2 in tables 1 and 2 satisﬁes the relations

aˆx = 0, bˆy + cˆz = yz. (6) These are in essence bipartite boxes. • GYNI boxes (boxes 25 and 29). These are the two no-signaling boxes associated with the tripartite ‘guess your neighbor inputs’ (GYNI) nonlocal game [15]. GYNI is a nonlocal game whose winning probability corresponds to the Bell expression = 1 | | | | w 4 [P(000000) + P(110011) + P(011101) + P(101110)]. (7) Quantum correlations achieve at most w = 1/4, which is not better than classical strategies. No-signaling correlations, however, can outperform classical and quantum strategies and achieve w = 1/3. Boxes in classes 25 and 29 are the two boxes achieving the maximum no-signaling winning probability w = 1/3. • Full-correlation boxes (boxes 44, 45, and 46). These boxes are the only full-correlation boxes, for which all one-party and two-party correlation terms vanish. They can thus be written as | = 1 P(abc xyz) 8 [1 + abc Ax By Cz ], (8) with xyz Ax By Cz=(−1) for box 44, (9)

x(y+z) Ax By Cz=(−1) for box 45, (10)

xy+xz+yz Ax By Cz=(−1) for box 46. (11) These boxes correspond to a situation with perfect correlations: for instance in the case of box 44, the outcomes satisfy abc =−1 if all parties use measurement ‘1’, and they satisfy abc = +1 in all other cases. These are the only genuine tripartite boxes with this property. These three boxes were already introduced in [4]. Boxes 46 were called ‘Svetlichny’ boxes because they violate Svetlichny’s original inequality [21] = − xy+xz+yz S ( 1) Ax By CZ 4 (12) xyz up to the algebraic maximum S = 8. Boxes 44 and 46 also violate the Mermin inequality [20] = − M3 A1B0C0 + A0B1C0 + A0B0C1 A1B1C1 2 (13)

up to its algebraic maximum M3 = 4. Note that it is also possible to violate maximally the Mermin inequality using quantum systems. By measuring the Greenberger–Horne–Zeilinger (GHZ) state √1 (|000 + |111) 2 | = 1 in suitable local bases [19], one obtains correlations of the form PGHZ(abc xyz) 8 [1 + abcAx By Cz] with

A1B0C0=A0B1C0=A0B0C1=−A1B1C1=1, (14)

A0B0C0=A0B1C1=A1B0C1=A1B1C0=0. (15)

8 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al

These correlations return M3 = 4 for the Mermin inequality. The quantum point PGHZ is not extremal, however, but it can be decomposed using extremal points in a very simple way: = 1 1 PGHZ 2 P46 + 2 P46. (16) Here, P46 is the extremal point given by (11), while P46 is another point in the same class. = − 1+x+y+z+xy+xz+yz P46 is defined by Ax By Cz ( 1) and can be obtained from P46 using the following local relabeling of the outputs: a → (−1)1−x a, b → (−1)1−y b, c → (−1)1−zc, i.e. in words, the parties flip their output for the input ‘0’ and leave it unchanged for the input ‘1’. It can be checked that P46 and P46 are the only two extremal points of the 46th class that reach M3 = 4 for the representative Mermin inequality (13). More generally, for each version of the Mermin inequality, there are 118 extremal points that reach M3 = 4: 2 in the class 46 (as we have seen), 8 in the class 44, 12 in the class 2, and 32 in each of the classes 21, 22, and 34. The GHZ correlations defined by (14) and (15) can be reproduced by mixing these strategies in an uniform way within each of these classes. Before embarking further in characterizations of the extremal boxes, let us make some additional observations. Out of the 46 types of extremal boxes, only 13 have their correlation terms that are either perfect or uniformly random, i.e. only 13 boxes have correlation terms that only take as possible values 0, 1, or −1. These are boxes 1–8, 41, 42, and 44–46. Finally, the boxes which are the most symmetric under relabeling of parties, inputs, and outputs are boxes 46, with only 16 representatives in their equivalence class, while the least symmetric are boxes 14, 17, 32, 35, 37, and 38, with 3072 different representatives.

3. Classiﬁcation through nonlocality

As a ﬁrst attempt at classifying these 46 different boxes, let us consider their basic nonlocal properties. Multipartite nonlocality is more intricate than the bipartite case, so we start by deﬁning several possible criteria.

3.1. Notions of multipartite nonlocality It is helpful to think about nonlocality in operational terms and ask what type of classical resources (shared randomness, communication) are needed by classical observers to simulate a particular kind of nonlocal box. A box is said to be local if it can be simulated by non-communicating classical observers using shared randomness only. A local box thus admits a decomposition of the form P(abc|xyz) = qλPλ(a|x)Pλ(b|y)Pλ(c|z), (17) λ where the variable λ has the probability distribution qλ and can be thought of as the shared randomness determining the local response of each party. We denote the set of local boxes as L. A box is said to be nonlocal if it cannot be written in the above way, which implies that some communication between the parties is required to simulate it. Among nonlocal boxes, one can distinguish further between those that require for their simulation only communication between two of the parties and those that require all three parties to communicate. Following Svetlichny’s original deﬁnition [21], we thus say that a box is two-way Svetlichny nonlocal if it admits a decomposition of the form AB/C AC/B BC/A P(abc|xyz) = q1P (abc|xyz) + q2P (abc|xyz) + q3P (abc|xyz) (18) where AB/C P (abc|xyz) = qλPλ(ab|xy)Pλ(c|z) (19) λ

9 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al corresponds to a nonlocal term involving communication only between parties A and B, and where P AC/B (abc|xyz) and P BC/A(abc|xyz) are similarly defined. We denote the set of two-way Svetlichny nonlocal boxes as S2. A box is said to be three-way Svetlichny nonlocal if it cannot be written in the above form. Classical communication models in the manner of Svetlichny presuppose that all parties receive their inputs at the same time5. This is followed by one or several rounds of communication after which all parties produce an output. Inputs into no-signaling boxes need not, however, be given simultaneously to all parties. For instance, in quantum theory, measurements on an entangled state can be performed in a sequence or on a subset of systems only. When one subsystem is measured, the outcome is obtained immediately, and one does not have to wait until all the other subsystems have also been measured. In analogy with the quantum case, the same feature can be thought of no-signaling boxes: once a party introduces an input into a box, an output is obtained immediately, irrespective of whether inputs have been introduced by other parties. This is possible thanks to the no-signaling condition which ensures that the marginal output probability distribution for any subset of the parties is well defined and is independent of the inputs for the other parties. When inputs are given in a sequence rather than simultaneously, it is necessary to consider communication models more restricted than Svetlichny’s ones, since a party’s output can depend on communications already received, but cannot depend on communications from parties later in the sequence. For instance, the nonlocal term Pλ(ab|xy) in equation (19) should be replaced by Pλ(a|x)Pλ(b|xy) if party A receives his inputs before party B, and by Pλ(a|xy)Pλ(b|y) if it is party B who receives it first. We consider here two alternative notions of two-way nonlocality based on such communication models with inputs given in a sequence. In the first model, inputs are given according to an arbitrary sequence which is known beforehand by the parties. We say that a box is two-way KS nonlocal (where KS stands for ‘known sequence’) if it can be reproduced using communication between at most two of the parties, irrespective of the predetermined input sequence. It is easy to show that a box is two-way KS nonlocal if it admits a decomposition of the form (18) where

AB/C P (abc|xyz) = qμPμ(a|x)Pμ(b|xy)Pμ(c|z) (20) μ = qν Pν (a|xy)Pν (b|y)Pν (c|z) (21) ν and similarly for the terms P AC/B (abc|xyz) and P BC/A(abc|xyz). Equation (20) speciﬁes the response of the parties when A precedes B in the sequence and equation (21) when it is B who precedes A. The response of the parties in each case is determined, respectively, by different set of random variables {μ} and {ν}. The set of two-way KS nonlocal boxes is denoted as KS2. In the second model, inputs are given according to an arbitrary sequence which is not known in advance by the parties and we say that a box is two-way US nonlocal (where US stands for ‘unkown sequence’) if it can be reproduced using communication between at most two of the parties, irrespective of the unknown input sequence. A box is two-way US nonlocal if it admits a decomposition of the form (18) with

5 The following discussion is a concise summary of a forthcoming paper on the deﬁnition of genuine multipartite nonlocality [22]. In particular, it will be argued in [22] that the proper notion of genuine tripartite nonlocality should be based on US nonlocality, see the deﬁnition below, rather than Svetlichny’s one.

10 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al

AB/C P (abc|xyz) = qλqμ|λPμ(a|x)Pμ(b|xy)Pλ(c|z) (22) λμ = qλqλ|ν Pν (a|xy)Pν (b|y)Pλ(c|z) (23) λν and similarly for the terms P AC/B (abc|xyz) and P BC/A(abc|xyz). As before, equation (22) specifies the response of the parties when A precedes B in the sequence and equation (23) when it is B who precedes A. In each case, the behavior of the parties is specified by two sets of shared variables {λ, μ} and {λ, ν}. The difference with the previous definition is that party C has no way to know the relative order between A and B as the input sequence is not specified in advance and as it is not communicating with the other parties. The response of party C is thus identical in each case and depends only on a variable λ common to both sets {λ, μ} and {λ, ν}. We denote the set of two-way US nonlocal boxes as US2. Finally, we also consider models where parties are allowed to use other no-signaling boxes as a resource rather than communication. We define the set NS2 as the set of tripartite boxes that correspond to convex combinations of bipartite no-signaling boxes, i.e. that admit a decomposition of the form (18) with Pλ(ab|xy) in (19) being restricted to be no-signaling (and similarly for other partitions of the parties). This represent the set of tripartite boxes that are in essence only bipartite. In the case of binary inputs and outputs, extremal no-signaling boxes of the form Pλ(ab|xy) correspond either to local deterministic boxes or PR-boxes. The set NS2 thus corresponds to the boxes that can be simulated using shared randomness and a single PR-box shared between any two parties. We clearly have the inclusions L ⊆ NS2 ⊆ US2 ⊆ KS2 ⊆ S2 (furthermore, these inclusions are strict, see [22]). Each of these sets corresponds to a polytope that can be characterized using linear programming, making it easy to determine whether a given box belongs to any of them.

3.2. Nonlocality of the extremal boxes In table 3, we computed the resistance to white noise of extremal boxes according to all these different notions of nonlocality. That is, we computed (using linear programming) the minimal value q such that the noisy box characterized by the probability distribution (1 − q)P(abc|xyz) + q/8 belongs to any one of the sets. The series of inclusions L ⊆ NS2 ⊆ US2 ⊆ KS2 ⊆ S2 implies that for a given type of box (corresponding to a given row of table 3) the corresponding noise resistances can only decrease. Note that the extremal boxes in tables 1 and 2 have been ordered according to the noise resistance computed in table 3. That is, boxes have been ﬁrst ordered in ascending order with respect to their noise resistance for S2. Boxes with the same noise resistance for S2 have been ordered with respect to their noise resistance for KS2, and so on. (The relative ordering between boxes with the same noise resistance according to all notions is arbitrary.) From table 3, one obtains the following. • L boxes (local boxes). As noted earlier, boxes 1 are the only local extremal boxes and they correspond to the vertices of the local polytope L. All other extremal boxes are nonlocal. • NS2 boxes (bipartite no-signaling boxes). As noted above, this class comprises boxes 1 (the deterministic ones) and boxes 2 (the PR-boxes). These two types of boxes deﬁne the vertices of the bipartite no-signaling polytope NS2. All other extremal boxes are genuine tripartite no-signaling boxes in the sense that they can only be reproduced

11 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al

Table 3. Amount of noise that can be tolerated before entering in the local (L), bipartite no-signaling (NS2), Svetlichny with unknown input sequence (US2), Svetlichny with ﬁxed input sequence (KS2), and original Svetlichny (S2) polytopes. The asterisk denotes the highest resistance in each column.

No L NS2 US2 KS2 S2

100000 22/3∗ 0000 31/21/30 0 0 42/52/50 0 0 51/22/50 0 0 61/21/31/30 0 71/21/31/30 0 81/22/52/50 0 91/23/81/31/60 10 3/53/71/31/40 11 1/24/11 2/72/70 12 1/24/11 1/31/30 13 1/28/23 4/13 4/19 4/37 14 1/28/23 4/13 1/41/7 15 3/53/71/31/31/7 16 1/28/23 1/34/19 4/25 17 4/72/58/23 8/29 4/23 18 3/53/71/31/31/5 19 8/15 4/11 4/11 1/34/19 20 16/31 16/41 1/32/73/13 21 1/27/19 4/11 1/34/17 22 1/28/23 16/49 4/13 1/4 23 1/28/23 1/34/13 1/4 24 4/72/55/14 4/13 1/4 25 4/72/52/516/49 1/4 26 4/72/58/23 1/31/4 27 1/24/11 4/11 1/31/4 28 4/72/520/53 8/23 1/4 29 4/72/52/52/512/47 30 16/31 8/23 8/23 8/23 2/7 31 16/31 8/23 8/23 8/23 2/7 32 16/31 16/41 8/23 8/23 2/7 33 16/31 16/41 32/87 8/23 2/7 34 1/24/11 8/23 1/31/3 35 1/24/11 8/23 1/31/3 36 1/24/11 6/17 1/31/3 37 1/23/84/11 1/31/3 38 4/72/54/11 1/31/3 39 4/72/556/155 1/31/3 40 1/22/52/51/31/3 41 1/22/52/51/31/3 42 1/22/52/51/31/3 43 8/15 32/81 48/125 4/11 4/11 44 3/53/83/83/83/8 45 1/21/2∗ 1/2∗ 1/2∗ 1/2∗ 46 1/21/2∗ 1/2∗ 1/2∗ 1/2∗

12 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al

using no-signaling resources shared between all three parties; they are the analogues of the quantum correlations that can be obtained only by measuring genuine three-partite entangled states. Not all genuine tripartite no-signaling boxes, however, are three-way nonlocal, as shown below. • US2 boxes (two-way nonlocal boxes). This class comprises boxes 1–5. All these boxes can be reproduced using classical communication between only two of the parties, even if inputs are given in an arbitrary sequence unknown to the parties. Consider box 3 for instance which is deﬁned by the relations

aˆ0 + bˆy = 1, aˆ1 + cˆ0 = 1, aˆ1 + bˆy + cˆ1 = y, (24) all other correlation terms being uniformly random. Here is a model reproducing it, involving only communication between Alice and Bob. The model uses two shared random variables λ0 and λ1 both taking the values 0 or 1 with equal probability. Charles produces his outputs according to cˆz = λz. If Alice receives her input ﬁrst, she outputs aˆ0 = λ0 + λ1 or aˆ1 = λ0 and Bob outputs bˆy = λ0 + λ1 if Alice’s input is x = 0, and bˆy = λ0 + λ1 + y if Alice’s input is x = 1. If Bob receives his input ﬁrst, he outputs bˆy = λ0 + λ1 + y and Alice outputs aˆ0 = λ0 + λ1 + y or aˆ1 = λ0. It is easy to see that this model correctly reproduces box 3. All extremal boxes that do not belong to US2 manifest genuine tripartite nonlocality, in the sense that their simulation requires communication between all three parties in at least one experimental situation (corresponding to inputs given in an unknown sequence). • KS2 boxes. This class comprises boxes 1 to 8. • S2 boxes. This class comprises boxes 1 to 12. Finally, note that the boxes that are the most nonlocal according to all notions of two-way nonlocality are boxes 45 and 46.

4. Interconversion between boxes

The classification through nonlocality that we have just presented exhibits a rich structure, which we may not have fully exploited in the discussion above. Moreover, it is not the only possible approach. A different classification, for instance, may be based on the possibility of simulating some boxes using other ones. We do not attempt a systematic study here, but want to point out that this classification will look astonishingly different from the one based on nonlocality. Indeed, consider just the question of whether a given tripartite box can be simulated by sharing any number of bipartite PR-boxes between each pair of parties. Boxes 44, 45, and 46, which are the most nonlocal according to the previous criterion, are easily simulated using at most one PR-box between each pair of parties [4]. (More generally, it was shown in [13] that any n-partite full correlation box can be simulated with PR-boxes.) In contrast, box 4 cannot be simulated even by sharing infinitely many PR-boxes between the pairs [13, 23], but it is pretty weak on the nonlocality scale and, as we shall mention below, it does not violate maximally any of Bell’s inequalities.

5. Extremal points and Bell-type inequalities

5.1. Overview of the inequalities In this section, we focus in greater detail on how extremal boxes differ from local correlations. For three parties and binary inputs and outputs, the local set was fully characterized by Pitowsky

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Table 4. Boxes attaining the maximal violation of each Bell-type inequality. The inequalities are listed from 1 to 46, following [25]. Boxes are numbered according to tables 1 and 2. Inequality Extremal boxes Inequality Extremal boxes 1 None 24 29 2 2, 21, 22, 34, 44, 46 25 25 3 2, 7, 9, 16, 19, 20, 23, 35, 37, 42, 44, 45 26 29 4 2 27 25, 28, 29 52282 62292 7 44302 8 2, 7, 11, 13, 16, 19, 21, 42, 44 31 19, 37, 42, 44, 45 2, 6, 12, 14, 17, 19, 23, 24, 27, 9 32 42 28, 35, 37, 38, 41, 42, 44, 45 10 25, 29 33 21, 44, 46 11 2, 41, 45 34 41 12 2 35 41, 45 13 2, 42 36 2 142372 15 2, 40 38 2 16 2, 19, 21, 27, 28, 42, 44 39 46 17 2, 8, 12, 19, 23 40 40, 42, 45 18 2, 41 41 2, 19, 44 19 2, 40, 42 42 2, 21, 29, 44 20 2 43 2, 19, 25, 28, 35, 37, 38, 42, 43, 44, 45 21 43 44 2 22 2, 19, 21, 27, 34, 36, 39, 41, 44, 46 45 2 23 25, 45 46 25

and Svozil [24] and Sliwa´ [25]. The local polytope has 53 856 facets deﬁning 46 different classes of inequalities that are inequivalent under relabeling of parties, inputs, and outputs. Only a few of these inequalities have been studied more or less thoroughly. Inequality 1 (we follow Sliwa’s´ numbering) is a trivial facet, and it corresponds to the condition that probabilities must be comprised between 0 and 1; obviously, no point can violate it. Inequality 2 is the Mermin inequality (13), inequality 4 is the CHSH inequality [12], and inequality 10 is the GYNI inequality (7).

5.2. Violations by the extremal points One of the most obvious questions to address is: for any inequality, ﬁnd the extremal points that return the highest no-signaling violation. The boxes that violate maximally each inequality6 are given in table 4. Some remarks on this table are as follows. • Boxes of class 2 (i.e. PR-boxes, the least nonlocal ones according to the criteria of section 3) violate maximally 28 of the inequalities, and for 14 inequalities they are the only ones that do so. • Boxes 3, 4, 5, 10, 15, 18, 26, 30, 31, 32 and 33 do not violate any inequality maximally.

6 Of course, there is no guarantee that the representative point written in tables 1 and 2 is the optimal one for the representative inequalities as written in [25].

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Table 5. For each class of boxes, the maximum amount of noise tolerated before ceasing to violate a given family of Bell-type inequalities. Each line corresponds to a box and each column to an inequality. The inequality numbers refers to the ones chosen in [25]. Table for inequalities 1 to 23. No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 100000000000000000000000 1 1 2 4 4 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 20 2 2 3 7 7 2 2 2 0 2 2 2 2 2 2 2 2 2 5 2 2 3 − 1 2 2 1 1 1 1 1 1 1 1 1 1 30 10 2 5 5 003 003 3 3 3 0 3 3 3 5 005 − 2 2 1 1 1 1 40 100 5 5 000 0 0 3 000003 0 5 005 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 50 0 02 5 5 0 3 0 3 3 3 0 3 3 003 3 5 3 3 5 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 60 0 02 5 5 3 3 2 0 3 0 3 3 3 3 3 3 3 3 3 3 3 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 70 0 2 2 5 5 3 2 3 0 3 3 3 003 3 3 3 3 3 0 5 − 1 2 2 1 1 1 1 1 1 1 1 1 1 1 80 10 2 5 5 003 003 3 3 3 3 2 0 3 5 3 0 5 1 1 2 1 1 3 3 1 1 1 1 1 1 1 3 1 1 5 1 1 5 90 0 2 3 5 2 3 7 7 5 5 5 3 5 5 3 7 5 3 13 3 5 13 3 2 2 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 5 10 0 0 0 5 5 5 3 5 7 5 3 5 3 5 5 3 3 3 3 3 3 3 13 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 0 3 3 2 5 5 3 2 3 3 3 3 3 3 0 3 3 3 3 3 3 3 5 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 0 0 3 2 5 5 3 3 2 0 3 3 3 3 3 3 2 3 3 3 3 3 3 2 2 2 8 8 2 1 2 1 1 1 2 1 1 2 1 1 2 2 2 1 1 13 0 5 5 5 17 17 5 2 5 4 4 4 5 4 4 5 4 4 5 5 5 4 3 1 2 2 8 8 2 2 1 1 2 1 2 1 1 2 2 2 2 2 2 2 2 14 0 4 5 5 17 17 5 5 2 4 5 4 5 4 4 5 5 5 5 5 5 5 5 1 1 3 2 2 3 1 3 1 1 1 1 1 1 3 1 1 1 1 3 3 5 15 0 5 5 5 5 5 7 3 7 5 3 5 3 5 5 7 3 3 3 3 7 7 13 1 1 2 8 8 2 1 2 1 1 1 2 1 1 2 2 1 2 2 2 1 2 16 0 4 2 5 17 17 5 2 5 4 4 4 5 4 4 5 5 4 5 5 5 4 5 1 1 4 8 8 2 2 1 1 2 1 2 1 1 2 2 2 2 2 2 2 2 17 0 4 4 7 17 17 5 5 2 4 5 4 5 4 4 5 5 5 5 5 5 5 5 1 1 3 2 2 3 1 3 1 1 1 1 1 1 1 1 1 1 1 3 3 5 18 0 5 5 5 5 5 7 3 7 5 3 5 5 5 5 3 3 3 3 3 7 7 13 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 19 0 3 2 2 5 5 2 2 2 0 3 3 3 0 3 2 2 3 3 3 2 2 3 1 1 4 4 16 4 4 4 2 3 2 2 1 2 3 4 3 3 3 4 3 7 20 0 6 2 9 9 31 9 9 9 7 8 7 7 6 7 8 9 8 8 8 9 8 17 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 21 0 2 3 2 5 5 2 2 3 003 3 002 3 3 3 3 2 2 5 1 2 8 4 2 2 1 1 1 1 2 2 1 1 2 2 22 0 2 0 5 17 13 5 5 4 004 4 0 4 5 0 5 4 3 5 5 0 1 2 8 8 2 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 5 23 0 0 2 5 17 17 5 5 2 4 5 4 4 4 5 5 2 5 5 5 5 5 11 1 4 4 8 2 1 1 1 2 1 1 1 1 2 2 2 2 1 2 2 5 24 0 0 4 7 13 17 5 4 2 4 5 4 4 4 4 5 5 5 5 3 5 5 11 2 4 8 8 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 25 0 0 5 7 17 17 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 2 1 4 8 4 2 1 2 1 1 1 1 2 1 2 1 1 2 2 1 26 0 4 0 7 17 13 5 4 5 0 4 4 4 0 4 5 4 5 4 3 5 5 3 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 27 0 3 3 2 5 5 2 3 2 0 3 0 3 0 3 2 3 3 3 3 2 2 3 1 2 4 8 8 1 2 1 1 2 1 2 1 1 1 2 2 2 2 1 2 2 28 0 4 5 7 17 17 2 5 2 4 5 4 5 4 4 2 5 5 5 5 2 5 5 2 4 8 8 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 29 0 5 0 7 17 17 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 2 4 16 8 4 4 4 2 2 2 2 4 2 4 2 3 4 4 2 30 0 9 7 9 31 23 9 9 9 0 7 7 7 0 7 9 7 9 7 8 9 9 7 2 4 16 8 2 2 2 2 4 2 4 2 2 4 2 31 0 7 0 9 31 23 7 7 7 0 7 9 0007 0 9 0 7 7 9 7 4 4 8 16 4 2 4 2 4 2 2 2 2 4 4 4 4 2 4 4 4 32 0 0 9 9 23 31 9 7 9 7 9 7 7 7 7 9 9 9 9 7 9 9 9 2 4 4 16 16 4 4 4 2 4 2 2 2 4 4 4 4 4 3 4 4 4 33 0 7 9 9 31 31 9 9 9 7 9 7 7 7 9 9 9 9 9 8 9 9 9 1 1 2 8 4 1 2 2 1 1 1 2 1 2 1 1 2 1 1 34 0 2 4 5 17 13 2 5 5 0 4 4 4 005 4 5 4 3 5 2 7 1 1 2 4 8 1 2 1 1 2 1 1 1 1 2 2 2 2 1 1 2 5 35 0 4 2 5 13 17 2 5 2 4 5 4 4 4 4 5 5 5 5 3 2 5 11 3 1 1 1 2 1 3 3 1 1 1 3 1 3 1 1 3 1 1 36 0 7 3 2 2 5 2 7 7 0 3 3 5 007 3 7 5 3 7 2 3 1 1 1 2 1 1 3 1 1 3 1 1 1 1 3 3 1 3 5 1 3 3 37 0 5 2 2 5 2 2 7 2 5 7 5 3 5 3 7 7 3 7 13 2 7 7 1 2 4 4 8 1 2 1 1 2 1 1 1 1 2 2 2 2 1 1 2 5 38 0 4 5 7 13 17 2 5 2 4 5 4 4 4 4 5 5 5 5 3 2 5 11 2 1 4 8 4 1 2 2 1 1 1 2 1 2 1 1 2 1 1 39 0 5 4 7 17 13 2 5 5 0 4 4 4 005 4 5 4 3 5 2 3 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 40 0 0 3 2 5 5 3 3 3 0 3 0 3 3 2 3 3 3 2 3 2 3 7 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 3 41 0 3 3 2 5 5 2 3 2 0 2 3 3 003 3 2 3 5 2 2 7

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Table 5. (Continued.) No1 2 3 4 5 6 7 8 9 1011121314151617181920212223

1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 3 1 1 3 42 0 3 2 2 5 5 2 2 2 0 3 0 2 3 3 2 3 3 2 7 2 3 7 4 6 8 16 16 8 6 6 4 4 2 4 6 4 4 6 4 8 6 5 43 0 11 13 15 37 37 15 13 13 0 11 0 11 9 11 13 11 11 13 11 15 13 12 1 1 1 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 44 0 2 2 2 5 5 5 2 2 0 3 0 3 002 3 3 3 3 2 2 3 1 1 2 1 1 1 1 1 1 1 1 1 1 1 45 0 0 2 2 0 5 2 3 2 0 2 00 0 0 3 3 3 3 0 2 3 2 1 1 2 1 1 1 − − − 1 1 1 1 46 0 2 0 2 5 0 2 3 3 1000 1 1 3 0 3 003 2 0

Table 6. Same as table 5, but for inequalities 24 to 46. No 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 100000000000000000000000 4 4 4 4 4 4 4 2 2 2 2 2 4 4 4 2 2 8 1 1 3 3 4 2 9 9 9 9 7 7 7 5 5 5 5 5 7 7 7 5 5 15 2 2 5 5 9 2 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 307 0 7 0 4 4 4 4 0 4 4 0 4 4 0 4 9 0 5 5 5 6 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 407 0 7 0 4 4 4 4 0 4 4 0 4 4 0 4 005 5 5 7 2 2 4 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 5 7 7 9 7 4 4 4 4 4 4 4 4 4 4 4 4 4 0 3 5 5 5 7 2 2 2 1 1 2 1 2 1 2 2 1 2 1 1 1 4 1 1 1 1 2 6 7 7 0 7 4 4 5 4 5 4 5 5 4 5 4 4 4 11 5 3 5 3 7 2 4 2 4 1 2 2 2 2 1 1 1 1 2 2 2 4 1 3 1 1 3 7 7 9 7 9 4 5 5 5 5 4 4 4 4 5 5 0 5 11 3 7 3 3 8 2 2 4 1 1 1 1 1 1 1 1 1 1 2 4 1 1 1 2 8 7 7 0 9 0 4 4 4 4 0 4 4 4 4 4 4 5 11 0 3 5 5 7 2 4 2 4 1 5 2 5 2 1 1 2 1 2 5 5 5 1 3 5 5 7 9 7 9 7 9 3 11 5 11 5 7 3 5 3 5 11 0 11 12 3 7 13 13 17 3 2 2 3 1 1 2 1 2 1 2 2 1 2 1 1 2 4 1 5 1 1 3 10 8 7 7 8 3 3 5 3 5 3 5 5 3 5 3 3 5 11 3 13 3 3 8 4 4 4 4 2 2 2 2 2 2 1 1 2 2 2 2 2 4 3 1 1 1 3 11 9 9 9 9 5 5 5 5 5 5 4 4 5 5 5 5 5 11 7 3 3 3 8 2 4 4 1 2 2 2 2 1 2 2 1 2 2 1 2 6 1 3 1 1 3 12 7 9 0 9 4 5 5 5 5 4 5 5 4 5 5 4 5 13 5 7 3 3 8 4 4 4 8 2 2 8 2 8 2 4 4 2 8 2 4 2 4 5 2 2 2 8 13 9 9 9 23 5 5 17 5 17 5 13 13 5 17 5 13 5 11 11 5 5 5 23 4 4 8 4 2 2 8 2 8 4 8 8 2 8 2 4 2 16 2 5 2 2 4 14 9 9 23 9 5 5 17 5 17 13 17 17 5 17 5 13 5 37 5 11 5 5 9 3 3 2 4 2 2 2 2 2 2 2 2 1 5 1 2 2 5 5 3 1 5 7 15 8 8 7 9 5 5 5 5 5 5 5 5 3 11 3 5 5 12 13 7 3 13 17 8 4 8 4 2 8 8 8 8 4 4 2 2 8 8 2 8 16 2 5 2 2 4 16 23 9 23 9 5 17 17 17 17 13 13 5 5 17 17 11 17 37 5 11 5 5 9 4 8 8 4 2 2 8 2 8 2 8 8 2 8 2 2 2 16 2 5 2 2 4 17 9 23 23 9 5 5 17 5 17 5 17 17 5 17 5 5 5 37 5 11 5 5 9 3 3 2 3 2 2 2 2 2 2 2 2 1 2 1 2 2 5 5 3 1 1 7 18 8 8 7 8 5 5 5 5 5 5 5 5 3 5 3 5 5 12 13 7 3 3 17 4 4 2 4 2 1 2 1 2 2 2 2 2 1 2 2 2 8 3 1 3 3 4 19 9 9 7 9 5 2 5 2 5 5 5 5 5 2 5 5 5 15 7 2 7 7 9 12 4 12 4 8 14 4 14 4 4 8 4 8 14 4 2 4 11 3 9 7 7 4 20 37 9 37 9 23 29 9 29 9 19 23 9 23 29 9 17 9 23 8 19 17 17 9 4 4 4 4 1 2 2 2 2 1 1 1 2 2 2 2 2 6 1 3 3 3 3 21 9 9 9 9 2 5 5 5 5 2 4 4 5 5 5 5 5 13 2 7 7 7 8 4 4 4 4 8 4 2 2 4 8 4 2 2 4 4 8 2 8 5 1 2 2 2 22 9 19 9 19 17 13 5 11 13 17 13 11 5 13 13 17 11 29 11 4 5 5 17 8 4 4 4 4 8 2 8 2 2 2 8 4 8 8 2 8 20 1 5 2 2 4 23 23 9 19 9 13 17 5 17 5 11 5 17 13 17 17 11 17 41 3 11 5 5 9 8 8 4 4 4 2 2 2 2 4 2 8 4 2 4 4 8 16 1 5 1 1 4 24 23 23 19 9 13 5 5 5 5 13 5 17 13 5 13 13 17 37 3 11 3 3 9 8 16 8 16 4 8 8 8 8 4 8 8 4 8 8 4 8 16 2 1 2 2 16 25 23 31 23 31 13 17 17 17 17 13 17 17 13 17 17 13 17 37 5 2 5 5 31 4 4 8 8 2 4 2 4 2 2 2 2 2 2 4 2 4 4 2 1 1 1 8 26 9 19 23 23 5 13 5 13 5 5 5 5 5 5 13 5 13 11 5 3 3 3 23 4 4 2 4 2 2 2 2 2 2 2 2 2 1 2 2 2 6 3 3 1 3 3 27 9 9 7 9 5 5 5 5 5 5 5 5 5 2 5 5 5 13 7 7 3 7 8 4 4 8 16 2 8 8 8 8 2 8 8 2 10 2 4 8 20 2 1 2 5 14 28 9 9 23 31 5 17 17 17 17 5 17 17 5 19 5 13 17 41 5 2 5 11 29 16 8 16 16 8 4 8 4 8 8 8 8 8 8 4 8 8 16 1 2 2 2 4 29 31 23 31 31 17 13 17 13 17 17 17 17 17 17 13 17 17 37 2 5 5 5 9 24 9 9 9 4 8 4 8 4 4 4 8 4 4 8 4 8 13 4 3 3 3 12 30 49 23 23 23 9 23 9 23 9 9 9 23 9 9 23 9 23 32 9 8 8 8 37 9 8 9 9 8 4 8 4 8 4 4 8 8 8 4 4 4 5 3 2 2 2 12 31 23 33 23 23 23 19 23 19 23 9 9 23 23 23 19 9 19 16 8 7 7 7 37 9 9 8 24 8 4 8 4 4 4 4 4 4 4 8 4 4 11 2 4 3 3 4 32 23 23 33 49 23 9 23 9 9 19 9 9 19 9 23 19 9 23 7 9 8 8 9

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Table 6. (Continued.) No 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

9 24 9 9 8 4 4 4 4 8 8 4 8 16 4 4 4 11 4 4 3 4 4 33 23 49 23 23 23 9 9 9 9 23 23 9 23 31 9 19 9 23 9 9 8 9 9 4 4 4 8 8 2 2 4 4 8 2 4 2 2 4 8 2 16 5 1 2 2 4 34 9 19 9 23 17 5 5 13 13 17 5 13 5 5 13 17 11 37 11 3 5 5 19 8 4 4 4 2 8 2 8 2 4 2 8 4 8 2 2 8 20 2 1 2 2 14 35 23 9 19 9 5 17 5 17 5 13 5 17 13 17 5 11 17 41 5 2 5 5 29 4 2 3 4 5 2 2 2 2 5 5 2 2 2 1 5 1 6 7 5 5 5 3 36 9 7 8 9 11 5 5 5 5 11 11 5 5 5 3 11 3 13 15 13 13 13 8 3 4 2 4 2 1 5 1 5 1 2 5 1 1 5 1 5 1 5 1 3 3 9 37 8 9 7 9 5 2 11 2 11 3 5 11 3 2 11 4 11 2 13 2 7 7 19 8 4 4 4 2 8 2 8 2 2 2 8 4 8 2 4 8 20 2 1 2 2 14 38 23 9 19 9 5 17 5 17 5 5 5 17 13 17 5 13 17 41 5 2 5 5 29 4 8 8 8 8 2 2 2 2 8 2 2 2 2 4 8 4 16 5 2 2 2 8 39 9 23 23 23 17 5 5 5 5 17 5 5 5 5 13 17 13 37 11 5 5 5 23 2 4 4 1 2 2 2 2 1 1 2 1 2 2 1 4 1 3 1 1 4 40 7 9 0 9 4 5 5 5 5 4 4 5 4 5 5 0 2 11 3 7 3 3 9 4 2 2 4 2 2 2 2 2 2 1 1 1 2 1 2 2 6 3 3 1 1 4 41 9 7 7 9 5 5 5 5 5 5 2 2 4 5 4 5 5 13 7 7 3 3 9 4 4 2 4 2 1 1 1 1 2 2 2 2 1 1 1 1 6 3 1 3 3 4 42 9 9 7 9 5 2 2 2 2 5 5 5 5 2 2 4 2 13 7 2 7 7 9 11 11 5 11 16 20 16 20 16 16 4 16 4 20 16 8 20 48 6 1 5 5 11 43 27 23 16 23 37 41 37 41 37 37 11 37 11 41 37 29 41 97 13 2 12 12 23 4 4 2 4 1 1 2 1 2 1 2 2 2 1 2 2 2 8 1 1 3 3 4 44 9 9 7 9 2 2 5 2 5 2 5 5 5 2 5 5 5 15 2 2 7 7 9 2 4 4 1 2 1 1 2 1 2 1 2 1 1 6 1 1 1 1 1 45 7 9 0 9 4 5 4 2 5 4 5 2 0 5 4 0 2 13 3 2 5 5 2 4 2 2 2 1 1 1 1 1 2 1 1 1 1 4 3 1 1 1 1 46 9 0 7 7 5 4 4 4 4 2 5 4 4 4 0 2 0 11 7 5 5 5 6

• Boxes 46, which, as we have seen above, are related to the GHZ argument, violate maximally only the inequalities 2 (Mermin), 22 and 33. Boxes 45, which are as nonlocal as boxes 46 according to table 3, violate a total of eight inequalities. For a given extremal point P(abc|xyz), we can also compute for each class of Bell inequalities, the minimal amount of white noise q such that the noisy point (1−q)P(abc|xyz)+ q/8 no longer violates any inequality in the class. Tables 5 and 6 report these values for each of the 46 extreme points. A positive value q>0 indicates that the corresponding extremal point violates some inequality in the class (i.e. we must add some noise q>0 so that it ceases to violate the inequalities). Among the extremal points that do not violate any inequality in a class, we can distinguish between those that lie on the border of the region deﬁned by these inequalities (i.e. those that reach the local bound of at least one inequality) and those that belong to the interior of this region (i.e. those that do not even reach the local bound of any of the inequalities). We distinguish these two situations by reporting values q = 0 and q<0, respectively. Some information that can be extracted from these tables is as follows: • By reading the tables column by column.Foragiven inequality, the boxes that give the largest violation, i.e. that have the largest resistance to noise, correspond to those listed in table 4. • By reading the table line by line. For 29 out of the 45 nonlocal boxes, the best resistance to noise is obtained with inequality number 4, i.e. CHSH. It comes as a surprise that an inequality, which is effectively tailored for two parties, is single out so markedly in a three-partite scenario; we recall that a similar situation is encountered for the resistance to noise when one stays in the bipartite case but increases the number of outcomes [26].

6. Conclusions

We have studied the no-signaling polytope corresponding to the Bell scenario with three parties, two inputs, and two outputs. Here, we summarize some of the properties that we discussed.

17 J. Phys. A: Math. Theor. 44 (2011) 065303 S Pironio et al

• The polytope has 53 856 extremal points, belonging to 46 non-equivalent classes upon relabeling of the parties, inputs, and outputs (tables 1 and 2). • The extremal points can be classiﬁed according to their nonlocality, measured in different ways (table 3). In this sense, the most nonlocal points appear to be boxes 45 and 46, the latter being related to the GHZ correlations in quantum physics. We also mentioned another criterion, based on interconversion of resources, and showed that it would lead to a very different classiﬁcation. • In this scenario, there are also 53 856 Bell-type inequalities (i.e. facets of the local polytope) belonging to 46 inequivalent classes, but we have not found any simple one-to- one correspondence with the classes of points. In fact, boxes 2 (the least nonlocal of all) violate maximally many of the inequalities, while several boxes do not violate maximally any of the inequalities (table 4). Much more information can certainly be extracted from the lists of points and from their properties presented here. For instance, a detailed study of the properties of each no-signaling point is lacking. It would also be interesting to understand how many inequivalent classes of boxes are there with respect to interconversions (for instance, boxes 2, 44, 45, 46 all belong to the same class because from boxes 2 we can obtain boxes 44, 45, and 46, and conversely from boxes 44, 45, and 46 we can obtain boxes 2). Finally, another open possibility is that some of the classes of extremal points may be ‘irrelevant’ for quantum correlations, in the sense that all quantum correlations could be decomposed in extremal no-signaling points without ever using any point belonging to those classes.

Acknowledgments

The list of extremal points was generated some years ago and we acknowledge discussions throughout these years with several colleagues, including Jonathan Barrett, Nicolas Gisin, Thinh Phuc Le, Serge Massar, Sandu Popescu, and David Roberts. This work was supported by the National Research Foundation and the Ministry of Education, Singapore, the Swiss NCCR Quantum Photonics, the European ERC-AG QORE, and the Brussels-Capital region through a BB2B grant.

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[16] Christof T and Loebel A PORTA(available at http://www.iwr.uni-heidelberg.de/groups/comopt/software/PORTA) [17] Fukuda K cdd (available at http://www.ifor.math.ethz.ch/∼fukuda/cdd_home/cdd.html) [18] http://homepages.ulb.ac.be/spironio/tripartite_boxes [19] Greenberger D M, Horne M and Zeilinger A 1989 Bells Theorem, Quantum Theory, and Conceptions of the Universe ed E Kafatos (Dordrecht: Kluwer) p 69 Mermin N D 1990 Am.J.Phys.58 731 [20] Mermin N D 1990 Phys. Rev. Lett. 65 1838 [21] Svetlichny G 1987 Phys. Rev. D 35 3066 [22] Bancal J-D, Barrett J, Gisin N and Pironio S in preparation [23] Scarani V 2006 AIP Conference Proceedings vol 844 (New York: AIP) pp 309–20 [24] Pitowsky I and Svozil K 2001 Phys. Rev. A 64 014102 [25] Sliwa´ C 2003 Phys. Lett. A 317 165–8 [26] Ac´ın A, Durt T, Gisin N and Latorre J I 2002 Phys. Rev. A 65 052325

Paper H

Practical private database queries based on a quantum-key-distribution protocol

M. Jakobi, C. Simon, N. Gisin, C. Branciard, J.-D. Bancal, N. Walenta and H. Zbinden

Physical Review A 83, 022301 (2011)

161

PHYSICAL REVIEW A 83, 022301 (2011)

Practical private database queries based on a quantum-key-distribution protocol

Markus Jakobi,1,2 Christoph Simon,1,3 Nicolas Gisin,1 Jean-Daniel Bancal,1 Cyril Branciard,1 Nino Walenta,1 and Hugo Zbinden1 1Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland 2Humboldt-Universitat¨ zu Berlin, D-10117 Berlin, Germany 3Institute for Quantum Information Science and Department of Physics and Astronomy, University of Calgary, Calgary T2N 1N4, Alberta, Canada (Received 23 February 2010; published 2 February 2011) Private queries allow a user, Alice, to learn an element of a database held by a provider, Bob, without revealing which element she is interested in, while limiting her information about the other elements. We propose to implement private queries based on a quantum-key-distribution protocol, with changes only in the classical postprocessing of the key. This approach makes our scheme both easy to implement and loss tolerant. While unconditionally secure private queries are known to be impossible, we argue that an interesting degree of security can be achieved by relying on fundamental physical principles instead of unveriﬁable security assumptions in order to protect both the user and the database. We think that the scope exists for such practical private queries to become another remarkable application of quantum information in the footsteps of quantum key distribution.

DOI: 10.1103/PhysRevA.83.022301 PACS number(s): 03.67.Dd, 03.67.Hk

I. INTRODUCTION science [3,4] and in quantum information. Classically, the problem seems like a logical contradiction. How could a As telecommunication gains steadily in importance, ques- database provider answer a question, which he is not supposed tions of security and privacy naturally arise. Indeed, private data are stored on a grand scale and have become a precious to know, without giving any additional information? One commodity. Unfortunately, as a matter of principle, classical might hope that quantum mechanics could solve this dilemma. information theory is not able to secure privacy in telecommu- Several quantum protocols were proposed (see, for example, nication against an unlimited adversary. It was hence found Refs. [5,6]), none of which were found to offer complete all the more extraordinary that quantum key distribution protection for both sides. Indeed, it was subsequently proven in (QKD) allows such “unconditionally” private communication, Ref. [7] that the described task cannot be implemented ideally, provided that the two parties trust each other. However, not even using quantum physics. The essential assumption the more general case of communication between distrustful in the impossibility proof is that the protocol is perfectly parties, who wish to protect not only their common privacy concealing, i.e., that Bob has no information whatsoever about against eavesdropping but also their individual privacy against which database element Alice has retrieved. Rephrased at each other, is maybe of even greater interest. the quantum level this is understood as the condition that Private queries are an important problem of this type. the density matrix of Bob’s subsystem must be completely Imagine that a user, Alice, wants to know an element of a independent of Alice’s choice. Reference [7] shows that under database held by a database provider, Bob, but does not want this condition Alice can always implement an attack based on him to know which element she is interested in. Bob in turn the Schmidt decomposition which allows her to read the entire wants to limit the amount of information that she can gain database. This argument is closely linked to the well-known about the database. In particular, he does not want to just hand impossibility proofs for quantum bit commitment [8,9]. over the whole database, which would trivially allow Alice Recently, Giovannetti, Lloyd, and Maccone [10] pointed to learn her bit of interest without giving any information on out that very interesting degrees of privacy are achievable her choice away. It is not hard to imagine scenarios (e.g., for protocols that are not perfectly concealing, because of in the ﬁnancial world) where the capability of implementing the possibility to catch dishonest parties due to the errors such private queries would be useful. The information stored they introduce (see also Refs. [11,13,14]. In the protocol of in the database may be both valuable and sensitive, such that Ref. [10] Alice encodes her question in a quantum state, Bob would like to sell it piece by piece, whereas the mere which she sends to Bob. She also sends a decoy state, fact of being interested in an element of the database might which gives her a chance to detect if Bob is cheating. The already reveal something important about Alice (e.g., that she security relies on the impossibility to perfectly discriminate is thinking about buying a certain company). Of course if there the nonorthogonal question and decoy states and on the were a cheap way of realizing the task, it would also be useful changes Bob’s measurement will introduce as a consequence. for protecting privacy in online bargaining and web search, for Unfortunately the protocol is very vulnerable in realistic example, as well as to construct other interesting cryptographic situations where there are signiﬁcant transmission losses, primitives from it [1]. such that Alice has to send the same question multiple The described task is also known as symmetrically pri- times. If some of the losses are in fact due to Bob tapping vate information retrieval and as 1 out of N oblivious the line, then he can learn Alice’s question without being transfer [2]. It has attracted much attention both in computer detected.

II. CLAIM of a first phase, where a large number of quantum states are prepared, exchanged, and measured, and then a second In this paper we present a new approach to the private query phase, where Alice and Bob extract a key from the quantum problem. Our protocol is explicitly not perfectly concealing in communication part with the help of an apriorichosen coding the above sense, so that the impossibility proof of Ref. [7] does and interpretation process. The key is then known to both Alice not apply. We show that the following statements hold for our and Bob entirely and can be used to encrypt the actual message, protocol. which is sent via a classical channel. The quantum states and (1) Database security is very good. Even for relevant the postprocessing procedure are chosen such that the key multiqubit joint measurements Alice’s accessible information cannot be eavesdropped on without introducing errors, thus is restricted to a well-defined small percentage of the database protecting Alice’s and Bob’s common privacy. elements. The concrete limits for different attacks are shown The basic idea of our protocol is to use QKD in combination in the security discussion. Moreover the additional elements with adequate postprocessing to generate an N-bit string Kf Alice learns are randomly distributed over the database and that will serve as an oblivious key [18] for a database of N bits. therefore of little use to her. In general, database security For this purpose, Kf must be distributed in such a way that is ensured by the impossibility of perfectly distinguishing (1) Bob knows the key entirely, (2) Alice knows only a few bits nonorthogonal quantum states. of Kf —ideally exactly one (database security), and (3) Bob (2) User privacy is also very high. We study several natural does not know which bits are known to Alice (user privacy). attacks and derive a simple limit on the information Bob can In order to use Kf to encrypt the database, Bob adds key obtain. In general, we show that the no-signaling principle and database bitwise with a relative shift chosen by Alice and implies that every malicious action of Bob’s will introduce sends her the encrypted database. The relative shift is needed errors and can hence be detected by Alice—systematic in order to ensure that Alice’s bit of interest is encoded with cheating is impossible. an element of Kf she knows, so that she can decipher the bit The protocol relies on QKD with changes only in the post- and thus receive the answer to her private query. processing and can hence profit from many of the advantages of Within our approach, the case of Alice knowing exactly one this well understood and commercially available technology. bit cannot be realized deterministically. So in general Alice will In comparison to Ref. [10] it offers the advantage of practical know a few bits of Kf , which means that database privacy is feasibility, in particular, loss tolerance and scalability to large good but not perfect. As the number of Alice’s elements is databases. Poisson distributed, there is also a small probability of Alice Note that the incorporation of security assumptions such having no bit in the end. The protocol then needs to be repeated. as the bounded storage model [15] could make the protocol This can be done without loss of privacy for either party: The completely secure, under the condition that those assump- created string Kf does not contain any information on tions are fulfilled. However, even in the absence of such the database, so database security is not touched, and likewise assumptions, our protocol’s basic security is guaranteed by the shift (which maps Alice’s known key element onto the fundamental physical principles, namely, the impossibility of database element she needs) is only communicated once a perfectly discriminating nonorthogonal quantum states and the correct key has been established. Of course, Alice could claim impossibility of superluminal communication. to have obtained no element of Kf with the hope of having It should be underlined that we do not propose an ideal more elements after a repetition. However, this strategy can be cryptographic primitive, which would furthermore allow one made ineffective by choosing the parameters of the protocol to construct other ideal cryptographic primitives such as user such as to make the case of Alice having no element very identification, bit commitment, and coin flipping [1], but rather unlikely (cf. also Sec. V). a new practical and potentially very useful application of As already mentioned, the generation of Kf can be based quantum communication. on QKD techniques. Consider for instance four-state BB84- Our protocol is similar to the proposal of Bennett et al. [5], type QKD. After Bob has sent the states (without further which can be interpreted to rely on the Bennett-Brassard 1984 information), Alice, choosing measurement bases at random, (BB84) QKD [16]. It is well known that the proposal of will measure half of the bits she receives in the correct Ref. [5] is susceptible to a quantum memory attack by the user, basis—without yet knowing for which ones her choice was which corrupts database security entirely. The crucial point is correct. When Bob subsequently announces the bases, we have that Ref. [5] is perfectly concealing, hence Lo’s impossibility the situation that (I) Bob knows the entire “raw key,” (II) Alice proof [7] implies that the user can learn the entire database—in knows half of the bits, and (III) Bob cannot know which ones this case with the help of a quantum memory. We show that this Alice has measured correctly. Alice’s limited information on type of attack can be forestalled by using the Scarani-Acin- the raw key can now be further diluted by adequate processing Ribordy-Gisin 2004 (SARG04) QKD scheme [17] instead in order to generate the oblivious key Kf , and this is indeed of the BB84 protocol. Then user privacy is slightly weakened, the way Ref. [5] essentially works. However, if Alice has a but the quantum memory attack is no longer feasible. Moreover quantum memory this protocol is no longer secure. She can the errors a cheating provider introduces largely guarantee user then store the received states and postpone all measurements privacy. until after Bob’s announcement. By doing so, she can learn Kf entirely—there is hence actually no database security III. APPROACH at all. In order to better understand our approach it is very Fortunately this attack can be largely forestalled rather useful to compare it to QKD. In general QKD consists easily if one uses a SARG-QKD scheme instead of the BB84

022301-2 PRACTICAL PRIVATE DATABASE QUERIES BASED ON A ... PHYSICAL REVIEW A 83, 022301 (2011) protocol. The SARG04 protocol uses the same states as the inconclusive results are kept. Alice and Bob now share a string four-state-BB84 protocol. The main difference lies in the which is known entirely to Bob and in a quarter to Alice. attribution of bit values to the quantum states. Whereas in (6) The created string must be of length k × N (with k the BB84 protocol one state from each of the two bases codes being a security parameter). It is cut into k substrings of length for 0 and the other one for 1, in the SARG04 protocol it is N. These strings are added bitwise in order to reduce Alice’s the basis itself that codes for the bit value. That is, if Bob information on the key to roughly one bit (cf. Fig. 1). sends a state in the “up-down” basis this signifies a 0, and (7) If Alice is left with no known bit after step 6, the protocol a state from the “left-right” basis ↔ means 1. During the has to be restarted. The probability for this to occur can be kept postprocessing Bob does not announce which basis he has small. See also the discussion in the previous and following used for each qubit. Instead Bob announces the state he has sections. sent plus one state from the other basis (in random order). Alice (8) If Kf has been established correctly, Alice will know at is thus faced with a state discrimination problem that cannot f least one element of it. Suppose she knows the jth bit Kj and be solved perfectly, i.e., unambiguously and deterministically wants the ith bit of the database Xi . She then announces the at the same time. This slight change has profound implications number s = j − i in order to allow Bob to encode the database for SARG04 QKD [19]. Here we show that it is also very useful by bitwise adding Kf , shifted by s. So Bob announces N bits for implementing private queries. A simple protocol based on = ⊕ f = ⊕ f Cn Xn Kn+s where Alice can read Ci Xi K and this approach consists of the following steps. j thus obtain Xi . The shift will hence make sure that Alice’s bit of interest is coded with a key element she knows so that the private query can be completed. IV. PROTOCOL (1) Bob sends a long random sequence of qubits (e.g., V. DISCUSSION |↑ |→ |↓ |← |↑ |↓ photons) in states , , , and . States and Steps 1 to 5 of the above protocol are completely identical |← |→ code for 0, and states and correspond to bit value to SARG04 QKD with the only difference that every bit is kept, 1. For instance, to send a bit 1 Bob can prepare a qubit in the regardless if it is conclusive or not for Alice. SARG04 QKD |→ state . was initially conceived to make QKD more resistant to photon ↔ (2) Alice measures each state in the or the basis at number splitting attacks when weak pulses are used instead of random. This alone does not allow her to infer the sent bit single photons for the sake of practical feasibility. In our case value. the use of SARG04 QKD not only provides us with the benefits (3) Alice announces in which instances she has successfully of loss tolerance, technological practicability, and conceptual detected the qubit; lost or not detected photons are disregarded. closeness to well-understood QKD, but it also prevents the The possibility to discard bits does not allow Alice to cheat, quantum memory attack that destroyed the security of the because after step 2 she still has no information whatsoever on protocol of Ref. [5]. Even using a quantum memory Alice the sent bit values (cf. step 5). As a consequence, the protocol is always confronted with the problem of discriminating two is completely loss independent. nonorthogonal quantum states and will hence always have (4) For each qubit that Alice has successfully measured, Bob incomplete knowledge on the raw key. This lack of information announces a pair of two states: the one that has actually been is subsequently further amplified by step 6. {|↑ |→} {|→ |↓} sent and one from the other basis, so , , , , Note that following the “honest” way of measuring and {|↓ |←} {|← |↑} |→ , ,or , .If has been sent, Bob could interpreting her results Alice will also gain probabilistic {|↑ |→} announce, for instance, , . This is exactly as in the information on nonconclusive bits. If Alice obtains no result SARG04 QKD protocol [17]. it is with probability 2/3 because she has chosen the same (5) Alice interprets her measurement results of step 4. basis for measurement as Bob has chosen for state preparation Depending on which basis she has chosen and which result she (which will never yield a conclusive result). Considering the has obtained she will be able to decipher the sent bit value or example of step 5, Alice can obtain the result |→ when |→ {|↑ |→} not. For instance, if has been sent and , has been measuring in ↔ both if Bob sent |→ (then with probability |↑ announced, Alice can rule out only if she has measured 1) and if Bob sent |↑ (then with probability 1/2 only). So, |↓ in the basis and obtained the result . She can then although |→ is not a conclusive result, Alice can infer that |→ conclude that the state was and the bit value is 1. Direct the sent state was |→ (bit 1) with probability 2/3 and |↑ measurement as under step 2 will yield 1/4 of conclusive (bit 0) with probability 1/3. This additional information can results and 3/4 of inconclusive ones. Both conclusive and be diluted to a negligible level by the postprocessing of step 6. After creation of the raw key of k × N bits, the string is divided into k substrings of length N. Following the protocol, after adding the substrings, Alice will on average know = 1 k n¯ N( 4 ) bits, where the number n follows approximately a Poisson distribution. On the other hand, the probability P0 that she does not know any bits at all and that the protocol must = − 1 k N ≈ −n¯ be restarted is P0 [1 ( 4 ) ] e .ForlargeN, which is FIG. 1. How to reduce Alice’s information: her information on a the most interesting case in practice, it is therefore possible to sum string is lower than that on the initial strings. Question marks ensure both n¯ N and small P0 by choosing an appropriate symbolize bits whose value is unknown to Alice. value of k. For instance, for a database of N = 50 000 elements

022301-3 MARKUS JAKOBI et al. PHYSICAL REVIEW A 83, 022301 (2011)

TABLE I. Example of possible choices of k for different database direct measurement. In the above example with N = 50 000 sizes N. We show the failure probability P0 and the expected number and k = 7 this will provide her with n¯ = 9.3 elements on of elements n¯ an honest Alice will obtain. average—only a small gain compared to n¯ = 3 and very little in relation to N = 50 000 for such a complex attack. So even N using a quantum memory, individual measurements will not 103 5 × 103 104 5 × 104 105 106 substantially increase her information on Kf . The reason for this is precisely the fact that our protocol is based on SARG04 k 456779coding rather than on BB84 coding. P0 0.020 0.008 0.087 0.047 0.002 0.022 A more general attack is to store the received photons n¯ 3.91 4.88 2.44 3.05 6.10 3.81 in a quantum memory and to postpone all measurements until the very end of the protocol after step 6, so that she k = 7 is a choice providing Alice with n¯ ≈ 3 elements of the knows which k qubits contribute to an element of the final final key on average whereas the probability of failure is only key. The individual bit values of the raw key are actually about 5% (see also Table I). The case of many repetitions of no interest to her. So, instead of performing the optimal individual measurement on each of the k qubits constituting (which might allow Alice to wait until she obtains a large f value of n by chance) is hence very unlikely. This is important an element of K , Alice should perform a joint measurement. for the protocol’s security. Since the states sent by Bob do not An example for this is Helstrom’s minimal error-probability contain any information about the database, and since Alice measurement, i.e., the measurement that distinguishes two only chooses and communicates the shift s to Bob once she quantum states with the highest information gain [22,23]. In knows at least one bit of the final key, a few repetitions will not the case of two equally likely quantum states ρ0 and ρ1,the compromise anybody’s security. Note that even if Alice knows probability to guess the state at hand correctly is bounded by P = 1 + 1 D(ρ ,ρ ), where D(ρ ,ρ ) is the trace distance. n>1 bits of the oblivious key, she has to pick a single shift guess 2 2 0 1 0 1 f s, which means that in general she can only learn one chosen For a joint Helstrom measurement on a bit of K one finds element of the database, since the other n − 1 bits known to her this probability to scale with the number k of added qubits 1 √1 as Pguess = + . So the more substrings are added to will be at random positions in the key and thus in the database. 2 2 2k However, the fact that Alice normally obtains additional, generate the final key, the harder it is for her to guess the bit less interesting bits should not be seen only as a drawback value, i.e., the parity of the k qubits. For example, for k = 7 of the protocol, as it also offers an interesting possibility to Alice will guess a key element correctly with 54.4% instead enhance her security: Alice can buy the extra bits in question of 50% for a random guess. Likewise, the success probability publicly (as opposed to privately), in order to compare them of unambiguously discriminating the two k-qubit mixed states with Bob’s answers. As explained in detail in the security corresponding to odd and even parity declines rapidly with section, a cheating Bob will always lose knowledge on Kf . the number of qubits k (see Fig. 2). In conclusion, it is clear The errors he thus introduces will then be detectable for Alice. that the impossibility to perfectly distinguish nonorthogonal This way what seems to be a flaw in the protocol can be used quantum states can effectively protect the database’s security to strengthen user privacy. and prevent Alice from knowing a substantial part of it, even when she uses perfect storage technology and realizes the theoretically optimal joint measurements. We see that VI. SECURITY incorporating a SARG04 state discrimination problem as We now turn to the question of which degree of privacy a vital part of the protocol, the Schmidt attack of Lo’s our protocol offers precisely. We study the most evident impossibility proof can be averted. The price to pay is a attacks and clarify the way in which two fundamental physical protection of the user that is not total. We now turn to the principles provide the basis for the protocol’s security. While question of user privacy. basic attacks are studied and the essential intuition is given, a complete security analysis remains work for the future.

A. Database security 0.3 Let us ﬁrst discuss database security. In general one must assume that Alice disposes of a quantum memory and is 0.25 hence not forced to measure directly as in step 2. Instead 0.2 she can keep the photon and, once Bob has announced the 0.15 state pair, apply the optimal unambiguous state discrimination 0.1 (USD) measurement [20,21] that will correctly tell her which of the two announced states has actually been sent. 0.05 0 The success probability of the USD measurement is, for the 1 2 3 4 5 6 7 8 9 10 11 − Upper Bound on USD case of two equally likely states, bounded by 1 F (ρ0,ρ1), combined qubits where F (ρ0,ρ1) is the ﬁdelity between the two quantum states one seeks to discriminate. Here, Alice’s measurement will FIG. 2. (Color online) The upper bound on the success probability hence only√ work with a success probability of 1 −| ↑|→ of the joint unambiguous state discrimination (USD) measurement | = 1 − 1/ 2 ≈ 0.29, only slightly more than the 0.25 of the on k qubits declines rapidly with k.

022301-4 PRACTICAL PRIVATE DATABASE QUERIES BASED ON A ... PHYSICAL REVIEW A 83, 022301 (2011)

B. User privacy matrices cannot be discriminated unambiguously for the As we have discussed above, a not perfectly concealing single-qubit case. The best chance to guess the state correctly protocol, i.e., a protocol where Bob can gain some information is 85.36%, as for the previous attack. The second given on Alice’s choice, is the prerequisite to prevent her from being measurement basis does indeed constitute Helstrom’s minimal able to compromise database security entirely [7]. For the given error probability measurement [22,23] for the conclusiveness protocol it may not be obvious at first sight how Bob can access of one of Alice’s bits. As a matter of fact, one can show that, information on Alice’s choice, in the absence of any classical given an arbitrary mixed qubit state, the likelihood to measure or quantum communication from her to him. It turns out that a conclusive result will be confined by the very same bounds he can indeed gather information on a bit’s conclusiveness and (85.36% and 14.64%). No qubit state can yield only conclusive hence infer if that particular bit is more or less likely to be a results upon the above measurement, or yield only inconclusive key element Alice knows. results. This individual attack is therefore optimal, yields The simplest attack for Bob is to send states other than information on the bit’s conclusiveness, and completely erases | the bit value information from Bob’s register. This last point those he announces, for instance, a state that is exactly f intermediate between |↑ and |→, while announcing a pair means that Bob will not know K correctly—a cheating {|↑,|→}. Alice’s probabilities to measure |↓ or |← are Bob can then be caught when providing wrong answers largely reduced. Indeed, she will find a probability of only [13]. In principle these results can be generalized to joint 14.64% to have such a conclusive result. Likewise sending the measurements on several qubits; however, these complicated state | (orthogonal to | ) while announcing {|↑,|→} attacks are beyond the scope of this paper. Instead we will raise the probability to interpret the result as conclusive to now clarify the conceptual reason why it is impossible for 85.36%. Bob can thus bias the probability of conclusive results Bob to have both the correct bit value and conclusiveness for Alice continuously between the above limits. However, information. every such attack will introduce errors, as Bob cannot predict Let us suppose that Bob can gain information on the her outcome with certainty. In the example above, Alice conclusiveness of one of Alice’s elements of the raw key, either registering |↓ and |←, i.e., both bit values, are equally likely by construction of the sent state or by some measurement events, and Bob’s bit error rate will therefore be as high as 50%. performed on his register at the end of the protocol. Let This evident example shows that Bob can gain information on us characterize this information by pc, the probability with the conclusiveness of Alice’s bits but will then lose information which Bob correctly guesses that Alice has a conclusive result. on the bit values she has recorded. (Remember that this likelihood is physically bounded by pc The presented attack is closely related to an attack that 0.8536 if a single qubit is sent.) Let us also assume that, either uses entanglement. Bob prepares a state of two qubits √1 {| ↑ by construction of the state or by some second measurement, 2 Bob can also guess the bit value b Alice has recorded (if |R +|→ |R }, where the first qubit is sent to Alice A 0 B A 1 B her measurement was conclusive) and is correct about it with and the second is kept in Bob’s register (with R |R = 0). 0 1 B the probability p . Recalling the way Alice interprets her Bob announces having sent |↑ or |→. Once Alice has b measurement results in step 5 of the protocol, it is clear that, if successfully measured and accepted her qubit, Bob can decide Bob correctly guesses that Alice’s result was indeed conclusive if he wants to measure honestly, i.e., recover the sent bit value, and correctly guesses which bit value she has obtained, then or gain some information on the conclusiveness of Alice’s he also correctly guesses which measurement basis she has measurement. In order to proceed honestly Bob measures used for this qubit in step 2. However, since there is no com- his register in the basis {|R ,|R }, which tells him which 0 1 munication whatsoever from Alice to Bob about her choice of of the two announced states has actually been sent [24]. He basis, the no-signaling principle dictates that his probability then knows which bit value Alice will record in case of a to guess her basis correctly has to be equal to 1/2. Otherwise conclusive outcome, but has gained no improved estimation the procedure would allow Alice to send signals to Bob that of the likelihood for this√ to happen. In√ contrast, measuring { | +| | −| } are faster than the speed of light. This immediately implies in the ( R0 R1 )/ 2,( R0 R1 )/ 2 basis provides the bound him with likelihood information on the conclusiveness of × a bit, but clearly yields no information at all on the sent pc pb 1/2. bit value. The inequality arises because even for inconclusive results This second measurement can also be seen from another Bob has a chance to guess Alice’s basis correctly. This angle. If Alice has obtained a conclusive result (probability simple upper bound illustrates the crucial point: Whenever 1/4) Bob’s register is in the state Bob tries to alter the conclusiveness probability of certain 1/20 bits in order to better judge which bits of Kf are (un)known ρ = ; c 01/2 to Alice, he will necessarily lose information on the bit value Alice records in order to comply with the no-signaling if Alice’s measurement was nonconclusive (probability 3/4) principle. This introduces errors in Kf and hence also in the he has √ encrypted database; i.e., he will run the risk of giving wrong 1/2 2/3 answers. = √ ρn . This shows that our protocol is cheat sensitive in the spirit of 2/31/2 Refs. [10,13]. In our scenario, Bob sells his database bit by bit. As ρc = ρn the protocol is not perfectly concealing. Using the Systematic cheating and hence giving wrong answers will ruin criteria of Refs. [20,21] one can show that these two density his reputation as a database provider. As we already mentioned

022301-5 MARKUS JAKOBI et al. PHYSICAL REVIEW A 83, 022301 (2011) above, one can now even make use of the fact that Alice line. Finally, it is possible to improve database security by normally obtains additional database elements. If she buys more sophisticated postprocessing, e.g., by taking a couple those elements from Bob in a regular, nonprivate way, she can of strings created in our probabilistic protocol (with P0 1) use them to check Bob’s honesty [25]. By doing so, Alice has a and allowing Alice to combine them, i.e., to freely choose powerful prompt privacy check at hand. One can thus turn what relative shifts to add them bitwise. Simulations show that seems to be a ﬂaw into an advantage, in order to make full use of she will be left with knowing exactly one bit of the ﬁnal the privacy, which, as we have seen, is guaranteed by the impos- key with overwhelming probability. Both error correction sibility of superluminal communication in quantum physics. and the described way of achieving tighter database security complicate the security analysis due to the necessary two-way communication. VII. OUTLOOK AND CONCLUSIONS The proposed protocol can be realized with any existing The above discussion has shown that practically very QKD system that is compatible with the SARG04 protocol. interesting levels of privacy in database queries can be Besides ensuring loss tolerance, this also makes it easy to scale achieved for both sides. The security of the presented protocol up to large databases. We hope that our proposal will stimulate relies on fundamental physical principles (the impossibility further work to clarify the open questions. Besides a more in- to deterministically discriminate nonorthogonal states and depth study of its security, these include the optimal classical the impossibility of superluminal communication), rather procedures for oblivious key generation and error correction. than on assumptions on quantum storage limitations [15], We think that there is the potential for private queries to become mathematical complexity [3], or noncommunication between a genuine application of quantum information technology in servers in multiserver protocols [4]. the footsteps of QKD. We have already emphasized that the protocol is completely loss resistant. We believe that error correction is possible as ACKNOWLEDGMENTS well. This requires additional classical two-way communication and still needs to be elaborated in more detail. Moreover, We thank G. Brassard, V. Giovannetti, S. Hastings-Simon, it is clear that the protocol can be implemented with weak U. Herzog, L. Maccone, S. Pironio, C. Schaffner, D. Stucki, coherent pulses as well. The acceptable amount of loss then S. Wolf, and J. Wullschleger for useful discussions and in- depends on the mean photon number per pulse, in order sightful comments. Financial support by the Swiss NCCR-QP, to safeguard database security. High mean photon numbers the European ERC-AG Qore, and the EU project QESSENCE largely facilitate unambiguous state discrimination for Alice, is gratefully acknowledged. C.S. was supported by an NSERC if one assumes that she is in control of the transmission Discovery Grant.

[1] J. Kilian, in Proceedings of the 20th STOC (Assoc. Comput. [14] F. de Martini, V. Giovannetti, S. Lloyd, L. Maccone, E. Nagali, Mach., New York, 1988), p. 20. L. Sansoni, and F. Sciarrino, Phys.Rev.A80, 010302 (2009). [2] M. O. Rabin, Technical Report TR-81, Aiken Computation Lab, [15] I. Damgaard, S. Fehr, L. Salvail, and C. Schaffner, in Proceed- Harvard University, 1981. ings of the 27th Annual International Conference on Advances [3] E. Kushilevitz and R. Ostrovsky, in Proceedings of the 38th in Cryptology (Springer, Berlin, 2007), p. 342. Annual Symposium on Foundations of Computer Science (IEEE, [16] C. H. Bennett and G. Brassard, in Proc. IEEE International New York, 1997), p. 364. Conference on Computers, Systems and Signal Processing [4] B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan, in (IEEE, New York, 1984), p. 175. Proceedings of the 36th Annual Symposium on Foundations of [17] V. Scarani, A. Ac´ın,G.Ribordy,andN.Gisin,Phys.Rev.Lett. Computer Science (IEEE, New York, 1995), p. 41. 92, 057901 (2004). [5] C. H. Bennett, G. Brassard, C. Crepeau,´ and M.-H. [18] D. Beaver, Lecture Notes in Computer Science, Vol. 963 Skubiszewska, in Lecture Notes in Computer Science, Vol. 576 (Springer, London, 1995), p. 97. (Springer, London, 1992), p. 351. [19] C. Branciard, N. Gisin, B. Kraus, and V. Scarani, Phys.Rev.A [6] G. Brassard, C. Crepeau,´ R. Jozsa, and D. Langlois, in Proceed- 72, 032301 (2005). ings of the 34th Annual Symposium on Foundations of Computer [20] P. Raynal, e-print arXiv:quant-ph/0611133. Science (IEEE, Washington, 1993), p. 362. [21] U. Herzog and J. A. Bergou, Phys.Rev.A71, 050301 (2005). [7] H.-K. Lo, Phys.Rev.A56, 1154 (1997). [22] C. A. Fuchs, e-print arXiv:quant-ph/9601020. [8] D. Mayers, Phys.Rev.Lett.78, 3414 (1997). [23] C. W. Helstrom, Quantum Detection and Estimation Theory [9] H.-K. Lo and H. F. Chau, Phys.Rev.Lett.78, 3410 (1997). (Academic Press, New York, 1976). [10] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 100, [24] This procedure is equivalent to rolling a quantum die in order to 230502 (2008). randomly decide which state to send. [11] This implies that one has to accept a certain trade off between [25] Once Bob has sent the encrypted database over a classical user privacy and database security. This has been studied channel, he must have measured his quantum register. At this quantitatively for bit commitment [12]. point it is hence no longer possible for Bob to decide to [12] R. W.Spekkens and T. Rudolph, Phys.Rev.A65, 012310 (2001). measure certain bits honestly and others not. The cheating is [13] A. Jakoby, M. Liskiewicz,´ and A. Madry,´ in Lectures Notes in then detectable in the bit error rate of the key, i.e., in the wrong Computer Science, Vol. 5155 (Springer, Berlin, 2008), p. 121. answers he gives.

022301-6 Paper I

Various quantum nonlocality tests with a commercial two-photon entanglement source

E. Pomarico, J.-D. Bancal, B. Sanguinetti, A. Rochdi and N. Gisin

Physical Review A 83, 052104 (2011)

169

PHYSICAL REVIEW A 83, 052104 (2011)

Various quantum nonlocality tests with a commercial two-photon entanglement source

Enrico Pomarico,* Jean-Daniel Bancal, Bruno Sanguinetti, Anas Rochdi, and Nicolas Gisin Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland (Received 7 March 2011; published 3 May 2011) Nonlocality is a fascinating and counterintuitive aspect of nature, revealed by the violation of a Bell inequality. The standard and easiest conﬁguration in which Bell inequalities can be measured has been proposed by Clauser- Horne-Shimony-Holt (CHSH). However, alternative nonlocality tests can also be carried out. In particular, Bell inequalities requiring multiple measurement settings can provide deeper fundamental insights about quantum nonlocality, as well as offering advantages in the presence of noise and detection inefﬁciency. In this paper we show how these nonlocality tests can be performed using a commercially available source of entangled photon

pairs. We report the violation of a series of these nonlocality tests (I3322, I4422, and chained inequalities). With the violation of the chained inequality with 4 settings per side we put an upper limit at 0.49 on the local content of the states prepared by the source (instead of 0.63 attainable with CHSH). We also quantify the amount of true randomness that has been created during our experiment (assuming fair sampling of the detected events).

DOI: 10.1103/PhysRevA.83.052104 PACS number(s): 03.65.Ud, 42.50.Xa, 42.65.Lm

I. INTRODUCTION the moment both loopholes have been closed [5–9], but not yet within the same experiment. Nonlocality is one of the most counterintuitive and fascinat- Nowadays, performing nonlocality tests with entangled ing aspects of nature revealed by quantum theory. Indeed, the photons is relatively easy. Indeed, the technology necessary fact that two separated systems appear to work in a joint way, for generating entanglement is well known and entanglement independently of the distance separating them, does not have sources can be extremely compact, cheap, and stable in time. a counterpart in the classical world. In particular, this bizarre Moreover, polarization entanglement can be analyzed in a effect is predicted by quantum mechanics for two entangled practical and accurate way. Therefore, the experimental study systems and manifests itself in the correlations of the outcomes of entanglement is not confined to a few privileged labora- of the measurements performed on the two systems. This kind tories, but can be also carried out as part of undergraduate of “spooky action at a distance” was the argument that Einstein, laboratory courses. Podolski, and Rosen used to claim the incompleteness of The CHSH inequality requires a rather simple config- quantum mechanics [1]. They pointed out that one could uration: two binary measurement settings on each of two restore locality in physics by assuming the existence of entangled particles. However, in recent years, new types of local variables determining the results of the measurements. Bell inequalities, involving a larger number of measurement However, in 1964 John Bell showed that the correlations of the settings or outcomes with respect to CHSH, have been results of the measurement on entangled particles are stronger investigated from a theoretical point of view. These inequalities with respect to what is expected by any physical theory of local can obtain conclusions unattainable with CHSH: From a fun- hidden variables. By measuring them, a simple inequality (i.e., damental point of view, they allow for a deeper understanding the Bell inequality [2]), based on the locality assumption, can of the nonlocal correlations, whereas from a practical one, they be violated. Therefore, with the term nonlocality we strictly represent useful tools in the presence of noise and detection refer in this paper to the violation of a Bell inequality. inefficiency [10], as we show in the next paragraph. Testing experimentally nonlocal correlations predicted by In this paper, we show that, with a commercial source quantum mechanics became a concrete idea after the Clauser- of photon pairs entangled in polarization (QuTools [11]), a Horne-Shimony-Holt (CHSH) formulation [3] of the original series of nonlocality tests alternative to CHSH can be made. In Bell inequality. Since the CHSH-Bell tests performed by particular, we report the violation of Bell inequalities requiring Aspect in 1981 [4], several experiments have confirmed multiple binary measurement settings. To our knowledge these consistent violations of the CHSH inequality. However, the inequalities (with the exception of I [12]) have not been experimental imperfections, which all these tests are suscepti- 3322 measured before. ble to, open loopholes that can be exploited by a local theory to In Sec. II we justify the interest in these kinds of inequalities reproduce the experimental data. Two relevant loopholes are from a fundamental and practical point of view. In Sec. III the detection and the locality loophole. The former relies on we describe the partial tomography of the density matrix of the fact that particles are not always detected in both channels the states prepared by the source and the optimization of the of the experiment. The latter is related to the necessity of settings for enhancing the violation of the Bell inequalities. In separating the two sites enough to prevent any light-speed Sec. IV we report the violation of inequalities inequivalent to communication between them from the time measurement CHSH, in particular the I [13] and two I inequalities settings are set until the detection events have occurred. At 3322 4422 [14]. Then, we show the violations of chained inequalities [15] from 3 to 6 settings per side. In Sec. VA we interpret the violation of the chained inequalities according to a specific *[email protected] nonlocality approach introduced by Elitzur, Popescu, and

Rohrlich (EPR2) [16]. This allows us to put an upper limit on content of the quantum correlations. For instance, they have the local content of the prepared states that is stronger than the been used to prove that maximally entangled quantum states in one attainable by CHSH. Finally, we show that the observed arbitrary dimensions have a zero local component [17] and to violations allow one to certify that true random numbers have decrease the upper bound on the local content of nonmaximally been created during the experiment. Throughout this work we entangled states [18]. assume fair sampling of the detected events, which allows us From a more practical point of view, inequalities based on to avoid detection loophole issues. multiple settings are also interesting. Recently, it has been shown that in the presence of high-dimensional entanglement, II. BELL INEQUALITIES WITH MULTIPLE that is, when the quantum systems sharing entanglement have MEASUREMENT SETTINGS dimensions larger than two, these inequalities can tolerate a detection efficiency of 61.8% for closing the detection A CHSH-Bell test requires the measurement of each photon loophole [10]. This value is lower with respect to the limit of an entangled pair in two different bases and the estimation imposed by CHSH in experiments of entangled qubits. of the correlations in the four possible combinations of bases. Its practical implementation is conceptually easy and needs minimum experimental effort with respect to other nonlocality tests. Moreover, the CHSH test is particularly robust III. OPTIMIZATION OF THE MEASUREMENT SETTINGS against the noise present in real experiments. However, other FOR A SPECIFIC STATE Bell inequalities, requiring a larger number of measurement Entangled states prepared in the laboratory (or by a settings, schematically represented in Fig. 1, can lead to commercial source) are not perfect in terms of purity and conclusions about the quantum correlations that are nontrivial degree of entanglement. In this case the measurement set- and sometimes inaccessible to CHSH. In general, finding tings needed to observe the largest possible violation of a all the Bell inequalities for a given setup is computationally given inequality do not necessarily coincide with those that difficult; therefore the research is limited to a small number are optimal for a maximally entangled state. In order to of settings per side. Even in this case, these tests can provide find these best measurement settings, a knowledge of the interesting insights about quantum nonlocality. In the case state is required. Usually, a complete reconstruction of the of three possible 2-outcome measurements per side, only one state is not possible or even not necessary. The QuTools inequality, called I3322, is inequivalent to CHSH [13]. Note that source setup [11], which we use in our experiment, projects the this inequality is relevant in the sense that it can be violated photons onto linear polarization states (see the description of by specific mixed 2-qubit states that do not violate CHSH. the experimental setup in the next section), so we can perform The I3322 inequality has also been used to show that three a partial tomography of the state. Once we know the state qubits can share bipartite nonlocality between more than two on the equatorial plane of the Bloch sphere corresponding to subsystems, a result that cannot be obtained with CHSH [13]. linear polarizations, we can optimize the measurement settings In the case of four 2-outcome measurements per side, only in this plane. a partial list of inequalities I4422 has been given [14]. Some of these inequalities are maximally violated, surprisingly, by nonmaximally entangled states, unlike CHSH. A. Partial tomography of a quantum state Another set of inequalities that have recently attracted attention is represented by the chained inequalities [15], which The density matrix ρ of a two-qubit state can always be are generalizations of CHSH with multiple settings. In recent written in the basis composed by the identity 1 and the Pauli { } years, several theoretical models have attempted to provide matrices σx ,σy ,σz as a better understanding of quantum nonlocality. One of these is the Elitzur, Popescu, and Rohrlich (EPR2) approach [16], 1 according to which the observed data could be explained ρ = 1 ⊗ 1 + a σ ⊗ 1 + b 1 ⊗ σ 4 i i j j assuming that only a fraction of the photon pairs produced in an i=x,y,z j=x,y,z experiment possesses nonlocal properties, while the remaining part gives rise to purely local correlations. In this scenario + c σ ⊗ σ , (1) chained inequalities are a useful tool for studying the local i,j i j i,j=x,y,z

where ai =σi ⊗ 1ρ , bj =1 ⊗ σj ρ , and ci,j =σi ⊗ σj ρ are 15 real coefficients which completely define the state. A complete tomography [19] allows the determination of the value of all these coefficients. In our case we only measure linear polarizations, so a complete knowledge of the state is not necessary in order FIG. 1. (Color online) Nonlocality tests where Alice and Bob to predict all possible measurement statistics. Indeed, only the 8 coefficients a , b , c with i,j = x,z are useful. These measure the N binary operators A1,...,AN and B1,...,BN , i j i,j respectively, on the photon pairs produced{ by the} entanglement{ source} coefficients can be measured with our setup, realizing a partial in the middle. tomography of the generated state.

052104-2 VARIOUS QUANTUM NONLOCALITY TESTS WITH A ... PHYSICAL REVIEW A 83, 052104 (2011)

B. Optimization of the settings The measurement of a qubit along a particular angle θ in the xz plane of the Bloch sphere can be represented by the measurement operator

O(θ) = cos θσz + sin θσx . (2) If the two-qubit state (1) is shared between Alice and Bob, the following marginal values are expected if Alice measures along angle α and Bob along β: FIG. 2. (Color online) Simple sketch of the polarization entanglement source. Linear polarizers at the Alice and Bob site are used for E(α) =A(α) ⊗ 1 = cos αa + sin αa , (3) ρ z x the settings necessary for the measurement of the Bell inequalities. E(β) =1 ⊗ B(β)ρ = cos βbz + sin βbx . (4) IV. NONOLOCALITY TESTS WITH MULTIPLE SETTINGS Moreover, the joint correlations are found to be A. The entanglement source = ⊗ E(α,β) A(α) B(β) ρ We use the commercial entanglement source sold by the = cos α cos βcz,z + cos α sin βcz,x QuTools company [11], to which we brought some minor modifications. This source generates photon pairs entangled + sin α cos βcx,z + sin α sin βcx,x. (5) in polarization from spontaneous parametric down conversion A Bell inequality I with N settings per side [20] can be (SPDC) in a bulk BBO crystal. The nonlinear crystal is cut generally defined by the following formula for a type II phase matching and is pumped by a continuous wave diode laser at 405 nm. Photon pairs at 810 nm are N N N generated, filtered, and collected into single-mode fibers. I = n E(α ) + m E(β ) + l E(α ,β ) I , i i j j ij i j L Linear polarizers at the Alice and Bob site can measure = = = i 1 j 1 i,j 1 polarizations respectively at the angles α and β with respect (6) to the vertical direction, as shown by the sketch in Fig. 2. They allow performing a partial tomography of the entangled where αi and βi are the respective angles of the measurements in Alice’s and Bob’s site, n , m , and l are coefficients state. Notice that a full tomography could be done by just i j ij inserting quarter wave plates. Photons are then detected using defining the inequality, and IL is the local bound. The inequality I can be represented schematically as a table: silicon avalanche photodiodes (Si-APDs) with efficiencies at ⎛ ⎞ the wavelength of the photons of 48% and 55%, respectively. m1 m2 ... mN The coincidences are measured by a time-to-digital converter ⎜ ⎟ ⎜n1 l11 l12 ... l1N ⎟ (TDC), and a coincidence rate of 4.2 kHz is measured for = ⎜ ⎟ 15 mW of input pump power in the absence of the linear I ⎝n2 l21 l22 ... l2N ⎠ IL. (7) ...... polarizers. Other details on this source can be found in [21]. nN lN1 lN2 ... lNN For a pair of polarization directions measured by the two polarizers, the number of coincidences in a time interval of 20 s Note that this table has the same form as the one introduced is measured. The error associated with the coincidence rate is in [13], but here coefficients correspond to expectations values estimated according to Poissonian statistics. The inequalities and not to probabilities. that we want to test require the measurement a large number It is clear from Eq. (6) that, for a given quantum state, the of correlations. One correlation term is given by measuring term I is a function of the measurement angles on Alice’s one photon in one basis and the other photon in another one. { } { } ( α1,...,αN ) and Bob’s site ( β1,...,βN ); that is, Measuring one photon in one basis means projecting it into two orthogonal polarization directions. So the measurement I = I(α ,...,α ,β ,...,β ). (8) 1 N 1 N of one correlation term requires four different settings for the This function is not linear in terms of its 2N variables, but two polarizers. numerical optimizations can be used in order to find the optimal measurement settings for Alice and Bob, which will B. The characterization of the state provide the largest possible value of I for the state under We perform a partial tomography of the state generated by consideration. the source on the plane of the Bloch sphere corresponding to For the nonlocality tests reported in this paper, we perform the linear polarizations. We determine the 8 correlation values a partial tomography of the polarization entangled states that indicated in Sec. III A, demanding to measure on each side the we prepare, as explained in the previous section. This allows operators in polarization: us to reconstruct the state in the plane of the Bloch sphere corresponding to linear polarizations. Then, we numerically σz ≡|HH|−|V V |,σx ≡|DD|−|AA|, find the optimal measurement settings for this state, restricting 1 ≡|HH|+|V V |≡|DD|+|AA|, them to lie on the xz plane. These settings are used for the experimental violation of the Bell inequalities taken in where |H and |V correspond to the states of horizontal and vertical polarization, respectively, and |D(A)= √1 (|H± consideration. 2

052104-3 POMARICO, BANCAL, SANGUINETTI, ROCHDI, AND GISIN PHYSICAL REVIEW A 83, 052104 (2011)

TABLE I. List of the expectation terms for the partial tomography the notation given in Sec. III B, we can represent the CHSH of the prepared entangled state. The measured values (Expt.) can be inequality by the following table: compared to the theoretical ones (Theor.) that are expected from a ⎛ ⎞ perfect singlet state. The values of the marginal terms obtained by 00 = ⎝ ⎠ measuring the identity 1 with the basis {|H ,|V } or {|D,|A} are ICHSH 0 11 2. (9) indicated with (z) and (x), respectively. 0 −11 For the CHSH test with optimal settings we obtain a value Expectation Term Expt. Theor. of 2.731 ± 0.015, clearly enhanced with respect to 2.691 ±

σz ⊗ σz−0.9649 ± 0.0012 −1 0.015, obtained by adopting the standard settings. These σx ⊗ σx −0.9344 ± 0.0017 −1 two values correspond to violations of the local bound of σz ⊗ σx 0.1053 ± 0.0045 0 respectively 49 and 46 standard deviations. This confirms the σx ⊗ σz−0.0201 ± 0.0048 0 validity of the settings’ optimization. Actually, we expected (z) σz ⊗ 1 0.0993 ± 0.0046 0 from the partial tomography a value for CHSH of 2.683 (x) σz ⊗ 1 0.0315 ± 0.0048 0 with optimal settings and 2.662 with the standard ones. This ⊗ (z) ± σx 1 0.0228 0.0047 0 little discrepancy could be explained by small variations of ⊗ (x) ± σx 1 0.0498 0.0045 0 the prepared state between the tomography and the final ⊗ (z) − ± 1 σz 0.0986 0.0046 0 measurements. (x) 1 ⊗ σz −0.0582 ± 0.0045 0 (z) 1 ⊗ σx −0.0339 ± 0.0047 0 (x) D. Inequalities inequivalent to CHSH 1 ⊗ σx 0.0033 ± 0.0048 0 We then measure Bell inequalities inequivalent to CHSH, in particular I3322 [13] and two different types of I4422, called AS and AS [14,22]. These inequalities are facets of their | 1 2 V ) to the diagonal and the antidiagonal ones. In Table I a list corresponding Bell polytope [20]. They are thus optimal to of the measured expectation terms for the partial tomography detect nonlocality for some correlations in scenarios involving of the prepared state is given. three and four settings. Note that I3322 asks to measure not only The errors associated with the expectation terms are joint correlations, but also four marginal probabilities. On the obtained simply by statistical propagation of the error in the contrary, the two I4422 inequalities that we want to measure are coincidences on which they depend. The marginal terms have the only two of this kind to require correlation terms uniquely. {| | } been measured twice, using the basis H , V (indicated This makes their measurement simpler from a conceptual and (z) {| | } (x) with )or D , A (indicated with ) in the site where the experimental point of view. In the following, the coefficients identity has to be measured. The full set of measurements has of these inequalities are given: been taken into account in the tomographic reconstruction. ⎛ ⎞ Note that errors on the measured expectation terms in Table I 110 ⎜ − − − ⎟ need to be considered in order to define from the partial = 1 1 1 1 I3322 ⎝ ⎠ 4, (10) tomography a density matrix which is definite positive. In 1 −1 −11 some cases, in order to avoid the negativity of the density 0 −11 0 matrix attainable from a tomography, some techniques, such as ⎛ ⎞ the maximum likelihood estimation, need to be employed [19]. 00 0 0 ⎜ ⎟ It is evident that the singlet state prepared with the commercial ⎜0 11 1 1⎟ = ⎜ − ⎟ source has some imperfections. There is a mixture component AS1 ⎝0 11 1 1⎠ 6, (11) in the state since the expectation values are slightly different 0 11−20 from the theoretical ones. The state is also unbalanced in 0 1 −10 0 {| | } {| | } the two orthogonal bases H , V and D , A , which ⎛ ⎞ confirms that it is not maximally entangled. 0000 The problem of optimization of the measurement settings ⎜ ⎟ ⎜0 2112⎟ for the nonlocality tests is necessarily limited to the plane = ⎜ − ⎟ AS2 ⎝0 11 2 2⎠ 10. (12) on which we have limited the tomography. We measure 0 12−2 −1 9 inequalities requiring in total 332 settings of the linear 0 2 −2 −1 −1 polarizers in the Alice and Bob site. These optimal settings Note that this notation is different with respect to that used have in some cases differences of some degrees with respect to expt the standard settings used for obtaining a maximum violation in [14]. We call I the experimental value of the Bell with a perfect singlet state. parameter. For each of the three different inequalities, we observe a violation of the local bound IL (Table II). In Table II the result for CHSH is that with optimal settings. The violations for the two I4422 inequalities are stronger than for I3322.We C. CHSH inequality evaluate this aspect by considering the resistance to noise of First of all, in order to check the validity of our optimization the three different inequalities. If some white noise were added method, we measure the CHSH inequality using the standard with probability pnoise to our state ρ, we would obtain a state = 1 + − settings for the singlet state, then the same inequality with with a density matrix ρnoise pnoise 4 (1 pnoise)ρ.Now, the settings optimized for the prepared state. According to it is easy to show that the violation would not be observed

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TABLE II. Measurement of the CHSH inequality and of the keep the violations decreases for an increasing number of expt inequalities inequivalent to CHSH. IL is the local bound, I settings. is the value of the Bell parameter obtained experimentally with the optimized settings, I tom is the expected value from the partial expt V. APPLICATION OF THE CHAINED INEQUALITIES tomography, I − IL is the difference between the obtained value and the local bound in terms of number of standard deviations σ ,and As recalled in Sec. II, the violation of the chained inequal- pnoise (%) is the critical level of white noise that can be added to the ities can be linked with numerous problems. In the following system while still keeping a violation. we brieﬂy mention what the violations we observed allow us to conclude with respect to the EPR2 model of nonlocality and expt − I IL pnoise to true randomness. expt tom IL I I (σ units) (%)

ICHSH 2 2.731 ± 0.015 2.683 49 27 A. EPR2 nonlocality ± I3322 4 4.592 0.024 4.769 25 13 Chained inequalities can be used to put an upper bound ± AS1 6 7.747 0.026 7.750 67 23 on the local content of the prepared state according to the ± AS2 10 12.85 0.030 12.819 95 22 EPR2 approach [16]. Indeed, as already explained in Sec. II, one can imagine that only part of the photon pairs produced by the source has nonlocal properties, while the other one − IL anymore if pnoise > 1 I expt .ForI3322 adding 13% of white behaves in a purely local way. In such a situation, the measured noise to the state compromises the violation. For the other (N) value of Ichain decomposes as a sum of the local bound IL inequalities the tolerance to the noise is higher and the CHSH with probability p and of some possibly larger value I inequality is confirmed as being the most robust to noise. L with probability 1 − pL. The latter satisfies the no-signaling In all the cases the obtained violations agree quite well with (NS) principle, i.e., the impossibility of faster-than-light what we expected with the tomography. communication. Therefore, I expt = p I (N) + (1 − p )I (N). (14) E. Chained inequalities L L L NS Finally, we measure chained inequalities with a number Since the local and the no-signaling values of the I quantity (N) = − (N) = of settings per side from 2 to 6. The chained inequality are at most IL 2(N 2) and INS 2N, respectively, the with N settings for Alice ({α1,α2,...,αN }) and N for Bob local part of the produced state is bounded by the maximum max ({β1,β2,...,βN }) can be written as a correlation inequality as value pL : expt (N) max I I = E(α1,β1) + E(β1,α2) + E(α2,β2) +··· p p = N − . (15) chain L L 2 + − − E(αN ,βN ) E(βN ,α1) 2(N 1). (13) To illustrate this bound, let us see how it applies to noisy sin- Contrary to the previous inequalities, only the chained inequal- = | − −|+ − 1 = glet states, i.e., Werner states ρW V ψ ψ (1 V ) 4 . ity with N 2, which is equivalent to CHSH, is a facet of the (N) π For these states the maximum value of I is 2NV cos( 2N ). Bell polytope. max Since the chained inequalities require 2N correlation terms, Thus, the maximum local probability pL associated with the number of settings required for their measurement scales a Werner state ρW depends on the visibility V of the state and max = ( − π ) linearly with N. We thus limit ourselves to 6 settings per is given by pL N 1 V cos( 2N ) . The behavior of this side. In Table III a list of the measured inequalities is given. function is shown in Fig. 3 for different values of the visibility = max For N = 2 we have the result for the CHSH inequality with V . For the pure singlet state (V 1), pL tends to 0 for an optimal settings. All the inequalities are violated in a way infinite number of settings, confirming the fact that the singlet consistent with the expected results and with a large number of standard deviations of difference with respect to the local bound. However, it is interesting to note that the larger the number of settings, the weaker the violation. Indeed, the fraction of noise that we can add to the system and still

TABLE III. Measurement of the chained inequalities with N settings per side.

expt I − IL pnoise expt tom N IL I I (σ units) (%)

2 2 2.731 ± 0.015 2.683 49 27 3 4 4.907 ± 0.019 4.925 48 18 4 6 7.018 ± 0.023 6.999 44 15 5 8 8.969 ± 0.026 8.996 37 11 6 10 10.91 ± 0.028 10.954 33 8 FIG. 3. (Color online) Largest local part associated with a Werner state for V = 1, 0.98, 0.96, 0.94, and 0.92 (from bottom to top).

052104-5 POMARICO, BANCAL, SANGUINETTI, ROCHDI, AND GISIN PHYSICAL REVIEW A 83, 052104 (2011)

Quantum bound No−signaling bound for all N 0.9

0.8 (a|x) * P 0.7

0.6

N=2 N=3 N=4 N=5 N=6 0.5 2N−2 2N−3/2 2N−1 2N−1/2 2N IN

FIG. 4. (Color online) Largest local part as a function of the FIG. 5. (Color online) Upper bound on the marginal probability number of settings per side. The red line joins the expected values distribution as a function of the inequalities violation. The bound max ± for our state. The minimum measured value of pL is 0.491 0.012 implied by the no-signaling principle is identical for all chained with a chained inequality of 4 settings per side. inequalities. state is fully nonlocal [17]. On the contrary, when V is smaller so that the detected pairs of particles fairly represent the ones max than 1, pL decreases until a certain limit value. Therefore, produced by the source. the number of settings which are useful for lowering the upper In order to quantify the amount of true randomness that bound on the local content of the state is limited and changes can be found in some experimental results, one must consider according to the visibility V . all the marginal probabilities P (a|x) that Alice finds the We observe a similar effect for the state produced by outcome a when she measures x, which are compatible with our source. In Fig. 4 we represent the violations of the the observed Bell inequality violation I expt. We searched for measured chained inequalities in terms of local probabilities, the largest marginal probability P (a|x) numerically among all as calculated from (15). The red line joins the expected values possible quantum correlations, i.e., correlations that can be max for pL obtained by the knowledge of the state. Our state is achieved by measuring a quantum state, as well as among more complex than a Werner state but it can be approximated all no-signaling correlations, denoting this largest quantity by a Werner state with visibility V = 0.94. The minimum by P ∗(A|X). The computed upper bounds P ∗(A|X)forthe max ± value of pL that we measure is 0.491 0.012 with a chained chained inequalities with up to 6 settings per party are shown inequality of 4 settings per side. This result means that at least in Fig. 5 together with our experimental results. half of the photon pairs produced by the source are nonlocal, a The lowest bound on the marginal probability that we result that cannot be shown with the CHSH inequality. To our can certify here is P ∗(A|X) = 0.684 ± 0.014, achieved for knowledge, this is the first experiment that fixes an upper bound the CHSH inequality. However, if we consider the bound on the local part of the quantum correlations according to the imposed by the no-signaling principle only, then the strongest EPR2 approach [23]. Similar work in progress is expected to one is P ∗(A|X) = 0.7455 ± 0.0057, achieved by the chained show a lower value of this quantity [24]. inequality with N = 4 settings per party. Note that in order to extract a truly random bit string out of measured outcomes, classical key distillation techniques B. Randomness certified by the no-signaling principle should be used [27]. The ratio between the number of measured When measuring a singlet state, locally random outcomes bits and the number of truly random, uniformly distributed can be observed. It has been recently shown that violation bits that is produced by this procedure is given by the min- of a Bell inequality can certify that this randomness truly entropy of Alice’s outcome A conditioned on her measurement | =− ∗ | emerges during the experiment, in the sense that no algorithm choice X: Hmin(A X) log2 maxa P (A X). In our case we can possibly predict the measured outcomes [25]. Indeed, if find Hmin = 0.55 ± 0.03 for the CHSH violation, meaning that such algorithm existed prior to the experiment, it could be approximately one random bit every two measurements has considered as a local hidden variable, and no violation of a Bell been created. inequality can be observed with only local hidden variables. Note that in order to discard any such algorithm, the detection VI. CONCLUSIONS loophole should be closed during the experiment. Indeed, such an algorithm could in principle inform detectors when they We have shown that nontrivial nonlocality tests alternative should click according to some detection loophole model [26]. to CHSH can be performed even with a rather simple In order to avoid this issue, we assume fair sampling of the commercial source. In particular, we have measured Bell detected events: The state of the particles coming onto a inequalities with multiple measurement settings. We have detector does not affect the fact that a detector fires or not, reported violations of I3322,oftwoI4422, and of chained

052104-6 VARIOUS QUANTUM NONLOCALITY TESTS WITH A ... PHYSICAL REVIEW A 83, 052104 (2011) inequalities with up to 6 settings per side. Violations of these produced in the experiment. This work emphasizes the richness inequalities have not been shown before. Moreover, by using of nonlocality and the importance of the no-signaling and could the chained inequalities, we have put an upper bound on the be of interest for undergraduate laboratory courses. local content of the prepared state at 0.491 ± 0.012, meaning that at least half of the photon pairs produced by the source ACKNOWLEDGMENTS have nonlocal correlations. We have also quantiﬁed the amount of true randomness created in the experiment. Therefore, even This work is supported by EU project Q-ESSENCE and with an extremely simple setup, it is possible to implement by the Swiss NCCR-Quantum Photonics. We would like to nontrivial nonlocality tests and get interesting conclusions on thank Antonio Ac´ın, Valerio Scarani, and Henning Weier for the nonlocal properties of the source and on the randomness valuable discussions, suggestions, and remarks.

[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 [13] D. Collins and N. Gisin, J. Phys. A 37, 1775 (2004). (1935). [14] N. Brunner and N. Gisin, Phys. Lett. A 372, 3162 (2008). [2]J.S.Bell,Phys.1, 195 (1964). [15] S. Braunstein and C. Caves, Ann. Phys. (NY) 202,22 [3] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. (1990). Rev. Lett. 23, 880 (1969). [16] A. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A 162,25 [4] A. Aspect, P. Grangier, and G. Roger, Phys.Rev.Lett.47, 460 (1992). (1981). [17] J. Barrett, A. Kent, and S. Pironio, Phys. Rev. Lett. 97, 170409 [5] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (2006). (1982). [18] V. Scarani, Phys.Rev.A77, 042112 (2008). [6] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and [19] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. A. Zeilinger, Phys. Rev. Lett. 81, 5039 (1998). Rev. A 64, 052312 (2001). [7] W. Tittel, J. Brendel, N. Gisin, and H. Zbinden, Phys. Rev. A 59, [20] R. F. Werner and M. M. Wolf, Quantum Inf. Comput. 1, 1 (2001). 4150 (1999). [21] P. Trojek, C. Schmid, M. Bourennane, H. Weinfurter, and [8] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. C. Kurtsiefer, Opt. Express 12, 276 (2004). Itano, C. Monroe, and D. J. Wineland, Nature (London) 409, [22] D. Avis, H. Imai, and T. Ito, J. Phys. A 39, 11283 (2006). 791 (2001). [23] Note that existing experimental results could be reinterpreted to [9] D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, provide such a bound as well. and C. Monroe, Phys.Rev.Lett.100, 150404 (2008). [24] L. Aolita et al. (unpublished). [10] T. Vertesi,´ S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, [25] S. Pironio, A. Ac´ın, S. Massar, A. Boyer de la Giroday, D. N. 060401 (2010). Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. [11] See [http://www.qutools.com]. Manning, and C. Monroe, Nature (London) 464, 1021 (2010). [12] J. B. Altepeter, E. R. Jeffrey, P. G. Kwiat, S. Tanzilli, N. Gisin, [26] N. Gisin and B. Gisin, Phys. Lett. A 297, 279 (2002). and A. Ac´ın, Phys. Rev. Lett. 95, 033601 (2005). [27] S. Ronen, Bull. Eur. Assoc. Theor. Comput. Sci. 77, 67 (2002).

052104-7

Paper J

Device-Independent Witnesses of Genuine Multipartite Entanglement

J.-D. Bancal, N. Gisin, Y.-C. Liang and S. Pironio

Physical Review Letters 106, 250404 (2011)

179

week ending PRL 106, 250404 (2011) PHYSICAL REVIEW LETTERS 24 JUNE 2011

Device-Independent Witnesses of Genuine Multipartite Entanglement

Jean-Daniel Bancal,1 Nicolas Gisin,1 Yeong-Cherng Liang,1 and Stefano Pironio2 1Group of Applied Physics, University of Geneva, Geneva, Switzerland 2Laboratoire d’Information Quantique, Universite´ Libre de Bruxelles, Brussels, Belgium (Received 1 February 2011; published 24 June 2011) We consider the problem of determining whether genuine multipartite entanglement was produced in an experiment, without relying on a characterization of the systems observed or of the measurements performed. We present an n-partite inequality that is satisﬁed by all correlations produced by measurements on biseparable quantum states, but which can be violated by n-partite entangled states, such as Greenberger-Horne-Zeilinger states. In contrast to traditional entanglement witnesses, the violation of this inequality implies that the state is not biseparable independently of the Hilbert space dimension and of the measured operators. Violation of this inequality does not imply, however, genuine multipartite nonlocality. We show more generically how the problem of identifying genuine tripartite entanglement in a device- independent way can be addressed through semideﬁnite programming.

DOI: 10.1103/PhysRevLett.106.250404 PACS numbers: 03.65.Ud, 03.67.Mn

The generation of multipartite entanglement is a central Suppose, however, that the measurement Y3 carries a objective in experimental quantum physics. For instance, slight (possibly unnoticed) bias towards the x direction, entangled states of 14 ions and 6 photons have recently i.e., we actually measure Y3 ¼ cosy þ sinx. Then been produced [1,2]. In any such experiment, a typical one sees that, all other measurements being ideal, question arises: How can we be sure that genuine jc i¼1 ðj00iþeij11iÞ ðj0iþ the biseparable state 2 pABffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n-partite entanglement was present? An n-partite state is 2 j1iÞC, where ¼ arctanðsinÞ, yields hMi¼2 1 þ sin , n said to be (genuinely) -partite entangled if it is not bise- which is strictly larger than 2 for any 0. Thus, unless parable, that is, if it cannot be prepared by mixing states we measure all particles exactly along the x and y direc- that are separable with respect to some partition. For tions, we can no longer conclude that observing hMi > 2 instance, a tripartite state bs is biseparable if it admits a implies tripartite entanglement. Importantly, this is not a decomposition X X X unique feature of the above witness, but rather all conven- k k k k k k tional witnesses are, to some extent, susceptible to such bs ¼ AB C þ AC B þ BC A; (1) k k k systematic errors that are seldom taken into account. Furthermore, tomography and usual entanglement wit- where the weight of each state in the mixture has been nesses typically assume that the dimension of the Hilbert included in its normalization; a state that cannot space is known. For instance, in an experiment demonstrat- be written as above is tripartite entangled. Determining ing, say, entanglement between four ions, we usually view n whether -partite entanglement was produced in an experi- each ion as a two-level system. But an ion has many ment represents a difficult problem that has drawn much degrees of freedom (position, vibrational modes, internal attention lately (see, e.g., [3,4]). The usual approach con- energy levels, etc.). Given the inevitable imperfections in sists of measuring a witness of genuine multipartite entan- experiment, is it justifiable to treat the relevant Hilbert glement, or of doing a full state tomography followed by a space of each ion as two dimensional and how does this direct analysis of the reconstructed density matrix. simplification affect our conclusions about the entangle- Such approaches, however, not only rely on the observed ment present in the system [6]? Even if it is justified to statistics to deduce the presence of entanglement, but also view each ion as a qubit, is entanglement between four require a detailed characterization of the systems observed systems really necessary to reproduce the measurement and of the measurements performed. Consider, e.g., the data, or could they be reproduced with fewer entangled following witness of genuine tripartite entanglement: systems if qutrits were manipulated instead? M ¼ X1X2X3 X1Y2Y3 Y1X2Y3 Y1Y2X3; (2) These remarks motivate the introduction of entanglement witnesses that are able to guarantee n-partite entan- where Xj ¼ x and Yj ¼ y are the Pauli spin observables glement, without relying on the types of measurements in the x and y direction for particle j. For any biseparable performed, the precision involved in their implementation, three-qubit state hMi¼trðMÞ2 [5]. Thus if measure- or on assumptions about the relevant Hilbert space dimen- 1 ments on three spin- 2 particles give an average value sion. We call such witnesses device-independent entangle- hMi > 2, we can conclude that the state is tripartite ment witnesses (DIEW). This type of approach was entangled. already considered in [7,8]. Note that other solutions to

0031-9007=11=106(25)=250404(4) 250404-1 Ó 2011 American Physical Society week ending PRL 106, 250404 (2011) PHYSICAL REVIEW LETTERS 24 JUNE 2011 the above problems are possible, such as entanglement x; y; z 2f1; ...;mg (corresponding, e.g., to the values of witnesses tolerating a certain misalignment in the mea- macroscopic knobs on the measurement apparatuses) and surement apparatuses [9] or the characterization of realistic denote the corresponding classical outcomes a; b; c 2 measurement apparatuses through squashing maps [6]. f1; ...;dg. The correlations obtained in the experiment These other approaches, however, still require some partial are characterized by the joint probabilities PðabcjxyzÞ of characterization of the system and measurement appara- finding the triple of outcomes a; b; c given the measure- tuses, which is not necessary when using DIEWs. ment settings x; y; z. Any DIEW is a Bell inequality (i.e., a witness of non- We say that PðabcjxyzÞ are biseparable quantum corre- locality). Indeed, (i) the violation of a Bell inequality lations if they can be reproduced through local measure- implies the presence of entanglement, and (ii) any mea- ments on a biseparable state bs, i.e., if there exists a surement data that do not violate any Bell inequality can be biseparable quantum state (1) in some Hilbert space H , M M M reproduced using quantum states that are fully separable measurement operators ajx, bjy, and cjz (which with- [10]. The violation of a Bell inequality is thus a necessary out loss of generality we can takeP to be projections satisfy- M M M M 1 and sufficient condition for the detection of entanglement ing ajx a0jx ¼ a;a0 ajx and a ajx ¼ ), such that in a device-independent (DI) setting. This observation is PðabcjxyzÞ¼tr½Ma x Mb y Mc zbs: (3) the main insight behind DI quantum cryptography [11,12], j j j where the presence of entanglement is the basis of security. If given quantum correlations PðabcjxyzÞ are not bi- n The relation between DIEW for -partite entanglement separable, they necessarily arise from measurements on and witnesses of multipartite nonlocality is more subtle. a tripartite-entangled state, and this conclusion is inde- While there exist Bell inequalities that detect genuine pendent of any assumptions on the type of measurements n -partite nonlocality [13–15], not every DIEW for performed or on the Hilbert space dimension. n -partite entanglement corresponds to such a Bell inequal- Equivalently, biseparable quantum correlations can be ity. Consider, for instance, the expression (2). If no as- defined as those that can be written in the form sumptions are made on the type of systems observed and X k k measurements performed, the inequality hMi2 corre- PðabcjxyzÞ¼ PQðabjxyÞPQðcjzÞ k sponds to Mermin’s Bell-type inequality [16]; i.e., a value X hMi > 2 necessarily reveals nonlocality, hence entangle- k k pffiffiffi þ PQðacjxzÞPQðbjyÞ ment. Moreover, a value hMi > 2 2 guarantees tripartite k X entanglement [7,14]. The Mermin expression (2) can thus k k þ PQðbcjyzÞPQðajxÞ; (4) be used as a tripartite DIEW. Yet, it cannot be used k as a Bell inequality for genuine tripartite nonlocality, since k k a simple model involving communication between two where PQðabjxyÞ and PQðcjzÞ correspond, respectively, to parties only already achieves the algebraic maximum arbitrary two-party and one-party quantum correlations; M 4 Pk ab xy tr Mk Mk k h i¼ [14]. i.e., they are of the form Qð j Þ¼ ½ ajx bjy AB The objectives of this Letter are to formalize the concept Pk c z tr Mk k and Qð j Þ¼ ½ cjz C for some unnormalized quan- of DIEW for genuine multipartite entanglement and initiate k k Mk Mk tum states AB, C and measurement operators a x, b y, a systematic study that goes beyond the early examples j j Mk [and similarly for the other terms in (4)]. Here, the [7,8]. We will start by introducing the notion of quantum cjz biseparable correlations. We then present a simple measurement operators for different bipartitions need not DIEW for n-partite entanglement which is stronger for be the same (though this can always be achieved as shown Greenberger-Horne-Zeilinger (GHZ) states than all the in- in Sec. D of [17]). Clearly, from the definition (1)of equalities introduced in [7,8]. For n ¼ 3, we also provide a biseparable states, any correlations of the form (3) are of general method for determining whether given correlations the form (4). Conversely, any correlations of the form (4) are also of the form (3), see Sec. A of [17]. reveal tripartite entanglement and apply it to GHZ and W Let Q3 denote the set of tripartite quantum correlations states. Apart from yielding practical criteria for the char- and Q2=1 Q3 the set of biseparable quantum correla- acterization of entanglement in a multipartite setting, our tions. From (4), it is clear that Q2=1 is convex and that its results also clarify the relation between device-independent Pext ab xy Pext c z multipartite entanglement and mulitpartite nonlocality. extremal points are of the form Q ð j Þ Q ð j Þ, where Pext ab xy Q Biseparable quantum correlations.—For simplicity of Q ð j Þ is an extremal point of the set 2 of bipartite ext exposition, let us consider an arbitrary tripartite system quantum correlations and PQ ðcjzÞ an extremal point of the (the following discussion easily generalizes to the n-party set Q1 of single-party correlations (the extreme points of case). To characterize in a DI way its entanglement Q1 are actually classical, deterministic points). Since the properties, we consider a Bell-type experiment: on each set Q2=1 is convex, it can be characterized by linear in- subsystem, one of m possible measurements is per- equalities. Those linear inequalities separating Q2=1 from formed, yielding one of d possible outcomes. We label Q3 correspond to DIEWs for tripartite entanglement. Since the measurements on each of the three subsystems by Q2=1 has an infinite number of extremal points, there exist 250404-2 week ending PRL 106, 250404 (2011) PHYSICAL REVIEW LETTERS 24 JUNE 2011 an infinite number of such inequalities. Note that the set of proof of this statement is based on the decomposition (4) P k k k local correlations PðabcjxyzÞ¼ kP ðajxÞP ðbjyÞP ðcjzÞ and is given in Sec. B of [17]. The Svetlichny bound is contained in Q2=1. Hence, any DIEW for tripartite en- associated to the expression In, on the other hand, is easily tanglement is a Bell inequality (though not necessarily a found to be 4 3n2 > 2 3n3=2 (see Sec. B of [17]); for tight one). Note also that the decomposition (4) corre- the local bound of In, see [18]. sponds to a Svetlichny-type decomposition [13] where all We now illustrate how this DIEW can be used to detect bipartite factors are restricted to be quantum, whereas less genuine multipartite entanglement. For this, let us consider n restrictive constraints [or even none in Svetlichny’s origi- the noisy GHZ state ¼ VjGHZinnhGHZjþð1 VÞ1=2 P k k nal definition PðabcjxyzÞ¼ kP ðabjxyÞP ðcjzÞþ] characterized by the visibility V. Carrying out the mea- xi1 1 xi1 1 are imposed on these bipartite terms in the definitions of surements cos½ð 3 6nÞx þ sin½ð 3 6nÞy on n1=2 multipartite nonlocality [13]. It follows that the set of all parties, we obtain In ¼ 3 V, which violates (5) genuinely tripartite nonlocal correlations is larger than provided that V>2=3. The DIEW (5) can thus detect in the set of biseparable quantum correlations as illustrated a DI way n-partite entanglement in a noisy GHZ state for in Fig. 1. Thus, while any Bell inequality detecting genuine V 2=3 visibilities as low as ¼ . This significantlypffiffiffi improves tripartite nonlocality is a DIEW for tripartite entanglement, over the threshold visibility V ¼ 1= 2 required to violate the converse is not necessarily true. All these observations the DIEW based on the Mermin expression (2) or the n extend to the -party case. different inequalities introduced in [8]. Note that in a DI n A DIEW for -partite entanglement.—We now present setting it is not possible to detect the tripartite entangle- n i a DIEW for parties, where each party performs a ment in tripartite GHZ states below V ¼ 1=2 using x 1; 2; 3 measurement i 2f g and obtains an outcome projective measurements; in this case, there exists a bise- a 1; 1 E x i 2f g. We denote ð Þ the correlator associated parable model reproducing all GHZ correlations (see x x ...;x to the measurement settingsP ¼ðQ1 nÞ, i.e., the Sec. C of [17]). E x P a x n a a expectation value ð Þ¼ a ð j Þ i¼1 i, where ¼ n 3 I a ...;a n Ek In the case ¼ , the DIEWpffiffiffi (5) takes the form 3 ¼ Pð 1 P nÞ denotes an -tuple of outcomes. Let n ¼ 3 4 6 7 9 n E3 þ E3 E3 E3 þ E3 6 3. It therefore involves ð xi ¼ kÞEðxÞ be the sum of correlators EðxÞ for x i¼1 only 18 expectation values, compared to 27 for a full which the measurement settings xi of the n parties sum up tomography of a three-qubit system. Let us stress, however, to k. Let fk be a function such that fkþ3 ¼fk and take that contrary to usual entanglement witnesses I3 is not successively the values ½1; 1; 0 on the integers k ¼ 0; 1; 2. restricted to two-dimensional Hilbert spaces, even though Then the inequality it uses observables with binary outcomes. For instance, if 3n X all parties perform the measurements 2jðxiÞihðxiÞj 1 k n3=2 In ¼ fk nEn 2 3 (5) 1 ið6x 7Þ=18 with jðxiÞi ¼ pffiffi ðj0iþe i j1iÞ on the three-qutrit k¼n 2 ffiffiffi 1 p state pffiffi ðj000iþj111iþj222iÞ, then I3 ¼ 6 3 þ 8=3, is satisfied by all biseparable quantum correlations, and is 3 thus a DIEW for genuine n-partite entanglement. The showing that the state is tripartite entangled. General characterization of biseparable quantum correlations in the case n ¼ 3.—Though the DIEW (5) seems particularly well adapted to GHZ states, we cannot expect a single nor a finite set of DIEW to completely characterize the biseparable region, as illustrated in Fig. 1. It is thus desirable to derive a general method to decide whether arbitrary correlations are biseparable. Here we show how the semidefinite programming (SDP) techniques introduced in [19,20] can be used to certify that the correlations observed in an experiment are tripartite entangled. Our approach is based on the observation that the tensor product separation AB C at the level of states, cf. definition (1), can be replaced by a commutation relation FIG. 1. A particular slice of the space of tripartite correlations at the level of operators. Specifically, let s ¼fAB=C; with three settings and two outcomes representing schematically AC=B; BC=Ag denote the three possible partitions of the the sets of general quantum (Q3), Svetlichny (S2=1), and bisepar- parties into two groups. Then, PðabcjxyzÞ are biseparable able quantum correlations (Q2=1). The point 1 corresponds to quantum correlations if and only if there exist three arbitrary random correlations and P to the GHZ correlations maximally (not necessarily biseparable) states s and three sets of violating the DIEW (5), which is represented by the straight line Ms ;Ms ;Ms measurement operators f ajx bjy cjzg such that I3; note that a DIEW can be violated by Svetlichny-local correlations. (The Svetlichny polytope S2=1 can be determined X Q Q P abc xyz tr Ms Ms Ms s ; exactly using linear programming, while 3 and 2=1 can be ð j Þ¼ ½ ajx bjy cjz (6) approximated efficiently using SDP techniques; see main text.) s 250404-3 week ending PRL 106, 250404 (2011) PHYSICAL REVIEW LETTERS 24 JUNE 2011 where measurement operators corresponding to an isolated TABLE I. Summary of numerical investigations. MAB=C;MAB=C 0 party commute, i.e., ½ cjz c0jz0 ¼ , and similarly for State Vmin with two settings Vmin with three settings pffiffiffi the other partitions. The equivalence between (3)and(6)is GHZ 0:7071 ’ 1= 2 0:6667 ’ 2=3 established in Sec. D of [17]. The problem of determining W 0:7500 ’ 3=4 0.7158 whether given correlations PðabcjxyzÞ are biseparable thus amounts to finding a set of operators satisfying a finite number of algebraic relations (the projection defining rela- s s s P s This suggests that the DIEWs based on (2) and (5) actually tions of the type M M 0 ¼ a;a0 M , M ¼ 1 and ajx a jx ajx a ajx form part of a larger family of m-settings DIEWs. This the commutation relations mentioned above) such that (6) deserves further investigation. A second problem is to holds. Such a problem is a typical instance of the SDP derive simple DIEWs that are adapted to W states and approach introduced in [19,20] (see Sec. D of [17]). that, in particular, reproduce the threshold visibilities ob- Specifically, it follows from the results of [20] that one can tained in Table I. Finally, we have shown a practical define an infinite hierarchy of criteria that are necessarily method to characterize tripartite biseparable correlations satisfied by any correlations of the form (6) and which can be using SDP. It would be interesting to understand how this tested using SDP. If given correlations do not satisfy one of generalizes to the n-partite case. A possibility would be these criteria, we can conclude that they reveal tripartite to combine the approach of [19,20] with the symmetric entanglement. Further, it is possible in this case to derive extensions of [21]. This question will be investigated an associated DIEW from the solution of the dual SDP. elsewhere. Modulo a technical assumption, it can be shown that the This work was supported by the Swiss NCCRs QP and hierarchy of SDP criteria is complete; that is, if given QSIT, the European ERC-AG QORE, and the Brussels- correlations are not biseparable this will necessarily show Capital region through a BB2B grant. up at some finite step in the hierarchy. Application to GHZ and W states.—Using finite levels of this hierarchy and optimizing over the possible measurements, we investigated the minimal visibilities above [1] T. Monz et al., Phys. Rev. Lett. 106, 130506 (2011). GHZ W W which the GHZ state jpffiffiffi i and the state j i¼ [2] W. Wieczorek et al., Phys. Rev. Lett. 103, 020504 (2009). ðj001iþj010iþj100iÞ= 3 exhibit correlations that are [3] O. Gu¨hne and G. To´th, Phys. Rep. 474, 1 (2009). not biseparable (and thus reveal genuine tripartite entan- [4] O. Gu¨hne and M. Seevinck, New J. Phys. 12, 053002 glement) in the case of two and three measurement settings (2010); B. Jungnitsch, T. Moroder, and O. Gu¨hne, Phys. per party. Our results are summarized in Table I. For GHZ Rev. Lett. 106, 190502 (2011); M. Huber et al., Phys. Rev. A 83, 022329 (2011). states, the reported visibility Vmin ¼ 2=3 for three mea- [5] G. To´th et al., Phys. Rev. A 72, 014101 (2005). surements per party corresponds to the threshold required [6] T. Moroder et al., Phys. Rev. A 81, 052342 (2010). to violate the DIEW (5), suggesting that this DIEW is [7] M. Seevinck and J. Uffink, Phys. Rev. A 65, 012107 (2001). optimal in this case. In the case of two measurements per [8] K. Nagata, M. Koashi, and N. Imoto, Phys. Rev. Lett. 89, party, we couldpffiffiffi not lower the visibility below the threshold 260401 (2002); J. Uffink, ibid. 88, 230406 (2002);M. Vmin ¼ 1= 2, which corresponds to the visibility required Seevinck and G. Svetlichny, ibid. 89, 060401 (2002). to violate the DIEW based on Mermin expression (2) and [9] M. Seevinck and J. Uffink, Phys. Rev. A 76, 042105 (2007). the DIEWspffiffiffi introduced in [7,8]. Note, however, that for V>1= 2 the GHZ state violates Svetlichny’s inequality [10] R. F. Werner, Phys. Rev. A 40, 4277 (1989). [11] A. Acıń et al., Phys. Rev. Lett. 98, 230501 (2007). [13] and thus exhibits genuine tripartite nonlocality. Thus [12] S. Pironio et al., Nature (London) 464, 1021 (2010). for the GHZ state the DIEWs introduced in [7,8] do not [13] G. Svetlichny, Phys. Rev. D 35, 3066 (1987). improve over what can already be concluded using the [14] D. Collins et al., Phys. Rev. Lett. 88, 170405 (2002). standard notion of tripartite nonlocality. On the other [15] J.-D. Bancal et al., Phys. Rev. Lett. 106, 020405 (2011). hand, our numerical explorations suggest that the visibil- [16] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990). ities Vmin ¼ 2=3 for GHZ states with three measurements [17] See supplemental material at http://link.aps.org/ and Vmin ¼ 3=4 for W states with two measurements supplemental/10.1103/PhysRevLett.106.250404 for our cannot be attained using the notion of genuine tripartite biseparable model and the proofs related to Eqs. (4)–(6). nonlocality, illustrating the interest of the weaker notion [18] M. Z˙ ukowski and D. Kaszlikowski, Phys. Rev. A 56, of DIEW. R1682 (1997). Discussion.—To conclude, we comment on some pos- [19] M. Navascue´s, S. Pironio, and A. Acıń, Phys. Rev. Lett. 98, 010401 (2007); New J. Phys. 10, 073013 (2008). sible directions for future research. First of all, note that by [20] S. Pironio, M. Navascue´s, and A. Acıń, SIAM J. Optim. identifying the measurement settings ‘‘Xi’’ with xi ¼ 1 and Y x 2 20, 2157 (2010). ‘‘ i’’ with i ¼ , the two-setting DIEW based onffiffiffi Mermin [21] B. M. Terhal, A. C. Doherty, and D. Schwab, Phys. Rev. 3 5 p expression (2) can be written as E3 E3 2 2, which is Lett. 90, 157903 (2003); A. C. Doherty, P.A. Parrilo, and of the same general form as the three-setting DIEW (5). F. M. Spedalieri, Phys. Rev. Lett. 88, 187904 (2002).

250404-4 Supplementary material for “Device-independent witnesses of genuine multipartite entanglement”

Jean-Daniel Bancal1, Nicolas Gisin1, Yeong-Cherng Liang1, Stefano Pironio2 1Group of Applied Physics, University of Geneva, Switzerland 2Laboratoire d’Information Quantique, Universit´eLibre de Bruxelles, Belgium (Dated: May 20, 2011)

Appendix A: Biseparable quantum correlations can be fixed by introducing local ancillas on each system acting as labels that indicate which term Here, we show that the definitions (3) and (4) of bisep- in the decomposition is considered: defining ρbs = kt kt ( t, kt t, kt )1 ... ( t, kt t, kt )n ρ ρ , arable correlations are equivalent. For the sake of gener- t kt | ih | ⊗ ⊗ | ih | ⊗ t ⊗ t′ ality, we prove this equivalence in the n-partite scenario. M = ( t, k t, k ) M kt we finally get ai xi t kt t t i a x P P| | ih | ⊗ i| i Let t denote a subset of 1,...,n containing from one P (¯a x¯) = tr M ... M ρ , and thus any { } P Pa1 x1 an xn bs to n 1 elements, and let t′ = 1,...,n t denote the | | ⊗ ⊗ | − { }\ correlations of the form (A4) can be written as in complementary subset. The pair (t,t′) thus represent a Eq. (A2). partition of the n parties into two non-empty groups. Let Let Qt denote the set of quantum correlations defined ρt ρt be an (unormalized) n-partite state that is prod- ⊗ ′ for the parties belonging to t and let Qbs denote the set uct across the partition (t,t′). An n-partite state ρbs is of biseparable quantum correlations, which is a subset then biseparable if it can be written as a mixture of the n-partite quantum correlations. From (A3), it is clear that Q is convex and that its extreme points are ρ = ρkt ρkt . (A1) bs bs t t′ ext ext ext ⊗ of the form PQ (¯at x¯t)PQ (¯at′ x¯t′ ) where PQ (¯at x¯t) is t kt | | | X X an extremal point of the set Qt. A state that cannot be written in the above form is genuinely n-partite entangled. Let P (a1 ...an x1 ...xn) = P (¯a x¯) be an n-partite Appendix B: DIEWs for genuine n-partite probability distribution| characterizing| a Bell experiment entanglement with measurement settingsx ¯ = (x1 ...xn) and measurement outcomesa ¯ = (a1 ...an). We say that the cor- Here we derive the maximum value that the expres- relations described by P (¯a x¯) are biseparable quantum | sion In, see Eq. (5), can take for measurements made correlations iff on a biseparable quantum state (the biseparable bound), and for Svetlichny models (the Svetlichny bound). From P (¯a x¯)=tr Ma x ... Ma x ρbs (A2) | 1| 1 ⊗ ⊗ n| n (A3), it follows that the biseparable bound is obtained for some biseparable state ρbs and measurement oper- by maximizing In over all correlations of the form ators Ma x . Equivalently, as we will see, biseparable i| i quantum correlations can be defined as those that can P (¯a x¯)= P kt (¯a x¯ )P kt (¯a x¯ ) . (B1) | Q t| t Q t′ | t′ be written in the form t X Xkt P (¯a x¯)= P kt (¯a x¯ )P kt (¯a x¯ ) , (A3) | Q t| t Q t′ | t′ In a Svetlichny model, on the other hand, one considers t X Xkt correlations of the form where we writea ¯t andx ¯t for the outcome and measure- P (¯a x¯)= P kt (¯a x¯ )P kt (¯a x¯ ) , (B2) ment settings of the parties belonging to the subset t | t| t t′ | t′ k k k t kt and where PQ(¯at x¯t)=tr i t Ma x ρt are arbitrary X X | ∈ i| i (unnormalized) quantum correlationshN for thei parties in t. i.e. no constraints (apart from positivity and normal- kt That any correlations of the form (A2) is of the form ization) are imposed on the joint terms P (¯at x¯t). But (A3) is immediate using the definition (A1) of biseparable more refined models àla Svetlichny can be introduced,| states. Conversely, (A3) can be rewritten in the form see e.g. [1], the more constraining one being the one where the joint terms P kt (¯a x¯ ) are assumed to be no- t| t kt kt kt signalling. We will compute the bound on In assuming P (¯a x¯)= tr Ma x ρt ρt . (A4) | i| i ⊗ ′ these no-signalling constraints and see that even in this t k " i n # X Xt O∈ case there exists a gap between the quantum biseparable This gives a representation that is almost of the and Svetlichny bounds (note also that the no-signalling form (A2), except that a different set of measure- bound that we will derive actually coincides with the ment operators corresponds to each term in the de- more general unconstrained Svetlichny bound because In composition (A1) of the biseparable state ρbs. This is an inequality involving only full n-partite correlators). 2

k xn+1 We will now prove the biseparable bound given in where the n-partite quantity En− (an+1, xn+1) Eq. (5) for n 3 using induction on the number of parties depends on the values of an+1 and xn+1 n. Let us start≥ with the first step of the induction, which since it is evaluated on the distribution consists of showing that the inequality holds for n = 3. P (a x )P (a x ,a , x ). Q 1,...,k| 1,...,k Q k+1,...,n| k+1,...,n n+1 n+1 By linearity of I3 in the probabilities, convexity of the de- Inserting this expression in the definition In+1 = composition (4) for tripartite biseparable quantum cor- 3(n+1) k fk nEn+1, we obtain k=n+1 − relations and the fact that I3 is invariant under any permutation of the parties, it is sufficient to prove the bound P 3 1 xn+1 for correlations of the form P (ab xy)P (c z). Moreover, In+1 = an+1PQ(an+1 xn+1)In (an+1, xn+1) Q Q | | | x =1 a = 1 it is sufficient to consider the case where PQ(c z) is ex- nX+1 n+1X− tremal, i.e., one where every c is determined unambigu-| (B5) j 3n+3 k j ously as a function of z. To this end, let us label the eight where we have defined In = fk nEn− = k=n+1 − distinct deterministic assignments of outcome c for given 3n k k k=n fk n+j En (note that En is different from zero input z by γ = (γ ,γ ,γ ) 1 3, where the determin- − P j 1 2 3 only if n k 3n) and write In(an+1, xn+1) istic (quantum) probability∈ {± distributions} are such that P ≤ ≤j to remind that In is evaluated on the distribu- PQ(c z) = Pγ (c z) = 1 if γz = c and Pγ (c z) = 0 oth- tion P (a x )P (a x ,a , x ) | | | Q 1,...,k| 1,...,k Q k+1,...,n| k+1,...,n n+1 n+1 erwise. Substituting these eight possible strategies into which depends on an+1 and xn+1. Using the fact that I3 [see (5)] gives eight different bipartite Bell expressions an+1 = 1 and that the probabilities P (an+1 xn+1) are for the system AB: bounded± between 0 and 1, it follows from (B5)| that

6 3 k I3,γ = gγ(k)E2 (B3) xn+1 In+1 In (an+1, xn+1) . (B6) k=2 ≤ | | X xn+1=1 3 X where gγ(k)= z=1 γzfk+z 3. Determining the bisepa- j j − Let max In = max In denote the maxi- rable bound of I3 therefore amounts to finding the max- | | ± j P mal value of the quantity In taken by any imum of the above Bell expressions over arbitrary bipar- biseparable correlations. Since± the distribution tite quantum correlations PQ(ab xy) = tr[Ma x Mb yρ], | | ⊗ | PQ(a1,...,k x1,...,k)PQ(ak+1,...,n xk+1,...,n,an+1xn+1) is i.e., finding the corresponding Tsirelson bounds. It can | | xn 1 biseparable, we clearly have that In − (an+1, xn+1) be easily verified that the eight expressions I3,γ are either j | | ≤ max In, independently of the values of an+1 and xn+1. vanishing or equivalent, under relabelling of inputs and We thus± find outputs, to two times the 3-input chained Bell inequal- 3 ity [2]. The biseparable bound on I3 is thus equal to the I max Ij . (B7) two times the Tsirelson bound of the 3-input chained Bell n+1 ≤ ± n inequality [3], i.e., I 6√3, which is in agreement with j=1 3 ≤ X Eq. (5). An important point to note now is that Ij for all Next, we need to show that whenever inequality (5) ± n j = 1, 2, 3 are equivalent to In, i.e., one can obtain the holds for n parties, it must also hold for (n + 1) par- j expression for In by starting from any of In and ap- ties. By linearity of In in the probabilities and convex- plying a different labeling for the inputs and/or± outputs. ity of the decomposition (B1), it is sufficient to com- To see that this is the case, we first note from the def- pute the bound over all product correlations of the 3 inition of fk that In = In. In other words, we can form PQ(¯at x¯t)PQ(¯aˆ x¯ˆ). Moreover, since In+1 is sym- 3 − | t| t obtain In by starting from In and applying the mapping metric under any permutation of parties, it is suffi- k an an to each of the En. Clearly, the same argument cient to consider the biseparations t = 1, 2,...,k , →− j j { n } also demonstrates the equivalence between In and In. t′ = k + 1,...,n + 1 where 1 k . With − { } ≤ ≤ ⌊ 2 ⌋ Next, note that if the following mappings are applied to obvious notation, we write the corresponding corre- k the definition of En, namely xn xn′ = (xn mod 3)+1 lations as PQ(a1,...,k x1,...,k)PQ(ak+1,...,n+1 xk+1,...,n+1). → | | and an an′ = an except for xn = 3 in which case Bayes’ rule and the fact that quantum correlations sat- → j an′ = an, then we can also obtain the expression for In isfy the no-signaling principle allow us to write the −j+1 from In , thus showing their equivalence. We thus have quantum correlations for parties k +1 to n +1 as j that max I max In and thus P (a x ,a )P (a x ) = ± n ≤ Q k+1,...,n| k+1,...,n+1 n+1 Q n+1| k+1,...,n+1 P (a x ,a , x )P (a x ). With In+1 3max In . (B8) Q k+1,...,n k+1,...,n n+1 n+1 Q n+1 n+1 ≤ straightforward| algebra, it can be seen that| the quantity k n+1 Now, by the induction hypothesis we know that max In En+1 = x¯ δ( i=1 xi = k)E(¯x) for this type of distri- n 3/2 ≤ 2 3 − for all biseparable correlations . It then follows bution can be rewritten as × n 1/2 P P from Eq. (B8) that In+1 2 3 − , which completes 3 1 the proof. ≤ × Ek = a P (a x ) n+1 n+1 Q n+1| n+1 × Note that, following the above reasoning, any witness x =1 a = 1 (B4) nX+1 n+1X− for n parties and m inputs per party which can be writ- k xn+1 k E − (an+1, xn+1) ten in the form fkE , where fk : Z R is a function n k n → P 3 satisfying fk+m = fk, can be generalized to more par- We check that the outcomes produced by the model are ties. ± identical to the ones found when measuring state (C1) in Note also that the above biseparable bound for the the given bases. witness (5) is tight, as it can always be achieved by performing the following local measurements 1. The model cos φ(x )σ + sin φ(x )σ : i =1,...,n 1 i x i y − , σz : i = n Before the parties receive their inputs, a common (B9) source chooses a party p at random, say Charlie (p = C), x 1 1 with φ(xi) = ( −3 + 6(n 1) )π on the biseparable state and sends him the vector −

ψn = GHZn 1 0 . (B10) ~λ = (sin α cos β, sin α sin β, cos α), (C2) | i | − i ⊗ | i Finally, let us note that the same procedure as the uniformly chosen on the sphere S2. The source then also one detailed above can be followed to obtain the (no- provides the two other parties, Alice and Bob in this case, signalling) Svetlichny bound of In+1. The only differ- with the quantum state ence is that in the first step of the induction proof, we α α iβ must compute the no-signalling bound of the I3 inequal- ΦAB = cos 00 + sin e− 11 . (C3) ity instead of the quantum biseparable bound. As in | i 2 | i 2 | i the quantum biseparable case, this reduces to computing Moreover, the source sends to all parties the signs twice the (bipartite) no-signalling bound of the 3-input s ,s ,s = 1, which are independently and iden- AB AC BC ± chained Bell inequality, which gives I3 2 6 = 12 and tically distributed with Prob(s = +1) = 2+√3 . n 2 ≤ × 4 thus I 4 3 − . To show that this bound is tight con- n ≤ × At the time of measurement, Alice and Bob measure sider a strategy where the n-th party is separated from their system according to ~x and ~y and get outcomes a,b = the rest and always outputs 1. Using an analysis similar 1, while Charlie calculates c = sg(~λ ~z). The parties to the one leading to Eq. (B6), we can then see that the ±then output respectively A = s s a· , B = s s b remaining (n 1) parties are playing an effective game AB AC AB BC 1 − 2 3 and C = sBC sAC c. defined by In 1 + In 1 + In 1 which can be shown to be − − − equivalent to twice the expression In 1. It is a simple − exercise to show that the algebraic maximum of In 1 is 2. Correlations obtained by the model n 2 − 2 3 − and that this is always achievable by (n 1) players× that are constrained only by the no-signalling− princi- Here we compare the correlations that are created ple. Therefore, with this particular strategy, one achieves n 2 when Alice, Bob and Charlie apply the biseparable model 4 3 − , which saturates the bound derived above. × above to the ones that they would get by measuring state (C1). Since the model treats equally each party, its Appendix C: Biseparable model for projective correlations are symmetric under exchange of the parties measurements on the tripartite GHZ state and we only need to check the following three relations: A =0 (C4) Here we present a biseparable model that simulates von h i 1 Neumann measurements on noisy tripartite GHZ states AB = cos θ cos θ (C5) h i 2 x y 000 + 111 000+ 111 11 1 ρ = V | ih | + (1 V ) (C1) ABC = sin θx sin θy sin θz cos(ϕx + ϕy + ϕz). (C6) 2 − 8 h i 2 1 Here we parametrized the parties’ measurements in terms of visibility V = 2 . Clearly this allows to simulate states 1 of spherical coordinates: with V < 2 as well, by mixing this model with another one in which all parties produce uniformly random out- ~x = (sin θ cos ϕ , sin θ sin ϕ , cos θ ) (C7) comes. x x x x x The model is presented as a protocol in which a source and similarly for ~y and ~z. distributes quantum states and random variables to the For definiteness, in the following we use the brackets parties. All parties can share the random variables, but q to express the expectation value of a quantity q with since only biseparable states may be used in the model, respecth i to a quantum state, and the bar not more than two parties at a time can share a quantum state. After distribution, the parties receive their 1 2π π respective measurement directions ~x, ~y or ~z belonging f(α, β)= dβ dα sin αf(α, β) (C8) 4π 0 0 to the Bloch sphere, and measure their quantum system Z Z and/or process the information they received accordingly for the average of a function f(α, β) over the random in order to produce binary outcomes A, B, C = 1. variable α and β. We always start by considering that ± 4

Charlie is the special party chosen by the source, to Similarly one can check that bc = 1 cos θ cos θ , and h i 2 y z whom the hidden vector ~λ is sent, and perform the so once averaged over the choice of the party being alone, the bipartite correlations are given by symetrization afterwards. Symmetrized quantities q = 1 (q + q + q ) are indexed by the symbol . 3 p=A p=B p=C 2 For simplicity we average over the depolarization signs ab = cos θx cos θy. (C12) h i 3 sAB,sAC ,sBC only at the end of the calculation.

a. Single-party expectation value One can easily ver- Finally, we need to apply the signs sAB, sAC , sBC . ify that the outcomes of each party is locally random ac- Since the expectation value of the product of two signs 3 cording to the model, in agreement with equation (C4). is sABsAC = 4 , it follows that overall the correlation betweenh twoi outcomes produced by the model is b. Two-party correlators Let us ﬁrst calculate cor- 1 relations ab , ac and bc for the case in which Charlie AB = cos θx cos θy, (C13) h i h i h i h i 2 receives the vector ~λ, and average over the choice of the party p being alone afterwards. in agreement with equation (C5). The ﬁrst term is easily found from equation (C3), which gives c. Three-party correlators We now proceed to calculate the tripartite correlations that are created by the model. Let us consider the three preceding cases sepa- ab = cos θx cos θy + sin α sin θx sin θy cos(β + ϕx + ϕy) h i (C9) rately again: and thus ab = cos θx cos θy. π h i 1. α 2 θz. In this case c = +1, so abc = ab as given≤ by− equation (C9). h i h i

To compute the correlation ac , it is useful to write π h i 2. α + θz. In this case c = 1, so abc = ab . Alice’s state as ≥ 2 − h i −h i π π α α 3. 2 θz α 2 + θz. In this case one has abc = ρ = tr ( Φ Φ ) = cos2 0 0 + sin2 1 1 . − ≤ ≤ h i A B AB AB ab (χ + (β) χ (β)). | ih | 2 | ih | 2 | ih | h i I − I− (C10) In total, the tripartite correlation are found after inte- The expectation value for Alice’s outcome is then a = h i gration over α and β to be cos α cos θx. Concerning Charlie, his outcome c is totally deter- 1 abc = sin θ sin θ sin θ cos(ϕ + ϕ + ϕ ), (C14) mined by ~λ and ~z. For simplicity we assume that h i 2 x y z x y z 0 θz π/2, but the other situation can be treated ≤ ≤ π which does not depend on which party p is alone. similarly. In this case, for α 2 θz, Charles al- ≤π − One can check that the application of the signs sij ways has c = +1, and for α 2 + θz, he always has c = 1. Now for α [ π θ≥, π + θ ], we can write has no influence on the tripartite correlations since they − ∈ 2 − z 2 z cancel out, and so ABC = abc . The model thus re- + = [ Φc + ϕz, Φc + ϕz] the interval of β for which the I − produces the expectedh correlationsi h i (C6). product ~λ ~z 0, and = [ϕz π, ϕz + π] + its com- plement, with· ≥ Φ = arccos(I− cot−α cot θ ). We\I thus have c − z the three following cases: Appendix D: Characterizing biseparable correlations through SDP in the tripartite case (n = 3) π 1. α 2 θz. In this case c = +1, which gives ac≤= cos− α cos θ . h i x It has been shown in [4] how to define a hierarchy of SDP that characterizes the set Q of bipartite quantum 2. α π + θ . In this case c = 1, which gives 2 2 z correlations. Note that biseparable quantum correlations ac≥= cos α cos θ . − h i − x P (abc xyz) can be written as a finite sum of such bipar- | k π π tite quantum correlations using the fact that each P (c z) 3. θz α + θz. In this case one has ac = 2 − ≤ ≤ 2 h i in (4) can be taken to be extremal and using the fact that| cos α cos θx(χ + (β) χ (β)) where I − I− there are a finite number of such extremal points corresponding to classical, deterministic strategies. It there- 1 if β χ (β)= ∈ I (C11) fore follows that biseparable quantum correlations can I 0 if β / ( ∈ I be characterized using a finite number of the SDP hier- archies introduced in [4]. Note however that the num- is the indicator function. ber of single-party deterministic strategies, and thus the number of terms in Eq. (4), grows exponentially with In total, after integration over α and β, this gives the number of measurement settings, making such an ac = 1 cos θ cos θ . approach impractical even for small problems. Here we h i 2 x z 5

s s s introduce an alternative SDP approach that has better trC[Mc ρ Mc ] is a proper (unormalized) state of sys- c s s scaling properties. tem AB. Define PQ(ab xy) = tr[Ma x Mb yρc] and | | ⊗ | Our approach is based on the observation that the ten- P c(c z) as the deterministic point satisfying P c(c z) = sor product separation ρ ρ at the level of states | | AB ⊗ C δ(cz,c). It is then easy to see using the properties of can be replaced by a commutation relation at the level of s s s s s the operators Mc z that tr[Ma x Mb y Mc zρ ] = operators. Specifically, let s = AB/C,AC/B,BC/A | s| ⊗ | ⊗ | { } c δ(cz,c)tr[Ma x Mb y Mc ρ ]= c δ(cz,c)tr[Ma x denote the three possible partitions of the parties into |c ⊗ | ⊗c | ⊗ Mb yρc] = c PQ(ab xy)P (c z). This last expression is two groups. Then, P (abc xyz) are biseparable quan- P | | | P | of the same form as the first series of terms in (4) (since tum correlations if and only if there exist three arbitrary P c s any deterministic point P (c z) can be seen as a single- (not necessarily biseparable) states ρ and three sets of | measurement operators M s ,M s ,M s such that (6) party quantum point PQ(c z)). A similar argument for { a x b y c z} the other values of s implies| that any correlations of the holds, i.e. | | | form (D1) are of the form (4). s s s s We thus have reduced the problem of determining P (abc xyz)= tr[Ma x Mb y Mc zρ ] , (D1a) | | ⊗ | ⊗ | whether given correlations P (abc xyz) are biseparable to s | X the problem of finding a set of operators satisfying a finite where measurement operators corresponding to an iso- number of algebraic relations (the projection defining re- s s s s lations of the type M M = δa,a M , M = 11 lated party commute, i.e., a x a′ x ′ a x a a x and the commutation| relations| (D1b))| such that| (D1a) P AB/C AB/C holds. Such a problem is a typical instance of the SDP [Mc z ,Mc z ]=0 | ′| ′ approach introduced in [5], which generalizes the results AC/B AC/B [Mb y ,Mb y ]=0 (D1b) of [4]. Specifically, it follows from the decomposition | ′| ′ BC/A BC/A (D1) and the results of [5] that it is possible to define [M ,M ]=0. j a x a′ x′ an infinite hierarchy of sets Q : j = 1,..., with | | { 2/1 ∞} the following properties: i) deciding whether given cor- Here, for convenience, the normalization factor of the j RHS of (D1a) has been absorbed in the quantum states relations belongs to Q2/1 can be determined using SDP ρs. (though the size of the SDP increases with j); ii) the Let us start by showing that any correlations of the j sets Q2/1 approximate better and better Q2/1 from the form (4) is of the form (D1). Note first the well- outside, i.e., Q Qj and Qj+1 Qj for all j. If known fact that if the dimension of the Hilbert space 2/1 2/1 2/1 2/1 ⊆ ⊆ j k is not fixed, any single-party correlations P (c z) admits given correlations are found not to belong to some Q2/1, k k | k a quantum representation P (c z) = tr[Mc z ρC ] where we can thus conclude that they do not belong to Q2/1 k | | and that they reveal genuinely tripartite entanglement. the operators Mc z commute among themselves. Indeed, | Further, it is possible in this case to derive an associated let c = (c ,...,c ) and P (c) = m P (c z), then a 1 m z=1 z DIEW from the solution of the dual SDP. Note that mod- quantum representation of P k(c z) is achieved| by defin- k k | Q ulo the assumption that the tensor product between the ing ρC = c P (c) c c and Mc z = c δ(cz,c) c c . We | ih | | | ih | three measurement operators in (D1a) is equivalent to the can thus write in Eq. (4), e.g., P k (ab xy)P k(c z) = P k PQ | | commutation of these operators — an assumption that is tr[Ma x Mb y Mc z(ρAB ρC )] with the operators Mc z true in finite-dimensional Hilbert space, but which still | ⊗ | ⊗ | ⊗ P | commuting between themselves. The decomposition (4) represents an open question in an infinite-dimensional is thus clearly a particular case of (D1). Hilbert space — it can be shown that the hierarchy of Let us now show the converse. Let us suppose first relaxations converges to the set of biseparable states, i.e., that s = AB/C and let c = (c ,...,c ). Since j 1 m limj Q2/1 = Q2/1. Thus if given correlations are not s →∞ the projectors Mc z commute, the projectors M(c) = biseparable this will necessarily show-up at some finite m s | z=1 Mc z defines a valid measurement and thus ρc = step in the hierarchy. z| Q

[1] S. Pironio, J.-D. Bancal, V. Scarani, J. Phys. A: Math. [4] M. Navascu´es, S. Pironio, A. Ac´ın, Phys. Rev. Lett. 98, Theor. 44, 065303 (2011). 010401 (2007); New J. Phys. 10, 073013 (2008). [2] S. Braunstein, C. Caves, Annals of Phys. 202, 22 (1990). [5] S. Pironio, M. Navascues, A. Ac´ın, SIAM J. Optim. vol [3] S. Wehner, Phys. Rev. A 73, 022110 (2006). 20, issue 5, 2157 (2010).

Paper K

Loophole-free Bell test with one atom and less than one photon on average

N. Sangouard, J.-D. Bancal, N. Gisin, W. Rosenfeld, P. Sekatski, M. Weber and H. Weinfurter

Physical Review A 84, 052122 (2011)

191

PHYSICAL REVIEW A 84, 052122 (2011)

Loophole-free Bell test with one atom and less than one photon on average

N. Sangouard,1 J.-D. Bancal,1 N. Gisin,1 W. Rosenfeld,2 P. Sekatski,1 M. Weber,2 and H. Weinfurter2 1Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland 2Fakultat fur Physik, Ludwig-Maximilians-Universitat Munchen, DE-80799 Munchen, Germany (Received 3 August 2011; published 28 November 2011) We consider the entanglement between two internal states of a single atom and two photon number states describing either the vacuum or a single photon and thus containing, on average, less than one photon. We show that this intriguing entanglement can be characterized through substantial violations of a Bell inequality by performing homodyne detections on the optical mode. We present the experimental challenges that need to be overcome to pave the way toward a loophole-free Bell test.

DOI: 10.1103/PhysRevA.84.052122 PACS number(s): 03.65.Ud

I. INTRODUCTION three-setting inequality. Furthermore, the photon is naturally used to distribute entanglement over long distances so that the Is quantum physics a complete theory, or does the descrip- choice of the measurement on one side and the measurement tion of nature’s laws require local-hidden-variable theories? result on the other side can easily be spacelike separated. Note The answer to this question, which was asked by Einstein, that the entanglement between internal states of an atom and Podolsky, and Rosen in 1935, can be found by realizing the polarization degree of freedom of a photon have already a Bell test [1]. On the one hand, two distant observers, been observed experimentally [14–16]. Such entanglement who have performed appropriate measurements on entangled has further been used to entangle remote atoms from an photon pairs, have observed correlated results violating a Bell entanglement-swapping operation [17]. We focus on the inequality even though the measurement choices were made entanglement between internal states of an atom and a partially long after the pair creation [2] and even though the photons filled optical mode, containing on average less than one photon, were too far from each other to agree on the results once as described in detail in Sec. II. We propose Bell-type scenarios they knew the measurement basis [3]. On the other hand, either combining a homodyne detection and a photon counting two ions close to each other have also exhibited the violation on the optical mode or using homodyne detections only to of a Bell inequality even though they were forced to give a characterize this special entanglement. Although homodyne result at each trial [4]. But to constitute a definitive answer, it detections are used, we show in Sec. III that unexpectedly large would be necessary to close all the loopholes in the same Bell violations of the CHSH inequality can be observed. We also experiment, i.e., to perform a Bell test both at a distance and present a feasibility study in Sec. IV. We provide the minimal with high detection efficiencies. entanglement generation and photon-counting efficiencies that Closing the detection loophole for the Clauser-Horne- are required to close the detection loophole. We then give the Shimony-Holt (CHSH) inequality [5] requires overall detec- typical distance that is necessary to close the locality loophole. tion efficiencies larger than 82.8% for a maximally entangled We also take the branching ratios into account, we analyze the state and larger than 66.7% using partially entangled states effect of the atomic motion, and we present the requirement [6] in the absence of other imperfections. This threshold on the optical-path-length stability. The last section is devoted detection efficiency can further be lowered using states with to the conclusion. a dimension higher than qubits. For example, in Ref. [7], it has been shown that a detection efficiency of 61.8% can be tolerated using four-dimensional states and a four-setting Bell inequality. However, considering realistic noise, achievable II. ENTANGLEMENT CREATION BETWEEN ONE ATOM coupling into the quantum channel (usually an optical fiber), AND LESS THAN ONE PHOTON and detection efficiencies, one rapidly becomes aware that Let us start with a description of the methods enabling closing the detection loophole in an optical Bell test is the creation of entanglement between two atomic states extremely challenging. and a single optical mode containing on average less than The problem of the single-photon detection efficiency might one photon. Consider an atom with a lambda-type level be circumvented by using homodyne measurements, which configuration (as depicted in Fig. 1), initially prepared in the are known to be very efficient [8–10]. In this framework, state g.A pump-laser pulse with the Rabi frequency partially theoretical proposals leading to substantial violations of Bell’s excites the atom in such a way that it can spontaneously decay inequalities and combining feasible states and measurements to the level s by emitting a photon [18]. Long after the decay have been put forth recently [11]. time of the atom, the atom-photon state is given by An attractive alternative is to use an asymmetric configuration involving, e.g., atom-photon entanglement. Since ψφ = cos θ|g,0+eiφ sin θ|s,1, (1) the atom can be detected with an efficiency close to 1, the detection efficiency on the photon side is lower than the case = 1 where θ 2 ds(s) refers to the area of the pump pulse. wherein the detections at both sides are inefficient [12,13]— The phase term is defined by φ = kprp − ksrs , where kp as low as 50% for the CHSH inequality and 43% for a (and ks ) correspond to the wave vector of the pump (and the

=− e positive and b1 1 if there is no photon. When he performs | the quadrature measurement, he gets a real number x. He then has to process this result to get binary outcomes. He decides =− to attribute the results b2 1 if the result is negative (x 0) and b2 =+1 otherwise. Ω spontaneous photon We now show that Alice and Bob can obtain a substantial violation of the CHSH inequality for appropriate settings. But let us first detail the calculation of probability distributions p(ai bj |Xi Yj ) for the four pairs of measurements separately. g s =− | When Bob measures n, he gets b1 1 with the probability | cos2 θ, and Alice’s qubit is projected into |g. Therefore, FIG. 1. Basic level scheme for the creation of entanglement between one atom and one optical mode containing on average less −→ 1 + cos αj p(+1, − 1|X Y ) = cos2 θ|v |g|2 = cos2 θ . than one photon. The branching ratio is such that when the atom is j 1 j 2 excited, it decays preferentially in s. Similarly, spontaneous photon) and r (and r ) are the atom position −→⊥ p s p(−1, − 1|X Y ) = cos2 θ|v |g|2 when the pump photon is absorbed (and the spontaneous j 1 j 1 − cos α photon is emitted). Note that φ may vary in practice, e.g., due = cos2 θ j . to atom-position variations. The requirements for the phase 2 stability are studied in detail below, but we first answer this Following similar lines for b1 =+1, one finds question: can the entanglement between an atom and a partially filled optical mode be measured from the violation of a Bell 1 − a cos α p(a , + 1|X Y ) = sin2 θ j j , inequality? j j 1 2 leading to III. HOMODYNE DETECTIONS IN AN ASYMMETRIC =− BELL TEST EXj Y1 cos αj . (3)

A. Principle of the Bell test When Bob measures Y2 and obtains b2 =−1, Alice’s state is First, let us recall the principle of a Bell-CHSH test. Two projected into distant observers, usually named Alice and Bob, share a 0 A = 2 | |2 | | quantum state. Each of them randomly chooses a measurement ρb =−1 cos θ dx 0(x) g g 2 −∞ among two projectors, {Xi } for Alice and {Yj } for Bob, where ∈ { } { } 1 0 i,j [1,2], and each obtains a binary result, ai and bj + −iφ iζ ∗ | | sin 2θe e dx 1(x) 0(x) g s for Alice and Bob, respectively. By repeating the experiment 2 −∞ several times, Alice and Bob can compute the conditional 1 0 | + iφ −iζ ∗ | | probabilities p(ai bj Xi Yj ). They can then easily deduce the sin 2θe e dx 0(x) 1(x) s g 2 −∞ value of the CHSH parameter 0 + 2 | |2 | | S = E + E + E − E , (2) sin θ dx 1(x) s s , X1Y1 X1Y2 X2Y1 X2Y2 −∞ = = | − = | where E(Xi Yj ) p(ai bj Xi Yj ) p(ai bj Xi Yj ). where 0(x) =x|0 and 1(x) =x|1 are the probability- Alice and Bob will conclude that the observed correlations amplitude distributions for the vacuum and single-photon Fock cannot be described by local-hidden-variable theories if they = √1 −x2/2 states, respectively [ n(x) n 1/2 Hn(x)e , where find measurement settings such that S>2. Note that all (2 n! π) Hn(x) is the Hermite polynomial]. p(+1, − 1|Xj Y2) is merely possible states leading to a violation of a Bell inequality are −→ A −→ deduced from v j |ρ =− | v j and p(−1, − 1|Xj Y2)from entangled. Therefore, a Bell test can be seen not only as a test b2 1 −→⊥ A −→⊥ v |ρ =− | v . One can check that p(a , + 1|X Y ) has of the laws of nature but also as a witness of entanglement. j b2 1 j j j 2 the same expression as p(aj , − 1|Xj Y2), but where the integration over dx runs from 0 to +∞, one finds B. Bell test with one atom and less than one photon Now, consider the specific case in which Alice and Bob 2 EX Y = sin αi sin 2θ cos(ϕj − φ + ζ ). share a state of the form (1). Alice applies projective measure- j 2 π ments on the atomic states and can freely choose projections on Interestingly, this expression is the same, up to a factor of −→ α α √ = j | + iϕj j | arbitrary vectors vj cos 2 g e sin 2 s of the Bloch 2/π, as the expression of the correlator when Bob applies sphere. For each measurement X ,j= 1,2, and Alice sets + j −→ a perfect qubit measurement along cos ζσx sin ζσy .This aj =+1 if she gets a result along vj and aj =−1ifthe invites us to interpret the homodyne measurement above (with −→⊥ → =− → =+ result is directed along vj . Bob applies measurements on the the binning x 0 b2 1 and x>0 b2 1inthe optical mode and chooses either to count the photon number {|0,|1} subspace) as a noisy qubit measurement√ in the xy Y1 = n or to measure the quadrature Y2 = cos ζ Xˆ + sin ζ P.ˆ plane of the Bloch sphere with visibility 2/π as was also When he measures n, he naturally sets b1 =+1 if the result is noticed in Ref. [19].

052122-2 LOOPHOLE-FREE BELL TEST WITH ONE ATOM AND ... PHYSICAL REVIEW A 84, 052122 (2011)

Substituting the correlators by their expressions into Eq. (2), 2.8 one obtains a value of the CHSH polynomial for any state of the form (1) and for any measurement√ of Alice. We found 2.7 =1 =− + ≈ d the maximal violation S 2 cos α1 2 2/π sin α1 2.56 =0.8 = = = = 2.6 d for θ π/4 and φ 0, i.e., the φ√+ state√ and for ϕ1 ϕ2 + + =0.6 π √ 2 π d ζ = 0, and α1 =−α2 = 2arctan( ). This violation is 2.5 2 =0.4 the largest that we know in a scenario involving a homodyne d detection in which both the measurements and the state can 2.4 2 x homodyne be realized experimentally (see Ref. [11] and the references CHSH 2.3 therein). 2.2 C. Bell test with homodyne detections only on the optical mode 2.1 A natural question is whether a violation of the CHSH inequality can also be observed by measuring the optical mode 2 0.5 0.6 0.7 0.8 0.9 1 with homodyne√ detections only. It turns out that a violation transmission S = 4/ π ≈ 2.26 can indeed be obtained if Alice and Bob t = share a maximally entangled state of the form (1) with θ π/4 FIG. 2. (Color online) Robustness of the CHSH violation with = and φ 0, provided that Bob’s measurements are performed respect to the transmission efficiency η . The dashed-dotted (purple) = ˆ = ˆ t in complementary quadratures Y1 X and Y2 P and that line is associated with the case in which Bob uses two homodyne Alice’s measurements correspond to projections along vectors detections (with unit efficiencies). The other lines correspond to the ◦ spanning the (xy) plane with angles ±45 between them. This cases in which Bob chooses either a photon counter or a measurement result can easily be understood using the analogy previously of a field quadrature. The upper, solid (blue) curve is associated with = mentioned. If Bob uses either√ σx or σy , the CHSH parameter a photon detector with unit efficiency ηd 1. The three dashed lines = would be saturated (S = 2 2). Since Xˆ and Pˆ correspond√ to are associated with inefficient counting (from ηd 0.8to0.4). such measurements but with the reduced visibility 2/π, S is reduced by the corresponding factor. of the single-photon detectors are inefficient. Let ηd be the efficiency of the photon-counting detector. From an optimiza- IV. IMPERFECTIONS tion similar to the previous one, we find that a threshold detection of ηd = 39% can be tolerated for a transmission So far, we have shown that an ideal realization would lead with unit efficiency. Our scheme is less sensitive to counting to significant violations of the CHSH inequality. However, inefficiency than transmission inefficiency since the former the story would not be complete without a discussion taking affects only one of Bob’s measurements. The two previous experimental imperfections into account. efficiency thresholds can be compared with a scheme that exhibits the same asymmetry but uses the entanglement with A. Transmission inefficiency the polarization mode [13] and for which the violation of the CHSH inequality requires η η 50%. (The effects of Let ηt be the transmission efficiency which accounts t d for all the coupling inefficiencies from the atom to Bob’s detection and transmission imperfections are the same in this | case since Bob uses two photon-counting detectors.) The latter location. The probability√ amplitude associated with s,1 is now multiplied by η , and Alice and Bob can share is less sensitive to inefficiency in the transmission (for ideal t √ = φ √1 iφ detectors with ηd 1) while the scheme we propose is less the state ψ = (cos θ|g,0+e sin θ ηt |s,1) with the ηt N sensitive to the detector inefficiency (for transmission with probability N = cos θ 2 + sin θ 2η . Alternatively, the photon t ηt = 1.) Note also that regarding the results presented in can be lost. Tracing out the lost photon, the resulting state Refs. [7,13] wherein the threshold efficiency has been lowered | is s,0 , and it contributes to the global state with a weight using inequalities with more settings, one could have possible 2 − sin θ(1 ηt ). To know the sensitivity of the CHSH inequality improvements using inequalities different from the CHSH with respect to the transmission inefficiency, we thus have to inequality. However, we could not find better resistances with compute S from the overall state additional binnings for Bob’s results and for more (up to three) φ φ 2 inputs. ρη = N ψ ψ + sin θ (1 − ηt ) |s,0s,0| (4) t ηt ηt In the case in which Bob uses only quadrature measure- for all possible values of θ, φ, ϕ1, ϕ2, ζ , α1, and α2 as a function ments, the CHSH inequality can be violated provided that the of ηt . The result is shown in Fig. 2. In the scenario in which transmission efficiency is larger than 79%. This scenario is thus Bob uses a photon counter and a homodyne detection with more robust to the transmission inefficiency than the proposals unit efficiencies, a transmission efficiency of ηt = 61% can of Refs. [7,13] if the single-photon detectors required in the be tolerated [see the solid (blue) curve of Fig. 2]. Although latter have an efficiency smaller than 63%. this is certainly demanding, recent results suggest that this might soon be within reach of experiments [20]. It is also interesting to study the sensitivity of the violation with respect B. Required distance between Alice and Bob to the detection inefficiency. The homodyne measurements In the previous section, we addressed efficiency issues can fairly be considered to have unit efficiencies, but most related to the photon detection and to the transmission. If we

052122-3 N. SANGOUARD et al. PHYSICAL REVIEW A 84, 052122 (2011) intend to close the locality loophole too, we have to determine is found to be how long the state detection takes. It is likely reasonable to = 1 + −2a2(n¯+1/2)k believe that the detection time is limited by the atom [21]. If F 2 1 e (7) the atomic states are read out on the basis of stimulated Raman √ = adiabatic passage, ultrafast laser ionization, and registration of in the weak-confinement regime [23,24]. a h/¯ (2mω)is the correlated electron-ion pairs with coincident counting via the size of the harmonic-trapping-potential ground state for an atom of mass m within a trap of frequency ω. n¯ is the average two opposing-channel electron multipliers [22], we can reach −→ a measurement time of less than 1 μs. Therefore, the locality number of thermal quanta of motion, and k =||k ||(1 − loophole can be closed if Alice and Bob are separated by cos θ), where θ is the angle between the pump beam and the 300 m. For 800-nm photons, the losses are of 2 dB/km. This emission direction. Hence, the problem of the atomic motion translates into a transmission of 93%. can not only be overcome by cooling the ions deeply within the Lamb-Dicke limit (where n¯ is small) but more simply by collecting the photons scattered in the forward direction where C. Branching ratio θ = 0. So far, we have considered that once the atom is excited, Let us also comment on the stability requirement on the it decays into the state s. Consider the more realistic case optical path lengths. The local oscillator that is required to in which the decay from e to s occurs with the probability perform the homodyne detections at Bob’s location can be fs . Let fg be the probability of a decay into g and faux the obtained by removing a fraction of the pump beam with decay probability into other auxiliary states such that fs + a beam splitter. In this case, the setup is made of a large fg + faux = 1. Taking these branching ratios into account, the Mach-Zehnder interferometer, and the path-length difference state long after the interaction with the pump pulse is between the two−→ arms of the interferometer L has to be stable so that || k ||L 1. Note that temperature variations −→ ρ = N ψφ ψφ + sin2 θf |g,0g,0| change both the refraction index (and thus || k ||) and the f fs fs g length of the fibers L. For commercial fibers, several tens of + sin2 θf |aux,0aux,0|. aux kilometers long, installed in an urban environment, the typical φ φ time needed for a mean-phase change of 0.1 rad (corresponding ψ is defined from ψ , where ηt is replaced by fs and fs ηt to a fidelity of 0.9) is of the order of 100 μs[25]. This lets = 2 + 2 N cos θ sin θ fs . We now present a strategy to make us believe that an active stabilization of the phase should be = S>2 (calculated from this state) as soon as fs 0. If Alice possible even for very long interferometers using available =− chooses to attribute the result aj 1 when she measures the technologies. This is well confirmed by recent experimental | atom in the state aux, the correlators calculated from aux,0 results [26]. = = ∀ ∈{ } are EXj Y 1 1,EXj Y 2 0 j 1,2 , and the resulting S value is equal to 2 in this case. Moreover, if she chooses two measurements very close to σz, i.e., (α1 = π − ,ϕ1 = 0) and (α2 =−π + ,ϕ2 = 0), one can check that the S value V. CONCLUSION computed from the component |g,0 is 2 − 2. If she further = = We proposed different scenarios to measure the entangle- excites the atom so that θ π/4 and φ 0 and if Bob√ chooses ment between the internal state of an atom and an optical mode ζ = 0, the S value from ψφ is roughly 2 + 4 √2fs .The fs 1+fs π containing, on average, less than one photon. We reported overall CHSH value is thus given by large violations of a CHSH inequality for both the case √ in which one homodyne detection and one-photon counting f S ≈ 2 + 4√ s + o(2). (5) are performed on the optical mode and for that involving 2π two homodyne detections. With homodyne detections only, a minimal entanglement-generation efficiency of 79% can be Therefore, in the absence of errors, the only requirement on the tolerated. This efficiency goes down to 61% if homodyne branching ratios to observe a violation of the CHSH inequality detections are combined with unit-efficiency photon counting. is that once the atom is excited, the probability that it decays We have also shown that in principle, a violation of the CHSH into s has to be nonzero. This Bell test can thus be applied to inequality can be obtained even for branching ratios favoring a large number of atomic species since it is very resistant to photon emission in undesired modes. There is no need to branching-ratio variations. However, the larger the decay into cool the atom deeply within the Lamb-Dicke regime if the s is, the larger the violation. scattered photons are collected close to the forward direction with respect to the pump propagation. Finally, the stability D. Stability requirements requirements for the optical path lengths are within reach of experiments. Let us now focus on the phase-stability constraints. The We believe that our work could provide motivations for phase term of the state (1) has to remain stable from trial to trial. several research groups. Much effort has already been devoted In practice, however, φ may vary, e.g., due to atom-position −→ to the characterization of single-photon Fock states with || || = variations.−→ −→ For wave vectors with the same norm k homodyne detections [27]. Moreover, although the setup ||kp || = || ks ||, the fidelity of the resulting entanglement recently developed by the Rempe group has been used to address squeezed light [28], it is of particular interest for ρ = F |ψφψφ|+(1 − F )|ψφ+π ψφ+π | (6) our proposal as it combines a single atom embedded in a

052122-4 LOOPHOLE-FREE BELL TEST WITH ONE ATOM AND ... PHYSICAL REVIEW A 84, 052122 (2011) high-ﬁnesse cavity with a homodyne detection. Note also ACKNOWLEDGMENTS that beyond its fundamental interest, our proposal might We thank M. Afzelius, N. Brunner, D. Cavalcanti, ﬁnd exciting applications in the framework of quantum- H. de Riedmatten, and R. Thew for interesting discussions. We information sciences, e.g., for device-independent quantum gratefully acknowledge support by the EU project Qessence, cryptography [29]. the EU ERC AG Qore, and the Swiss NCCRs QP and QSIT.

[1]J.S.Bell,Physics1, 195 (1964). [20] The main problem is likely to achieve a high efﬁciency of the [2] W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, Phys. Rev. Lett. photon collection. This requires embedding the atom within 81, 3563 (1998); N. Gisin and H. Zbinden, Phys. Lett. A 264, a cavity with a high ﬁnesse and a small mode volume. A

103 (1999). Purcell factor Fp of 2 has already been achieved experimentally [3] A. Aspect et al., Phys.Rev.Lett.28, 938 (1972); G. Weihs, for a trapped ion in a cavity [see A. B. Mundt, A. Kreuter, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, ibid. C. Becher, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and 81, 5039 (1998). R. Blatt, Phys. Rev. Lett. 89, 103001 (2002)], leading, in − [4]M.A.Roweet al., Nature (London) 409, 791 (2001); D. N. principle, to a collection efﬁciency Fp 1 = 50%. Fp Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and [21] The jitter of single-photon detectors can be shorter than 100 ps, C. Monroe, Phys. Rev. Lett. 100, 150404 (2008). and in our case, the photon duration is typically of the order of [5]J.F.Clauser,M.Horne,A.Shimony,andR.A.Holt,Phys. Rev. 10 ns. Regarding the results presented in H. Hansen et al., Opt. Lett. 23, 880 (1969). Lett. 26, 1714 (2001), one can reasonably conclude that pulsed [6] P. H. Eberhard, Phys.Rev.A47, R747 (1993). homodyne detection is possible within less than 1 μs. Therefore, [7] T. Vertesi, S. Pironio, and N. Brunner, Phys.Rev.Lett.104, the detection time is likely limited by the time it takes to choose 060401 (2010). the measurement. In the scenario in which Bob uses either a [8] H. Nha and H. J. Carmichael, Phys.Rev.Lett.93, 020401 (2004). photon counter or a homodyne detection, one needs an optical [9] R. Garcia-Patron, J. Fiurasek, N. J. Cerf, J. Wenger, R. Tualle- switch. We believe that switching times shorter than 1 μswith Brouri, and P. Grangier, Phys.Rev.Lett.93, 130409 (2004). negligible losses are feasible. [10] S. W. Ji, J. Kim, H. W. Lee, M. S. Zubairy, and H. Nha, Phys. [22] F. Henkel, M. Krug, J. Hofmann, W. Rosenfeld, M. Weber, and Rev. Lett. 105, 170404 (2010); D. Cavalcanti and V. Scarani, H. Weinfurter, Phys.Rev.Lett.105, 253001 (2010). ibid. 106, 208901 (2011). [23] C. Cabrillo, J. I. Cirac, P. Garcia-Fernandez, and P. Zoller, Phys. [11] D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and Rev. A 59, 1025 (1999). V. Scarani, e-print arXiv:1012.1916. [24] L. Luo, D. Hayes, T. A. Manning, D. N. Matsukevitch, P. Maunz, [12] A. Cabello and J.-A. Larsson, Phys. Rev. Lett. 98, 220402 S. Olmschenk, J. D. Sterk, and C. Monroe, Fortschr. Phys. 57, (2007). 1133 (2009). [13] N. Brunner, N. Gisin, V. Scarani, and C. Simon, Phys. Rev. Lett. [25] J. Minar, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, 98, 220403 (2007). Phys. Rev. A 77, 052325 (2008). [14] D. L. Moehring, M. J. Madsen, B. B. Blinov, and C. Monroe, [26] S.-B. Cho and T.-G. Noh, Opt. Express 17, 19027 (2009). Phys. Rev. Lett. 93, 090410 (2004). [27] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, [15] J. Volz, M. Weber, D. Schlenk, W. Rosenfeld, J. Vrana, and S. Schiller, Phys.Rev.Lett.87, 050402 (2001); A. Zavatta, K. Saucke, C. Kurtsiefer, and H. Weinfurter, Phys.Rev.Lett. S. Viciani, and M. Bellini, Phys. Rev. A 70, 053821 (2004); 96, 030404 (2006). S. R. Huisman et al., Opt. Lett. 34, 2739 (2009). [16] W. Rosenfeld, F. Hocke, F. Henkel, M. Krug, J. Volz, M. Weber, [28] A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. and H. Weinfurter, Phys.Rev.Lett.101, 260403 (2008). Pinkse, G. Rempe, and K. Murr, Nature (London) 474, 623 [17] D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge, D. N. (2011). Matsukevitch, L.-M. Duan, and C. Monroe, Nature (London) [29] For the principle, see e.g., J. Barrett, L. Hardy, and A. Kent, 449, 68 (2007). Phys. Rev. Lett. 95, 010503 (2005); A. Acin, N. Brunner, [18] Note that the pump-pulse duration is considered to be short N. Gisin, S. Massar, S. Pironio, and V. Scarani, ibid. 98, 230501 relative to the lifetime of the excited state in order to minimize (2007). For possible realizations, see e.g., N. Gisin, S. Pironio, two-photon emissions. and N. Sangouard, ibid. 105, 070501 (2010); N. Sangouard, [19] M. T. Quintino, M. Araujo,´ D. Cavalcanti, M. F. Santos, and B. Sanguinetti, N. Curtz, N. Gisin, R. Thew, and H. Zbinden, M. T. Cunha, e-print arXiv:1106.2486. ibid. 106, 120403 (2011).

052122-5

Paper L

A framework for the study of symmetric full-correlation Bell-like inequalities

J.-D. Bancal, C. Branciard, N. Brunner, N. Gisin and Y.-C. Liang

Journal of Physics A: Mathematical and Theoretical 45, 125301 (2012)

199

IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 45 (2012) 125301 (14pp) doi:10.1088/1751-8113/45/12/125301

A framework for the study of symmetric full-correlation Bell-like inequalities

Jean-Daniel Bancal1, Cyril Branciard2, Nicolas Brunner3, Nicolas Gisin1 and Yeong-Cherng Liang1

1 Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland 2 School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia 3 H H Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK

E-mail: [email protected]

Received 12 January 2012, in ﬁnal form 13 January 2012 Published 5 March 2012 Online at stacks.iop.org/JPhysA/45/125301

Abstract Full-correlation Bell-like inequalities represent an important subclass of Bell- like inequalities that have found applications in both a better understanding of fundamental physics and in quantum information science. Loosely speaking, these are inequalities where only measurement statistics involving all parties play a role. In this paper, we provide a framework for the study of a large family of such inequalities that are symmetrical with respect to arbitrary permutations of the parties. As an illustration of the power of our framework, we derive (i) a new family of Svetlichny inequalities for arbitrary numbers of parties, settings and outcomes, (ii) a new family of two-outcome device-independent entanglement witnesses for genuine n-partite entanglement and (iii) anew family of two-outcome Tsirelson inequalities for arbitrary numbers of parties and settings. We also discuss brieﬂy the application of these new inequalities in the characterization of quantum correlations.

PACS numbers: 03.65.Ud, 03.67.−a

1. Introduction

Bell inequalities [1] play a central role in quantum physics and in quantum information [2]. Initially discovered in the context of foundational research on quantum correlations, they are today used in a wide range of protocols for quantum information processing. For instance, they are naturally associated with communication complexity [3] and are the key ingredient in device-independent quantum information processing [4–10]. Thus, developing and harnessing Bell inequalities is fundamental toward a deeper understanding of the foundations of quantum mechanics, as well as for applications in quantum information.

The most famous and widely used Bell inequality is due to Clauser–Horne–Shimony–Holt (CHSH) [11]. The CHSH scenario, which is the simplest nontrivial Bell scenario, involves two parties each performing two possible binary-outcome measurements. Denoting by x and y the measurement settings of Alice and Bob, respectively, and by Ax and By their measurement + + − outcomes, the CHSH inequality reads E11 E12 E21 E22 2, where the two-party 4 correlators Exy are defined as Exy = P(Ax = By) − P(Ax = By) . Clearly, the value of the correlator does not depend on the individual values of Alice’s and Bob’s outcomes, but rather on how Ax and By relate to each other. Since the CHSH inequality is expressed in terms of these correlators only, it is said to be a correlation Bell inequality. It is natural and useful to investigate Bell tests beyond CHSH. Bell scenarios can indeed involve in general an arbitrary number of parties, each party having an arbitrary number of measurement settings and each of the corresponding measurements an arbitrary number of possible outcomes. Here we denote by the triple (n, m, k) a Bell scenario where n parties all have m possible measurement settings with k possible outcomes. Correlation Bell inequalities can also be naturally defined in these situations and represent powerful tools for investigating nonlocality (see, e.g., [12, 13]). In this regard, we will refer to a k-valued function of all parties’ measurement outcomes as a full-correlation function if the function can still take on all k possible values even when all but one of the parties’ outcomes (for given measurement settings) are fixed. A full- correlation Bell inequality is then one that can be written as a linear combination of probabilities associated with a full-correlation function taking particular values. These inequalities are natural generalizations of the Bell-correlation inequalities considered by Werner and Wolf in [12] to an arbitrary number of measurement outcomes. In this paper, we shall consider specifically inequalities where the full-correlation function involved is the sum (modulo k)of all parties’ measurement outcomes. Up until now, several families of full-correlation Bell inequalities have been discovered for specific cases. First, for the multi-input (2, m, 2) case, Pearle, followed by Braunstein and Caves, introduced the chained Bell inequalities [14]. In the multipartite (n, 2, 2) case, the CHSH inequality has then been generalized by Mermin and further developed by Ardehali, Belinskiˇı and Klyshko (MABK) [15]. In fact, a complete characterization of all the 22n full- correlation Bell inequalities present in this scenario was later achieved by Werner and Wolf [12], and independently by Zukowski˙ and Brukner [16]. For the (2, 2, k) case, Collins– Gisin–Linden–Massar–Popescu (CGLMP) derived correlation inequalities for scenarios with an arbitrary number of measurement outcomes [17](seealso[18]). Finally, Barrett–Kent– Pironio (BKP) presented in [19] Bell inequalities for the (2, m, k) case, unifying the CGLMP and the chained Bell inequalities. Beyond standard Bell inequalities, other types of inequalities are worth considering. These include Tsirelson inequalities [20], which are satisfied by all quantum correlations; Svetlichny inequalities [21], which can be used to detect genuine multipartite nonlocality; and device- independent entanglement witnesses (DIEWs) [10, 25], which detect genuine multipartite entanglement even with Svetlichny–local correlations. We shall refer to all these inequalities as Bell-like inequalities. For detecting genuine multipartite nonlocality, Collins et al [22] and Seevinck–Svetlichny [23] have derived full-correlation Svetlichny inequalities for the (n, 2, 2) case, generalizing those in [21]. Just like the BKP Bell inequalities, which can be seen as a generalization of the chained Bell inequalities to more outcomes, a generalization of the Svetlichny inequalities

4 Throughout the paper, the notation P denotes probabilities.

2 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al

Figure 1. Previously known families of Bell expressions are recovered from the n,m,k expression of equation (1)whenn, m or k = 2 (see the text); the CHSH expression is recovered,I in particular, when n = m = k = 2. n,m,k generalizes these expressions for n, m, k > 2, thus completing the vertex ‘?’ of the cube depictedI above. An even more general full-correlation Bell expression is n,m,k; f , deﬁned in equation (11) below.

[22, 23]tothe(n, 2, k) scenario was also achieved in [24], effectively unifying the CGLMP inequality and the generalized Svetlichny inequalities of [22, 23]. Since all the aforementioned families of Bell-like inequalities reduce to CHSH for n = m = k = 2, a natural question that one may ask is whether it is possible to unify all these inequalities into a single family of a mathematical expression (henceforth referred as Bell expression) for the general (n, m, k) scenario (see figure 1). In this paper, we provide an affirmative answer to this question. To achieve this, we will start in section 2 by presenting a unified Bell expression that, together with the appropriate bound, reduces to all the Bell-like correlation inequalities mentioned in the last paragraphs as limiting cases. This effectively provides a unified framework for the study of a large family of full-correlation Bell-like inequalities. After that, in section 3, we discuss how new multipartite Tsirelson inequalities, Svetlichny inequalities and DIEWs can be constructed within our framework, starting from the respective bipartite and tripartite bounds. Explicit examples of such Bell-like inequalities are then presented. We then conclude in section 4 with some possible avenues for future research.

2. A framework for symmetric full-correlation Bell-like inequalities

2.1. A uniﬁed Bell expression

In this section, we present a unified Bell expression that reduces to various known Bell expressions as special cases. For definiteness, let us label the measurement settings (inputs) for the ith party as si = 0, 1,...,m − 1 and denote the corresponding outcome (output) by = , ,..., − = ( , ,..., ) rsi 0 1 k 1. For convenience, we will also write s s1 s2 snand define the = n = n sums of all parties’ inputs and outputs, respectively, as s i=1 si and rs i=1 rsi . Finally, for any integers X and d, we denote the value of X modulo d by [X]d ∈{0,...,d − 1}. With these notations, let us define the following Bell expression:

k−1 k−1 s s n,m,k = r − P([rs]k = r) + −r + P([rs]k = r), I = m k = m k s s.t. [s]m=0 r 0 s s.t. [s]m=1 r 0 (1)

3 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al where · is the floor function. n,m,k is clearly a full-correlation Bell expression, with the I full-correlation function involved being the sum, modulo k of all parties’ outputs, i.e. [rs]k; furthermore, since for a given choice of settings s, it only depends on the sum of inputs s and the sum of outputs rs, n,m,k is symmetric under any permutation of parties. I The expression n,m,k unifies a few important classes of full-correlation Bell expressions, as illustrated in figureI 1:forthe(2, m, k) case, it reduces to the one appearing in the BKP inequalities (which, in turn, contains both the CGLMP inequalities and the chained Bell inequalities as special cases [19]); for the (n, 2, k) case it reduces to the generalized Svetlichny expression of [24] (which contains the expressions of [22, 23] as special cases); for the (n, 3, 2) case, it reduces to the DIEW of [10]. For details on how these known Bell expressions are recovered from n,m,k (and for an alternative way of writing n,m,k), see appendix A. I I

2.2. From Bell expressions to Bell-like inequalities

Clearly, as deﬁned in equation (1), n,m,k is only a linear combination of probabilities. In order to make use of it in, say, entanglementI detection, we need to specify the appropriate bounds that depend on the situation of interest. For example, in a theory where only shared randomness is allowed5, one would have

L L n,m,k β , , , (2) I n m k βL where the local bound n,m,k is the lower bound of the Bell expression admissible within such a theory.6 Here, we have used the symbol ‘L’ to remind that the inequality is a constraint that has to be satisﬁed by a locally causal theory [1]; analogous notations will be adopted in all subsequent discussions. Inequality (2) is generally called a Bell inequality. The CHSH inequality [11], the Pearle– Braunstein–Caves chained inequalities [14], the CGLMP inequalities [17] and the BKP inequalities [19] in ﬁgure 1 are examples of such inequalities that can be written explicitly as

L L L L 2,2,2 1, 2,m,2 1, 2,2,k k − 1, 2,m,k k − 1. (3) I I I I The violation of a Bell inequality is a signature of Bell nonlocality. Likewise, in a multipartite scenario, one could be interested in detecting genuine multipartite nonlocality (also known as true n-body nonseparability [21]). In this case, it S is necessary to establish the Svetlichny bound of n,m,k, which we shall denote by β .One I n,m,k can then write down a Svetlichny inequality in terms of n,m,k as I S S n,m,k β , , . (4) I n m k The inequalities due to Collins et al [22] as well as Seevinck–Svetlichny [23] and that presented in [24] are inequalities of this kind and can be written explicitly as

S S n−2 n−2 n,2,2 2 and n,2,k 2 (k − 1). (5) I I S n−2 From [10], it also follows that n,3,2 3 . The quantum violation ofI a Svetlichny inequality is a sufﬁcient condition for genuine multipartite entanglement. However, for the detection of such entanglement, it already sufﬁces

5 Such theories are also commonly referred to as local (hidden variable) theories. 6 For simplicity of presentation, we will only discuss the lower bounds on the Bell expressions. Clearly, one can also discuss the upper bounds on n,m,k analogously. I 4 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al to violate the weaker constraint given by a DIEW [10], which we can write in the context of n,m,k as I B B n,m,k β , , , (6) I n m k βB where n,m,k is the quantum biseparable bound on n,m,k. In this notation, the DIEW of [10] can be written as I B √ n−2 n,3,2 3 (3 − 3). (7) I Note ﬁnally that for any given scenario (n, m, k), the set of probability distributions allowed in quantum theory is bounded and thus the Bell expression n,m,k is also restricted in quantum theory to I Q Q n,m,k β . (8) I n,m,k βQ Such an inequality is often referred to as a Tsirelson inequality, whereas the bound n,m,k is known as a Tsirelson bound [20]. For the CHSH expression (corresponding to n = m = k = 2), for instance , one has Q √ 2,2,2 2 − 2. (9) I Note that for general Bell expressions, these lower bounds obey the following set of inequalities: βQ ,βS βB βL , n,m,k n,m,k n,m,k n,m,k (10) but the bounds arising from the quantum set and the Svetlichny constraints are not necessarily comparable. For example, three parties that share only a Popescu–Rohrlich [26] box between two of them can clearly generate non-quantum but Svetlichny–local correlations. Conversely, there are Svetlichny inequalities that can be violated quantum mechanically.

2.3. Generalization to include other Bell expressions

While n,m,k already embraces a large number of known Bell expressions, it can actually be furtherI generalized to include an even larger family of Bell expressions. To this end, let us now define s n,m,k; f = f [s]m, r − P([rs]k = r), (11) m k s r where the first sum runs over all possible combinations of settings s, the second sum runs from r = 0tok − 1, and where f : {0,...,m − 1}×{0,...,k − 1}→R is a real-valued function (defined by m × k real parameters) that fully characterizes n,m,k; f . As with n,m,k, n,m,k; f is clearly a symmetric, full-correlation Bell expression. Note the I ( , ) specific form of the arguments of f s r , and how the sums s and rs of inputs and outputs s play different roles, with s appearing also in the second argument through the quantity m . This very term, which is responsible for the minus sign in the CHSH expression, turns out to be crucial for the computation of multipartite bounds on n,m,k; f (see the following section). The Bell expressions n,m,k introduced previously simply correspond to the choice I ⎧ ⎨r if s = 0, ( , ) = ( , ) = − = , f s r f s r ⎩[ r]k if s 1 (12) I 0 otherwise,

5 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al

Table 1. A summary of some other known Bell expressions that can be recovered as special cases of n,m,k; f . nmkf(s, r) a Bell expression

b δ · 3, odd 22 s,0 r MABK [15] δ · 3 22 s,0 r DIEW [25] ( s− π)· 3 22cosm r DIEW [25] a Notation: δs,0 is the Kronecker delta (such that δs,0 = 1ifs = 0, δs,0 = 0 otherwise) and can be any arbitrary real number. b Note that the MABK expressions — often referred to as the Mermin expressions — are identical to the Svetlichny–Bell expressions of [22, 23]forevenn. They are thus already recovered by n, , . I 2 2

so that n,m,k = n,m,k; f . Expression (11) is thus more general than (1). However, not all symmetricI full-correlationI Bell expressions can be put in the form (11)7. Still, the generalized expression n,m,k; f now includes some other known families of Bell expressions (up to relabelings of inputs and outputs and possibly afﬁne transformations), such as those appearing in the MABK Bell inequalities [15] and the DIEWs from [25]. The parameters leading to these inequalities are summarized in table 1. On top of these, there are a handful of other known bipartite two-output Bell inequalities that are of the form 2,m,2; f . Some of these examples can be found in equation (5) of [27] and in its appendix A, as well as in [28]. As mentioned above, in order for Bell expressions to be useful in practice, one needs to determine their relevant bounds, so that L βL , n,m,k; f n,m,k; f (13a)

S βS , n,m,k; f n,m,k; f (13b)

B βB , n,m,k; f n,m,k; f (13c)

Q βQ , n,m,k; f n,m,k; f (13d) where the various bounds depend on the choice of a function f . In the following section we show how, starting from bipartite bounds on 2,m,k; f , one can construct bounds on n,m,k; f to obtain multipartite Bell-like inequalities.

3. From bipartite to multipartite bounds: how to generate new Bell-like inequalities

Determining the local bound or any of the other bounds described in section 2.2 for a given Bell expression is in general a highly nontrivial problem. Nonetheless, we will demonstrate in what follows that once the corresponding Tsirelson and local bounds for the bipartite expression 2,m,k; f are known (for any given choice of f ), one can immediately write down, respectively, a Tsirelson inequality and a Svetlichny inequality for n,m,k; f (for the same choice of f ). Analogously, we will also demonstrate, in the particular case where k = 2 and where the function f takes the form f (s, r) = g(s) · r, how a quantum biseparable bound on n,m,2; f can be obtained by solving a simple optimization problem, for a given m and a given function g(s).

7 = = = ( = ) For instance, for n m k 2, the trivial (single-term) expression P rs1=0 rs2=0 is not of the form (1), since ( = ) ( , ) in 2,2,2; f the term P rs1=1 rs2=1 must also come with the same coefﬁcient f 0 0 .

6 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al Our starting point is to note that for any n 2, one can rewrite n,m,k; f as a sum of m expressions involving effectively one less party. More precisely, let us decompose n,m,k; f as m−1 ( , ) = sn rsn , n,m,k; f n−1,m,k; f (14) sn=0 with ( , ) s sn rsn = , − ( = ). n−1,m,k; f f [s]m r P [rs]k r (15) m k s1,...,sn−1 r Defining ( ) s + s s sn = [s + s ] = s + s − 1 n m, 1 1 n m 1 n m n−1 ( ) s + s = sn + = − 1 n , s s1 si s m = m i 2 (16) ( , ) s1 + sn r sn rsn = r + r − , s1 s1 sn m k n−1 (sn,rs ) r = r n + r , s s1 si i=2 we obtain ( , ) s sn rsn = , − ( = ). n−1,m,k; f f [s ]m r P [rs ]k r (17) m (sn ), ,..., r k s1 s2 sn−1 ( , ) sn rsn Thus, every term n−1,m,k; f appearing in the decomposition (14) is of the general form (11)forthen − 1 first parties and for the same function f (s, r). This implies that, for all given ( , ) sn rsn ( − ) sn and rsn , n−1,m,k; f defines an n 1 -partite Bell expression. The invariance of n,m,k; f under permutation of the parties implies that the same decomposition can be carried out for any of the other parties. Bearing these in mind, we are now ready to construct some nontrivial multipartite Bell-like inequalities in terms of their bipartite bounds.

3.1. Tsirelson inequalities To derive a Tsirelson inequality for a general multipartite scenario, one can make use of equation (14) recursively and apply inequality (13d)forn = 2. This leads to Q n−2 βQ , n,m,k; f m 2,m,k; f (18) which is an n-partite Tsirelson inequality obtained as a function of the bipartite Tsirelson βQ bound 2,m,k; f .

3.2. Svetlichny inequalities To derive a Svetlichny bound for the general (n, m, k) scenario, we consider a Svetlichny scenario in which n − 1 parties are separated into two groups. By hypothesis, cf equation ( , ) sn rsn (13b), the value of n−1,m,k; f for any given value of sn and rsn is restricted by the Svetlichny βS bound n−1,m,k; f . Let us then introduce a new party (labeled by ‘n’), and (without loss of

7 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al generality8) let it join the same group as the first party; this does not change the total number of groups. Equations (16) and (17) can then be interpreted as follows: since the first and the nth parties are in the same group, they can collaborate, and thus the nth party can communicate to the first party his/her input and output (and vice versa). The first party can thus define ( , ) (sn ) sn rsn new effective inputs s1 and outputs rs1 as in equation (16): this allows the first party to account for every possible strategy of the new party. We thus see that in this new scenario, we must also have9 ( , ) S sn rsn βS . n−1,m,k; f n−1,m,k; f (19) βS = βL By repeating the above argument recursively and noting that 2,m,k; f 2,m,k; f , i.e. that the Svetlichny and local bounds coincide for n = 2, we thus obtain the Svetlichny inequalities

S n−2 βL . n,m,k; f m 2,m,k; f (20) Note that the bound corresponding to the situation in which the n parties are separated into G groups [29] can be derived in a similar way, from the local bound of G,m,k; f .

3.3. Two-output DIEWs Consider now the case where the outputs are binary (k = 2) and f (s, r) = g(s) · r for some function g : {0,...,m − 1}→R (as in the examples of table 1 for instance). The probabilities P([rs]2 = r) appearing in n,m,2;g·r can in this case be expressed in terms of the commonly 10 used n-partite correlators Es = P([rs]2 = 0) − P([rs]2 = 1), so that ( = ) = 1 + (− )r . P [rs]2 r 2 [1 1 Es] (21) We then obtain m−1 1 s 1 n−1 s , , ; . = g([s] ) [1 − (−1) m E ] = m g(s)− g([s] ) (−1) m E , n m 2 g r 2 m s 2 m s s s=0 s (22) where we used the fact that for each value of s = 0,...,m − 1, there are mn−1 lists of = β settings s such that [s]m s. Any lower bound n,m,2;g.r on n,m,2;g.r will thus be related to a corresponding upper bound on the last sum of equation (22) by an afﬁne transformation. In the case of biseparability in particular, we show in appendix B how to determine the biseparable bound on equation (22)forn = 3. A biseparable bound for general n can then be derived straightforwardly by invoking the recursive arguments employed in the previous subsections. This thus allows us to obtain, from equation (B.8), the following two-output DIEWs: m−1 m−1 B 1 η π n−2 ( ) − η j ( )ωs , n,m,2;g.r m m g s max j csc g s j (23) 2 j=0,...,m−1 2m s=0 s=0 i π (2 j+1) where η j is the greatest common divisor of 2 j + 1 and m, while ω j = e m .

8 This follows from the possibility to perform analogous decomposition as in equations (14)and(17) for any other party. 9 If the new party does not join any of the existing groups, the n parties can clearly only do worse in terms of ( , ) sn rsn minimizing n−1,m,k; f . 10 The correlator Es can be seen as the average value of the product of experimental outcomes, when these are labeled by ±1.

8 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al

3.4. Three explicit examples We showed in the previous subsections how to derive multipartite bounds on the general expression n,m,k; f , from bipartite or tripartite bounds. Applying the above results to the more specific case of n,m,k = n,m,k; f , cf equation (12), we now derive three explicit examples of new Bell-like inequalities.I I βL = − (1) For the expression n,m,k, we have the bipartite local bound 2,m,k k 1, cf equation (3). Substituting this intoI equation (20), we thus obtain the following Svetlichny inequality for arbitrary numbers of parties, inputs and outputs: S n−2 n,m,k m (k − 1). (24) I The case m = 2 of this expression, previously derived in [24], is marked as Svetlichny– CGLMP in figure 1. Inequality (24) represents the Svetlichny inequality for the vertex marked by ‘?’ in the cube shown in figure 1. (2) For binary outputs (k = 2), since [−r]2 = [r]2, the function f (s, r) specified in I equation (12) is of the form f (s, r) = g (s) · r, with g (s) = 1ifs = 0or ( ) = I I I m−1 ( ) = 1, and g s 0 otherwise. For this choice, one obtains s=0 g s 2 and m−1 I( )ωs = (2 j+1)π I s=0 g s j 2 cos . Substituting these into equation (23) and after some I 2m computation11, one arrives at the following two-output DIEWs for arbitrary numbers of parties and inputs: B π n−2 n,m,2 m m − cot . (25) I 2m (3) In a similar manner, it follows from the result of [30] that the Tsirelson bound for 2,m,2 βQ = − π I is 2,m,2 m 1 cos 2m . Substituting this into equation (18) then gives the following n-partite, m-setting Tsirelson inequality: Q π n−1 n,m,2 m 1 − cos . (26) I 2m

3.5. Tightness of our inequalities Evidently, it is desirable to understand if the Tsirelson inequalities, Svetlichny inequalities and DIEWs derived using the above procedures can be saturated. Note that a key common feature in these derivations involves equation (14). Hence, the n-partite bound can be saturated only ( , ) ( − ) sn rsn if all the n 1 -partite bounds on the expressions n−1,m,k; f involved in equation (14) can be simultaneously saturated. In general, one may thus expect that the n-partite bounds and hence the inequalities derived in sections 3.1–3.3 are not necessarily tight. Nonetheless, for all the examples that we have checked, all these bounds can indeed be saturated. For example, for the βS = n−2 expression n,m,k,√ both the Svetlichny bound n,3,2 3 and the biseparable bound βB = n−I2( − ) 12 n,3,2 3 3 3 obtained above can be saturated [10]; likewise, it can be verified that η π ( + )π 11 η j 2 j 1 η = m From (23), one needs to calculate max j j csc 2m cos 2m . By decomposing (when j ) the expression η π η π η j · (2 j+1)π j η to maximize in the form j cot 2m cos 2m sec 2m , the first absolute value is maximized for j as small as π possible, while the second one is upper bounded by 1. The maximum one needs to calculate is thus found to be cot 2m , obtained for j = 0. 12 This can be done, for example, using the optimization tools of [31] and the converging hierarchy of semidefinite programs discussed in [32].

9 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al √ Q Q Q the Tsirelson bounds for n,m,k satisfy β = 2β = 4β = 4 · (2 − 2), whereas I 4,2,2 3,2,2 2,2,2 βQ ≈ βQ ≈ βQ ≈ × − 11 13 4,2,3 2 3,2,3 4 2,2,3 4 3 3 .

4. Concluding remarks

Starting from a unified Bell expression n,m,k, we have shown that various important correlation Bell expressions can be recovered asI special cases (cf figure 1). A natural generalization of n,m,k to n,m,k; f has, in turn, allowed us to also recover other known correlation Bell expressionsI that have previously been investigated in the literature. Within the framework of n,m,k; f , we also demonstrated how multipartite Tsirelson inequalities, Svetlichny inequalities and device-independent witnesses for genuine multipartite entanglement (DIEWs) can be constructed. This, in particular, has allowed us to construct a new family of Svetlichny inequalities for arbitrary numbers of parties, inputs and outputs as well as a new family of two-output DIEWs that can be applied to a scenario involving arbitrary numbers of parties and inputs. Clearly, a natural question that one may ask is how useful the (new) inequalities that can be constructed within this framework are. To this end, we note that inequality (24) has recently also been discovered independently by Aolita et al [33] and used to show that the higher dimensional n-partite Greenberger–Horne–Zeilinger (GHZ) states can exhibit fully random genuinely multipartite quantum correlations. The DIEW given in inequality (25), on the other hand, can also be shown to detect the genuine multipartite entanglement of a noisy GHZ state up to the same level of noise resistance (visibility)—for any given m and n—as those given in [25]. Finally, it is worth noting that the existing techniques for computing Tsirelson bounds (such as those discussed in [32]) generally do not work very well beyond small values of n and/or k. Our general Tsirelson inequality (18) may thus serve as a useful tool for characterizing and understanding the extent of nonlocality allowed in quantum theory. We believe that our inequalities and the framework from which they were constructed, given their generality and simplicity, have the potential for many other interesting applications. Evidently, there are many open problems that stem from this work. An obvious question that we have not addressed, for instance, is whether there is any choice of the function f (s, r) for which the local bound on n,m,k; f can be easily determined, and whether the resulting inequalities correspond to facets of the respective local polytopes. As we already acknowledged, the framework that we have provided does not allow one to consider all possible full-correlation Bell-like inequalities. The expression n,m,k; f defined in equation (11) was constructed so that it has the nice property of being decomposable as in (14), namely as a sum of m (n−1)-partite Bell expressions of the same form; there are however symmetric full-correlation expressions which cannot be written in such a way (see, e.g. [34]). Besides, it could also be interesting to look at correlation Bell-like inequalities that do not have full symmetry with respect to permutation of parties. We shall leave these possibilities for future research.

Acknowledgments

We acknowledge useful discussions with Tamas´ Vertesi,´ Stefano Pironio and Antonio Ac´ın. This work is supported by the UK EPSRC, a UQ Postdoctoral Research Fellowship, the Swiss

13 This last set of equalities was only veriﬁed numerically, up to the numerical precision of 10−9.

10 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al

NCCR ‘’Quantum Photonics’, the Spanish MICINN through CHIST-ERA DIQIP and the European ERC-AG QORE.

Appendix A. Reduction of , , to known Bell expressions In m k In this appendix, we show that by appropriate relabeling of inputs and outputs, and possibly by applying some afﬁne transformation, n,m,k reduces to the respective Bell expressions given in ﬁgure 1. I We start by noting that n,m,k can alternatively be written using the bracket notation I k−1 introduced in [35] via the average values R = = rP(R = r): r 0 s s n,m,k = rs − + −rs + . (A.1) I m k m k s s.t. s s.t. [s]m = 0 [s]m = 1

A.1. Reduction to known two-party Bell expressions

For the case of n = 2, n,m,k simpliﬁes to I x + y x + y 2,m,k = Ax + By − + −Ax − By + , (A.2) I + = m k + = m k [x y]m 0 [x y]m 1 where for ease of comparison with the notation adopted in [19], we have written s1 = x, s2 = y, = = = − = − − rs1 Ax and rs2 By. Introducing the new output variables B0 [ B0]k and By [1 Bm y]k for y 1, the above expression becomes m−1 2,m,k = [A0 + B0]k + [Ax + Bm−x − 1]k + [−A0 − B1]k + [−A1 − B0]k I x=1 m−1 + [−Ax − Bm+1−x + 1]k , (A.3) x=2 m−1 m−1 = − + − + − − , [Ax Bx]k [Bx−1 Ax]k [Bm−1 A0 1]k (A.4) x=0 x=1 which is precisely the Bell expression due to BKP [19].

A.2. Reduction to known two-input Bell expressions For the case with two inputs, i.e. m = 2, all terms with all possible combinations of inputs appear in n,2,k: I s s n,2,k = (−1) rs − . (A.5) I 2 k s Deﬁning the new output variables r = [r − s + 1] and r = [r − s ] for all i = 2,...,n, s1 s1 1 k si si i k as well as the new sum r = n r , so that [r −s ] = [r +s−1−s ] = [r +s−1 ] , s i=1 si s 2 k s 2 k s 2 k we can rewrite n,2,k as I − s s 1 n,2,k = (−1) r + , (A.6) I s 2 s k which is precisely, up to a change of notation (r ↔ a , r ↔ a ), the n-partite s j=0 j s j=1 j Svetlichny–CGLMP Bell expression of [24](seealso[36]): one can indeed check that 2,2,k is I 11 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al thesameasS2,d in equation (6) of [24] and that n,2,k satisﬁes the recursive rules of equations (7) and (9) of [24]. (Note that the terms with (−I1)s = 1, respectively −1, in the sum above correspond to the terms denoted as [...] and [...]∗ in [24].)

A.3. Reduction to known two-output Bell expressions

In the case of binary outputs, by applying equation (22)to n,m,2, one finds that n,m,2 is s I I equivalent to (−1) m E .Form = 3, this is precisely the DIEW introduced in [10]. s,[s]m=0,1 s More generally, for k = 2 and when f (s, r) is of the form g(s) · r, one finds from equation (22) that n,m,2;g.r is equivalent to a symmetric full-correlation Bell expression which s is characterized by coefficients of the form g([s]m)(−1) m , i.e. a discrete function of s that is antiperiodic with antiperiod m. This is a characteristic shared by several previously known Bell expressions; in particular, the MABK Bell inequalities [15] and the DIEWs discussed in [25] can also be recovered from n,m,k;g.r (see table 1).

Appendix B. Computing the tripartite biseparable bound of equation (22)

From the definition of a biseparable bound, it follows that the quantum biseparable upper bound on the last sum of equation (22) can be written explicitly as (cf appendix B in the supplementary information of [10]) x+y+z ˆ ˆ max max g([x+y+z]m)(−1) m Ax By ⊗ Cz ρ , (B.1) A =±1 ρ x x,y,z ˆ ˆ where ρ is any quantum state shared by two parties, Bob and Charlie, and By and Cz are ˆ2 = ˆ2 = quantum observables that satisfy By 1, Cz 1. That is, the required biseparable bound is the Tsirelson bound for a bipartite Bell inequality between Bob and Charlie with coefficients defined by m−1 { } x+y+z Ax = ( + + )(− ) m , Myz g [x y z]m 1 Ax (B.2) x=0 but further maximized over all possible choices of the third party (Alice’s) strategies Ax =±1. It thus follows that a (not necessarily tight) biseparable bound on equation (22) can be obtained by solving the semidefinite programme formalized in [30] and optimizing over the choices of Ax. { } To this end, let us follow [25] and construct an m × m matrix M Ax with coefficients given by equation (B.2), but with z replaced14 by m−1−z (here, y and z represent, respectively, the { } row and column indices of M Ax ). By the weak duality of semidefinite programmes [37] and from the results of [30], it can be shown that an upper bound on the Tsirelson bound of any { } bipartite, m-input, two-output, Bell correlation inequality with coefficients defined by M Ax is { } given by m times the largest singular value of M Ax . { } Note that the matrix M Ax thus constructed from equation (B.2) is a Toeplitz matrix (more precisely, a ‘modified circulant matrix’ [25, 38]) with (orthogonal) eigenvectors v = ,ω ,...,ωm−1 j 1 j j and corresponding eigenvalues m−1 m−1 { } λ Ax = ωx ( )ωm−1−s, j Ax j g s j (B.3) x=0 s=0

14 This corresponds to a relabeling of the input of Charlie, which does not change the biseparable bound of the Bell expression.

12 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al

i π (2 j+1) {A } for ω j = e m . Furthermore, one can show that M x is normal, and therefore its singular values are given by the absolute values of its eigenvalues. The desired biseparable bound can {Ax} = ,..., − =± λ then be obtained by computing max j 0 m 1 maxAx 1 j , which can be achieved using the following lemma.

Lemma 1. For a given integer j = 0,...,m−1,letη j be the greatest common divisor of 2 j + 1 and m; then − m 1 η π ωx = η j . max Ax j j csc (B.4) Ax=±1 2m x=0

ωx Proof. To prove this, ﬁrst note that each j is a 2m-root of unity and can therefore be understood as a phase vector on the complex plane. The above optimization over Ax is thus ωx simply a maximization of the magnitude of (the vectorial sum) x Ax j, which can be achieved ωx by concentrating Ax j as much as possible, at most on half a plane. Hence, an optimal choice = ωx ,π) =− of Ax corresponds to setting Ax 1 when the argument of j is in [0 , and Ax 1 when its argument is in [π,2π).

ω ωx = ω j,x = ( + ) It then follows from the deﬁnition of j that Ax j 0 , where j,x 2 j 1 x mod m. Moreover, as x increases from 0 to m − 1 in steps of 1, the integer j,x is never repeated until m x hits , in which case j,x = j,0 = 0. Next, note that 2 j+1 and m are both integer multiples η j of η j; it thus follows that j,x must also be an integer multiple of η j. This together with the fact m m that there are distinct values of j,x as x varies from 0 to − 1 implies that we must have η j η j

m m η −1 η −1 { } j ={ ,η , η ,..., −η }={ η } j . j,x x=0 0 j 2 j m j x j x=0 (B.5) m η −1 { ωx} j Geometrically, this means that all neighboring phase vectors in the set Ax j x=0 are equally spaced. m η j m ω = ωx Finally, note that because A η 1, the phase vectors Ax for larger values of x will j j j be identical to those with 0 x m . Bearing all these in mind, the left-hand side of equation η j (B.4) can now be evaluated to give m η −1 m−1 j η η π A ωx = η ωx j = η j , max x j j 0 j csc (B.6) Ax=±1 2m x=0 x=0 thus completing the proof of lemma 1. m−1 ( )ωm−1−s = m−1 ( )ωs Putting all these together, and noting that s=0 g s j s=0 g s j ,we thus see that for n = 3, the last sum in equation (22) admits a biseparable (upper) bound of − η π m 1 × η j ( )ωs , m max j csc g s j (B.7) j 2m s=0 implying m−1 m−1 B 1 η π ( ) − η j ( )ωs . 3,m,2;g.r m m g s max j csc g s j (B.8) 2 j 2m s=0 s=0

13 J. Phys. A: Math. Theor. 45 (2012) 125301 J-D Bancal et al

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[1] Bell J S 2004 Speakable and Unspeakable in Quantum Mechanics 2nd edn (Cambridge: Cambridge University Press) [2] Werner R F and Wolf M M 2001 Quantum Inform. Comput. 1 1 [3] Buhrman H, Cleve R, Massar S and de Wolf R 2010 Rev. Mod. Phys. 82 665 [4] Mayers D and Yao A 1998 Proc. 39th IEEE Symp. on Foundations of Computer Science (Los Alamitos, CA: IEEE Computer Society Press) p 503 [5] Ac´ın A, Brunner N, Gisin N, Massar S, Pironio S and Scarani V 2007 Phys. Rev. Lett. 98 230501 [6] Pironio S et al 2010 Nature 464 1021 [7] Colbeck R and Kent A 2011 J. Phys. A: Math. Theor. 44 095305 [8] Bardyn C-E, Liew T C H, Massar S, McKague M and Scarani V 2009 Phys. Rev. A 80 062327 [9] Rabelo R, Ho M, Cavalcanti D, Brunner N and Scarani V 2011 Phys. Rev. Lett. 107 050502 [10] Bancal J-D, Gisin N, Liang Y-C and Pironio S 2011 Phys. Rev. Lett. 106 250404 [11] Clauser J F, Horne M A, Shimony A and Holt R 1969 Phys. Rev. Lett. 23 880 Bell J S 1971 Foundation of Quantum Mechanics: Proc. Int. School of Physics ‘Enrico Fermi’ course 49 (New York: Academic) pp 171–81 [12] Werner R F and Wolf M M 2001 Phys. Rev. A 64 032112 [13] Hoban M J, Wallman J J and Browne D E 2011 Phys. Rev. A 84 062107 [14] Pearle P M 1970 Phys. Rev. D 2 1418 Braunstein S L and Caves C M 1990 Ann. Phys., NY 202 22 [15] Mermin N D 1990 Phys. Rev. Lett. 65 1838 Roy S M and Singh V 1991 Phys. Rev. Lett. 67 2761 Ardehali M 1992 Phys. Rev. A 46 5375 Belinskiˇı A V and Klyshko D N 1993 Phys.—Usp. 36 653 Gisin N and Bechmann-Pasquinucci H 1998 Phys. Lett. A 246 1 [16] Zukowski˙ M and Brukner Cˇ 2002 Phys. Rev. Lett. 88 210401 [17] Collins D, Gisin N, Linden N, Massar S and Popescu S 2002 Phys. Rev. Lett. 88 040404 [18] Kaszlikowski D, Kwek L C, Chen J-L, Zukowski M and Oh C H 2002 Phys. Rev. A 65 032118 [19] Barrett J, Kent A and Pironio S 2006 Phys. Rev. Lett. 97 170409 [20] Cirel’son B S 1980 Lett. Math. Phys. 4 93 [21] Svetlichny G 1987 Phys. Rev. D 35 3066 [22] Collins D, Gisin N, Popescu S, Roberts D and Scarani V 2002 Phys. Rev. Lett. 88 170405 [23] Seevinck M and Svetlichny G 2002 Phys. Rev. Lett. 89 060401 [24] Bancal J-D, Brunner N, Gisin N and Liang Y-C 2011 Phys. Rev. Lett. 106 020405 [25] PalKFandV´ ertesi´ T 2011 Phys. Rev. A 83 062123 [26] Popescu S and Rohrlich P 1994 Found. Phys. 24 379 [27] Gisin N 2009 Quantum Reality, Relativistic Causality and Closing the Epistemic Circle: essays in Honour of Abner Shimony ed W C Myrvold and J Christian (The Western Ontario Series in Philosophy of Science) (Berlin: Springer) pp 125–40 (arXiv:quant-ph/0702021v2) [28] Bancal J-D, Gisin N and Pironio S 2010 J. Phys. A: Math. Theor. 43 385303 [29] Bancal J-D, Branciard C, Gisin N and Pironio S 2009 Phys. Rev. Lett. 103 090503 [30] Wehner S 2006 Phys. Rev. A 73 022110 [31] Liang Y-C and Doherty A C 2007 Phys. Rev. A 75 042103 [32] Navascues´ M, Pironio S and Ac´ın A 2007 Phys. Rev. Lett. 98 010401 Navascues´ M, Pironio S and Ac´ın A 2008 New J. Phys. 10 073013 Doherty A C, Liang Y-C, Toner B and Wehner S 2008 Proc. 23rd IEEE Conf. on Computational Complexity (College Park, MD: IEEE Computer Society Press) pp 199–210 Doherty A C, Liang Y-C, Toner B and Wehner S 2008 arXiv:0803.4373 Pironio S, Navascues´ M and Ac´ın A 2010 SIAM J. Optim. 20 2157 [33] Aolita L, Gallego R, Cabello A and Ac´ın A 2011 arXiv:1109.3163 [34] Liang Y-C, Lim C W and Deng D-L 2010 Phys. Rev. A 80 052116 [35] Ac´ın A, Gill R and Gisin N 2005 Phys. Rev. Lett. 95 210402 [36] Chen J-L, Deng D-L, Su H-Y, Wu C and Oh C H 2011 Phys. Rev. A 83 022316 [37] Boyd S and Vandenberghe L 2004 Convex Optimization (Cambridge: Cambridge University Press) [38] Gray R M Toeplitz and circulant matrices: a review http://www-ee.stanford.edu/gray/toeplitz.html

14 Paper M

Quantum nonlocality based on ﬁnite-speed causal inﬂuences leads to superluminal signaling

J.-D. Bancal, S. Pironio, A. Ac´ın, Y.-C. Liang, V. Scarani and N. Gisin

arXiv:1110.3795

215

Quantum nonlocality based on ﬁnite-speed causal inﬂuences leads to superluminal signaling

Jean-Daniel Bancal,1 Stefano Pironio,2 Antonio Ac´ın,3, 4 Yeong-Cherng Liang,1 Valerio Scarani,5, 6 and Nicolas Gisin1 1Group of Applied Physics, University of Geneva, Switzerland 2Laboratoire d’Information Quantique, UniversitéLibre de Bruxelles, Belgium 3ICFO-Institut de CiènciesFotòniques,Av. Carl Friedrich Gauss 3, E-08860 Castelldefels (Barcelona), Spain 4ICREA-InstitucióCatalana de Recerca i Estudis Avan¸cats,Lluis Companys 23, E-08010 Barcelona, Spain 5Centre for Quantum Technologies, National University of Singapore, 3 Science drive 2, Singapore 117543 6Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542 (Dated: October 19, 2011) The experimental violation of Bell inequalities using spacelike separated measurements precludes the explanation of quantum correlations through causal influences propagating at subluminal speed. Yet, it is always possible, in principle, to explain such experimental violations through models based on hidden influences propagating at a finite speed v > c, provided v is large enough. Here, we show that for any finite speed v > c, such models predict correlations that can be exploited for faster-than-light communication. This superluminal communication does not require access to any hidden physical quantities, but only the manipulation of measurement devices at the level of our present-day description of quantum experiments. Hence, assuming the impossibility of using quantum non-locality for superluminal communication, we exclude any possible explanation of quantum correlations in term of finite-speed influences.

Correlations cry out for explanation [1]. Our pre- with one another. quantum understanding of correlations relies on a com- But quantum non-locality is not only puzzling because bination of two basic mechanisms. Either the correlated of its apparent conflict with relativity, it also seems to events share a common cause — such as seeing a flash invalidate the more fundamental idea that correlations and hearing the thunder when a lightning strikes — or can be explained by causal influences propagating con- one event influences the other — such as the position of tinuously in space. Indeed, according to the standard the moon causing the tides. In both cases, we expect the textbook description, quantum correlations are achieved chain of events to satisfy a principle of continuity: that through the collapse of the wavefunction, a process that is is, the idea that the physical carriers of causal influences instantaneous and independent of the spatial separation propagate continuously through space. In addition, we between particles. Any explanation of quantum correla- expect them — given the theory of relativity — to prop- tions via hypothetical influences would therefore require agate no faster than the speed of light. The correlations that they “propagate” at speed v = . Clearly, one may observed in certain quantum experiments call into ques- ask whether infinite speed is a necessary∞ ingredient to action this viewpoint. count for the correlations observed in Nature or whether When measurements are performed on two entan- a finite speed v, recovering a principle of continuity, is gled quantum particles separated far apart from one an- sufficient. At first, this question seems unanswerable. other, such as in the experiment envisioned by Einstein, Indeed, provided that v is large enough, any model re- Podolosky, and Rosen (EPR) [2], the measurement reproducing the non-local correlations of quantum theory sults of one particle are found to be correlated to the through (hidden) influences propagating at a finite speed measurement results of the other particle. Bell showed v > c can always be made compatible with all exper- that if these correlated values were due to local common imental results observed so far. It thus seems like the causes, then they would satisfy a series of inequalities best that one could hope for is to put lower-bounds on v arXiv:1110.3795v1 [quant-ph] 17 Oct 2011 [1]. But theory predicts and experiments confirm that by testing the violation of Bell inequalities with systems these inequalities are violated [3], thus excluding any lo- that are further apart and better synchronized [6, 7]. cal common cause type of explanation. Moreover, since Here we show that there is a fundamental reason why the measurement events can be spacelike separated, any influences propagating at a finite speed may not account influence-type explanation must be based on influences for the non-locality of quantum theory. We demonstrate, propagating faster than light. following an original suggestion of [8, 9], that all mod- This non-local connection between distant particles els for quantum correlations where non-local influences presents profound interpretative difficulties and is a propagate at a given finite speed v > c give, for any v, source of tension between quantum theory and relativity predictions that can be used for faster-than-light commu- [4, 5]. However, it does not put the two theories in direct nication. More precisely, consider any such model that conflict thanks to the no-signaling property of quantum correctly reproduces the quantum prediction within the correlations. This property guarantees that spatially sep- range of its causal influences and ceases to violate Bell in- arated observers in an EPR-type experiment cannot use equalities beyond this range. Then we show that it will their measurement choices and outcomes to communicate also necessarily predict, in certain configurations, mea- 2

time future time a) b) K2 B

K1 A A B

past space space FIG. 2. Bipartite Bell experiments in a v-causal model. FIG. 1. Space-time diagram in the privileged reference frame. a) A is in the past v-cone of B. The variable λ, In the (shaded) light cone delimited by solid lines, causal in- with probability distribution q(λ), denote the joint state fluences propagate up to the speed of light c, whereas in the of the particles, or more generally a complete specifica- v-cone (hatched region), causal influences travel up to the tion of any initial information in the shaded spacetime speed v. An event K1 can causally influence a spacelike sep- region that is relevant to make predictions about a and arated event K2 contained in its future v-cone, but cannot b [15]. In this situation, we can write PA c on the speed of abilities PA K2) if K2 is in the future (past) will predict PA B(ab xy) = q(λ)P (a x, λ)P (b y, λ), ∼ | λ | | v-cone of K1 and K1 K2 if K1 and K2 are outside see Figure 2b; that is, the model will predict correlations ∼ P each other’s v-cones. An event at K1 can have a causal that are formally “local” and satisfy Bell inequalities. influence on points K2 > K1 in its future v-cone and can Intuitively, the two systems cannot influence each other, be influenced by points K2 < K1 in its past v-cone. But but can nevertheless be correlated through the variables there cannot be any direct causal relation between two λ specifying their common past. The causal structure events K1 K2 that are outside each other’s v-cones: that we consider thus simply corresponds to Bell’s no- any correlation∼ between them must originate from com- tion of local causality [17] but with the speed of light c mon causes in the intersection of their past v-cones. replaced by the speed v > c. 3

Quantum theory and v-causal models. rections x, y and counting detector clicks a, b). For instance, when the spacetime ordering between the systems According to quantum theory, measurements on two is such that the model reproduces the correlations pre- separated systems prepared in the quantum state ρ yield dicted by quantum theory, then these correlations can x obviously not be used for signaling. joint probabilities of the form PQ(ab xy) = tr(ρMa y | ⊗ A suﬃcient condition for the correlations P not to Mb ), regardless of the spacetime ordering between the measurements. A v-causal model for quantum correla- be exploitable for signaling is that they satisfy a series tions is then one that reproduces PQ(ab xy) in the sit- of mathematical constraints known as the “no-signaling uations A < B and B < A, i.e., a model| such that conditions”. In the bipartite case, these are the conditions that the marginal distributions P (ab xy) = PA

Then the following inequality is satisfied time C0 S = 3 A B B C 3 D − h 0i − h 0i − h 1i − h 0i − h 0i A B A B + A C BC − h 1 0i − h 1 1i h 0 0i + 2 A C + A D + B D h 1 0i h 0 0i h 0 1i B1D1 C0D0 2 C1D1 − h i − h i − h i (2) D + A B D + A B D + A B D h 0 0 0i h 0 0 1i h 0 1 0i A B D A B D A B D − h 0 1 1i − h 1 0 0i − h 1 1 0i A + A C D + 2 A C D 2 A C D h 0 0 0i h 1 0 0i − h 0 1 1i space 7, ≤ FIG. 3. Four-partite Bell-type experiment characterized by where we have introduced the correlators A = the spacetime ordering R = (A < D < B C). Let λ x ∼ 1 a 1 a+b h i describe any relevant information from the past of A, B, C, D ( 1) P (a x), AxBy = ( 1) P (ab xy), a=0 − | h i a,b=0 − | and in addition let µ be a (sufficiently complete) specification A B C = 1 ( 1)a+b+cP (abc xyz), and so on. hPx y zi a,b,c=0 − P | of the shaded region, which screens-off the intersection of the past v-cones of B and C. Note that µ may depend Proof. Let PADP (00 00) denote the AD marginal prob- on the value of the past variables a, x, d, w, λ and is thus abilities P (a = 0|, d = 0 x = 0, w = 0) and let characterized by a probability distribution q(µ axdwλ). Since | | PB AD(b y) denote the B AD probabilities P (b y, a = B C, we have as in Figure 2b P (b y, czµ) = P (b y, µ) and | | | | ∼ | | 0, x = 0, d = 0, w = 0), and define similarly PC AD(c z) P (c z, byµ) = P (c z, µ). We can thus write PR(abcd xyzw) = | | |q(λ)P (a x, λ)P| (d w, axλ) q(µ axdwλ)P (b y,| µ)P (c z, µ). and PBC AD(bc yz). Consider the following inequality λ | | µ | | | | | This implies that the correlations BC condi- Ptioned on AD are localP since P (bc yz, axdw) = I = P (1000 0000) + P (0001 0010) + P (0011 0011) R q˜(µ axdw)P (b y, µ)P (c z, µ) whereq ˜(µ axdw| )= q(λ) | | | µ | | | | λ × + P (0100 0011) + P (1000 0100) + P (0011 0110) P (a x, λ)P (d w, axλ)q(µ axdwλ)/ q(λ)P (a x, λ)P (d w, axλ). | | | P | | | λ | P | + P (0000 0111) + P (0111 0111) + P (0010 1000) Let us now show that the marginal correlations ABD are | | | quantum. For this, suppose thatP a choice is made af- + P (1100 1000) + P (0010 1100) + P (1100 1100) | | | (3) ter the shaded spacetime region (and in a way that is + PAD(00 00) 1 PB AD(0 0) PC AD(0 0) independent of the variables µ) to delay the measure- | | | − | − | ment on particle C, up to the point C0. Thus, B now + PBC AD(00 00) + PBC AD(00 01) | | | | lies in the past v-cone of C and the spacetime ordering is + PBC AD(00 10) PBC AD(00 11) 0 . T = (A < D < B < C0). We can then write PT (abcd xyzw) = | | − | | ≥ q(λ)P (a x, λ)P (d w, axλ) q(µ axdwλ)P (b y,| µ)P (c z, byµ), λ | | µ | | | This inequality is satisfied by any correlations P from which it follows that PT (abd xyw) = PR(abd xyw) = P P q(λ)P (a x, λ)P (d w, axλ) q(|µ axdwλ)P (b y, µ|). fulfilling condition i). Indeed, the first twelve λ | | µ | | But since by assumption PT = PQ, we deduce that terms and the term PAD(00 00) are clearly posi- P P | PR(abd xyw) = PQ(abd xyw). Similarly, it can be shown tive. Moreover, the term in square brakets is positive | | that PR(acd xzw) = PQ(acd xzw). since 1 PB AD(0 0) PC AD(0 0) + PBC AD(00 00) + | | − | | − | | | | PBC AD(00 01) + PBC AD(00 10) PBC AD(00 11) 0 is nothing| but| the Clauser-Horne-Shimony-Holt| | − | | (CHSH)≥ where U = cos( 4π )σ sin( 4π )σ and H is the Hadamard inequality [18] for the BC correlations conditioned on 5 z − 5 x a = 0, x = 0, d = 0, w = 0 and is thus non-negative matrix, yields S = 7.2014 > 7. according to condition i). Using the no-signaling con- Assume now that there exists a v-causal model for the ditions, it is now easy to see that S can be written as above quantum correlations, that is, that there exists a S = 7 8I, which implies S 7 [19]. model such that PT = PQ if T is a spacetime order- − ≤ ing that does not constrain causal influences between Note that inequality (2) is violated by quantum theory, any pair of systems. Consider the predictions of such since measuring the state a model in a configuration where the quantum predictions need not be reproduced completely such as in the 17 1 1 1 configuration of Figure 3, characterized by the ordering Ψ = 0000 + 0011 0101 + 0110 R = (A < D < B C), i.e., superluminal influences can | i 60| i 3 | i − √8 | i 10 | i propagate from system∼ A to all the other ones and from 1 1 1 1 + 1000 1011 1101 + 1110 . system D to systems B and C, but cannot propagate 4 | i − 2 | i − 3 | i 2 | i between systems B and C. By definition of the model, with the operators it then follows that the correlations BC AD are local, i.e., condition i) of the Lemma is satisfied,| see Figure 3 ˆ ˆ ˆ for details. Since B C, as noted above, we should A0 = UσxU †, A1 = UσzU †, B0 = H, ∼ − not expect, in general, that PR = PQ. However, in- Bˆ = σ Hσ , Cˆ = Dˆ = σ , Cˆ = Dˆ = σ , 1 − x x 0 − 0 z 1 1 − x equality (2) is completely determined by the value of the 5 tripartite terms P (abd xyw) and P (acd xzw), where the time model does give the same| predictions as| quantum theory for the spacetime ordering R, that is, any v-causal model A0 for these quantum correlations satisfies P (abd xyw) = R | PQ(abd xyw) and PR(acd xzw) = PQ(acd xzw), see cap- D0 tion of Figure| 3. Intuitively,| this is because| the marginal correlations ABD (ACD) are well-defined, that is independent of the measurements performed on C (B), and in particular independent of whether this measure- B C ment is delayed, in which case the correlations should reproduce the quantum ones. Therefore, any quantum D v-causal model for the above quantum correlations will space also yield S = 7.2014 > 7 in the configuration of Fig- A dB dC dD ure 3. Inequality (2) is thus violated and therefore one of the two conditions of the Lemma must be violated. But FIG. 4. Let the four systems of Figure 3 lie along some spatial 1 1 1 since the model satisfies condition i), it must necessarily direction at, respectively, a distance dB = 4 (1 + r ) + 1+r , 1 1 1 violate condition ii), i.e., the correlations P must violate dC = (1 + ) , dD = 1 form A, where r = v/c > 1, R 4 r − 1+r the non-signaling conditions (1). 2 and let them be measured at times tA = 0, tB = tC = c+v , This implies that the correlations PR can be used for tD = 1/r. Suppose that the correlations PR produced by a superluminal communications. Indeed, since they violate v-causal model are such that the BCD marginal correlations the no-signaling conditions, at least one of the tripartite depend on the measurement x made on the first system A. If correlations ABC, ABD, ACD, or BCD must depend parties B and C broadcast (at light-speed) their measurement on the measurement setting of the fourth party. This results, it will be possible to evaluate the marginal correlations is not the case for the marginal ABD (ACD) as it is BCD, at the point D0. Since this point lies outside the future light-cone of A (shaded area), this scheme can be used for defined independently of C (B) (and equal to the quan- superluminal communication from A to D0. Similarly, if the tum marginal). It thus follows that either the marginal ABC marginal correlations depend on the measurement w ABC must depend on the measurement setting w of sys- made on D, they can be used for superluminal communication tem D or that the marginal BCD must depend on the from D to the point A0. measurement setting x of system A. In both cases, these marginals can be evaluated outside the fourth party’s future light-cone and can thus be explicitly used for su- This work illustrates the difficulty to modify quantum perluminal communication, see Figure 4. physics while maintaining no-signaling. If we want to keep no-signalling, it shows that quantum non-locality must necessarily relate discontinuously parts of the uni- Conclusion verse that are arbitrarily distant. This gives further weight to the idea that quantum correlations somehow We have shown that if the non-local effects that we ob- arise from outside spacetime, in the sense that no story serve in Bell experiments were due to hidden influences in space and time can describe how they occur. propagating at any finite speed, then non-locality could Acknowledgments. We acknowledge Serge Massar and be exploited for superluminal communication. Our re- Tamas Vértesifor helpful discussions and Jonathan Sil- sults therefore uncover a new aspect of the complex rela- man for comments on the manuscript. This work was tionship between multipartite quantum non-locality and supported by the European ERC AG Qore and SG the impossibility of signaling [20–22]. PERCENT, the European EU FP7 QCS and Q-Essence Our results answer a question first raised in [8, 9]. Par- projects, the CHIST-ERA DIQIP project, the Swiss NC- tial progress on this problem was made in [23], where a CRs QP & QSIT, the Interuniversity Attraction Poles conclusion similar to ours was obtained for a particular Photonics@be Programme (Belgian Science Policy), the set of non-quantum correlations. The approach of [23], Brussels-Capital Region through a BB2B Grant, the based on the tripartite configuration considered in [8, 9], Spanish FIS2010-14830 project, the National Research does not seem, however, to generalize in a straightforward Foundation and the Ministry of Education of Singapore. way to the physically relevant case of quantum correlations. Our approach is based instead on the introduction of a general formulation of the concept of v-causality, which allowed us to go beyond the configuration orig- inally considered in [8, 9]. It would be interesting to understand if the conceptual framework presented here can be used to rule out finite-speed influences for any conceivable non-local theory or if there exist non-local theories compatible with finite speed influences. 6

[1] J. S. Bell, Speakable and Unspeakable in Quantum Me- A. J. Banday, P. Lubin, Astrophys. J. 470, 38 (1996). chanics: Collected papers on quantum philosophy (Cam- [12] D. Bohm, Phys. Rev. 85, 166 (1952); ibid., 85, 180 bridge University Press, Cambridge, 2004). (1952). [2] A. Einstein, B. Podolsky, and N. Rosen, Physical Review [13] G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 47, 777 (1935). 34, 470 (1986). [3] A. Aspect, Nature 398, 189 (1999). [14] However, for a fully relativistic collapse theory, see R. Tu- [4] T. Mauldin, Quantum Non-Locality and Relativity: mulka, pp. 340-352 in A. Bassi, D. Duerr, T. Weber and Metaphysical Intimations of Modern Physics (Blackwell N. Zanghi (eds), Quantum Mechanics: Are there Quan- Publishers, Oxford, 2002). tum Jumps? and On the Present Status of Quantum Me- [5] A. Shimony, Search for a Naturalistic World View: Sci- chanics, AIP Conference Proceedings 844, American In- entific Method and Epistemology (Cambridge University stitute of Physics (2006). Press, Cambridge, 1993). [15] Strictly, for a v-causal model, only the shaded region that [6] D. Salart, A. Baas, C. Branciard, N. Gisin, and is in the past v-cone of A can have a causal influence on A; H. Zbinden, Nature 454, 861 (2008). likewise for the other figures. However, all our arguments [7] B. Cocciaro, S. Faetti and L. Fronzoni, Phys. Lett. A still follow through even if we consider spacetime regions 375, 379 (2011). of the kind depicted. [8] V. Scarani and N. Gisin, Phys. Lett. A 295, 167 (2002) [16] By this, we mean that given the specification of λ in Λ, [9] V. Scarani and N. Gisin, Braz. J. Phys. 35, 2A (2005). specification of any information in the past v-cones of A [10] Or a particular foliation of spacetime into spacelike hy- and B become redundant. perplanes. One reason to assume a privileged frame is [17] T. Norsen, arXiv:0707.0401 (2007). that it is not easy to formulate a fully relativistic model [18] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, (i.e. with the Lorentzian metric as the only spacetime Phys. Rev. Lett. 23, 880 (1969), J. F. Clauser and M. A. structure) containing superluminal causal influences that Horne, Phys. Rev. D 10, 526 (1974). are limited in spacetime and that does not lead to causal [19] To see the equivalence between (2) and (3), one can write 1 a b paradoxes [4]. The results presented here, however, can P (abcd xyzw) = 16 (1 + ( 1) Ax)(1 + ( 1) By)(1 + c | hd − · − · probably be adapted to any model of this sort. Also, ( 1) Cz)(1 + ( 1) Dw) , expand the products, insert our results apply to models with many preferred frames into− (3)· and cancel− all· pairsi of terms with opposite signs. defined by the measuring devices, in this case even for [20] M. L. Almeida, J.-D. Bancal, N. Brunner, A. Ac´ın,N. v = [8]. Gisin, S. Pironio, Phys. Rev. Lett. 104, 230404 (2010). ∞ [11] In fact, even in a perfectly Lorentz-invariant theory, there [21] R. Gallego, L. E. Würflinger,A. Ac´ın,and M. Navascués, can be natural preferred frame due to the non-Lorentz- arXiv:1107.3738 (2011). invariant distribution of matter – a well-known example [22] T. H. Yang, D. Cavalcanti, M. Almeida, C. Teo, and V. of this is the reference frame in which the cosmic mi- Scarani, arXiv:1108.2293 (2011). crowave background radiation appears to be isotropic. [23] S. Coretti, E. Hänggi,and S. Wolf, Phys. Rev. Lett. 107, See C. Lineweaver, L. Tenorio, G. F. Smoot, P. Keegstra, 100402 (2011). Paper N

The deﬁnition of multipartite nonlocality

J. Barrett, S. Pironio, J.-D. Bancal and N. Gisin

arXiv:1112.2626

223

The deﬁnition of multipartite nonlocality

Jonathan Barrett,1 Stefano Pironio,2 Jean-Daniel Bancal,3 and Nicolas Gisin3 1Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX U. K. 2Laboratoire d’Information Quantique, UniversitéLibre de Bruxelles, Belgium 3Group of Applied Physics, University of Geneva, Switzerland In a multipartite setting, it is possible to distinguish quantum states that are genuinely n-way entangled from those that are separable with respect to some bipartition. Similarly, the nonlocal correlations that can arise from measurements on entangled states can be classified into those that are genuinely n-way nonlocal, and those that are local with respect to some bipartition. Svetlichny introduced an inequality intended as a test for genuine tripartite nonlocality. This work introduces two alternative definitions of n-way nonlocality, which we argue are better motivated both from the point of view of the study of nature, and from the point of view of quantum information theory. We show that these definitions are strictly weaker than Svetlichny’s, and introduce a suitable Bell- type inequality for the detection of 3-way nonlocality. Numerical evidence suggests that all 3-way entangled pure quantum states can produce correlations that violate this inequality.

Consider two quantum systems, prepared in a joint cal correlations enables the quantum advantage in com- quantum state ψ and located in separate regions of munication complexity problems [2], device independent space. Suppose| Alicei measures one system, obtaining quantum cryptography [3, 4], randomness expansion [5], outcome a, and Bob the other, obtaining outcome b. The and measurement-based quantum computation [6, 7]. joint outcome probabilities can be written P (ab XY ), With three or more systems, qualitatively diﬀerent where X is Alice’s measurement and Y is Bob’s measure-| kinds of nonlocality can be distinguished. For deﬁnite- ment. If the measurements are performed at spacelike ness, consider the tripartite case. If correlations can be separation, then Bell’s condition of local causality [1] im- written plies that even if the particles have interacted in the past P (abc xyz) = q P (a x) P (b y) P (c z), (3) (or were produced together in the same source), they are | λ λ | λ | λ | now independent. Therefore, even if the quantum state Xλ of the two particles is entangled, it ought to be possible with 0 qi 1 and qi = 1, then they are local. ≤ ≤ i to specify a more complete description λ of the joint state Otherwise they are nonlocal. But, as pointed out by P of the two particles, such that given λ, the probabilities Svetlichny [8], some correlations can be written in the can be written in the form hybrid local-nonlocal form

P (ab XY ) = P (a X)P (b Y ). (1) P (abc xyz) = qλ Pλ(ab xy) Pλ(c z)+ λ | λ | λ | | | | Xλ The state λ is conventionally referred to as a hidden state, q P (ac xz) P (b y) + q P (bc yz) P (a x), µ µ | µ | ν ν | ν | since it is not part of the quantum description of the µ ν X X experiment. Any hidden state λ which satisﬁes Eq. (1) (4) is local. If the observed correlations P (ab XY ) can be | where 0 q , q , q 1 and q + q + q = 1. explained by a locally causal theory, then they can be ≤ λ µ ν ≤ λ λ µ µ ν ν written Here, each term in the decomposition factorizes into a product of a probability pertainingP P to one party’sP out- P (ab XY ) = q P (a X)P (b Y ). (2) come alone, and a joint probability for the two other | λ λ | λ | arXiv:1112.2626v1 [quant-ph] 12 Dec 2011 λ parties. We say that correlations of the form (4) are S2- X local. If correlations cannot be written in this form, then On the other hand, if correlations P (ab XY ) violate a a term like P (abc xyz) must appear somewhere in the | λ Bell inequality [1], then they cannot be written in this decomposition. Such| correlations are often said to ex- form. Such correlations cannot be explained by a locally hibit genuine 3-way nonlocality, although we will refer causal theory, and are referred to as nonlocal correlations. to this as Svetlichny nonlocality. Svetlichny introduced Quantum nonlocality is a puzzling aspect of nature, an inequality, violation of which implies Svetlichny non- but also an important resource for quantum informa- locality. Svetlichny’s inequality can be violated by ap- tion processing. An information theoretic interpreta- propriate measurements on a GHZ or W state [9]. tion of quantum nonlocality is that two separated par- In further work, Seevinck and Svetlichny [10], and in- ties who wish to simulate the experiment with classical dependently, Collins et al. [11], generalized the tripartite resources cannot do so using only shared random data notion of Svetlichny nonlocality to n parties. In both - they must also communicate with one another. The Refs. [10] and [11], an inequality is derived that detects fact that entangled quantum states can produce nonlo- n-partite Svetlichny nonlocality. See also Refs. [9, 12, 13]. 2

The present work considers two alternative definitions of genuine multipartite nonlocality, which are different from Svetlichny’s. We argue that these definitions are Y1 = b2 better motivated, both physically, and from the point of X2 = a1 view of information theory. We show that the alternative definitions are strictly weaker than Svetlichny’s and 2 1 describe a Bell inequality such that its violation is suf- a ficient for genuine 3-way nonlocality according to both 2 b1 alternative definitions. Numerical evidence suggests that ¸1 ¸2 X1 Y2 any pure, 3-way entangled quantum state can produce correlations that violate this inequality. On the other 1 2 hand, there exist pure, 3-way entangled quantum states for which we have not been able to find any measure- a1 b2 ments giving rise to Svetlichny nonlocality. Different kinds of nonlocality. Consider again the case of bipartite correlations. There are various ways in which a hidden state λ might fail to be local. Let FIG. 1: Let X, Y, a, b 0, 1 . The particle pair labelled 1 ∈ { } Pλ(a XY ) = Pλ(ab XY ) be the marginal probabil- is independent from the pair labelled 2. The joint state λ is | b | 1 ity for Alice to obtain outcome a when the measure- such that if a1 = Y1, then Pλ (a1b1 X1Y1) = 0, whereas λ2 P 1 | ment choices are X and Y , and similarly let Pλ(b XY ) = is such that if b2 = X2, then Pλ2 (a2b2 X2Y2) = 0. Consis- | tent predictions are6 impossible if measurement| choices are as a Pλ(ab XY ) be the probability for Bob to obtain b. Suppose that| λ satisfies shown. P

P (a XY ) = P (a XY 0) a, X, Y, Y 0 (5) λ | λ | ∀ how signalling hidden states can lead to grandfather-style Pλ(b XY ) = Pλ(b X0Y ) b, Y, X, X0. (6) paradoxes, where no consistent assignment of probabili- | | ∀ ties to outcomes is possible. In this case, if Alice and Bob are in possession of two par- One solution to these problems would be to restict at- ticles, which they know to be in the hidden state λ, then tention to models that involve only non-signalling hid- even if λ is nonlocal, observing her own outcome gives den states. But a more general solution is to introduce a Alice no information about Bob’s measurement choice. notion of hidden state, according to which the outcome This is because the marginal probabilities for a are in- probabilities can vary according to the timing of the mea- dependent of Bob’s choice. Hence Bob cannot send sig- surements. In a fully general treatment, λ will be time nals to Alice by varying his measurement choice. Sim- dependent, or alternatively, λ will refer to the state of the ilarly, Alice cannot send signals to Bob. Such a λ is particles at some fixed time (perhaps just after creation) non-signalling. in some fixed frame, and the probabilities for outcomes If Eq. (5) is satisfied but Eq. (6) is violated, then Bob’s depend on the exact timing of the measurements. outcome gives him at least some information about Al- For now, let’s keep things simple. Consider a hidden ice’s measurement choice, hence Alice can send signals to state λ such that the probabilities do not depend on the Bob. The hidden state λ is 1-way signalling. Similarly if exact timing of measurements, but do depend on the time Eq. (6) is satisfied but Eq. (5) is violated. If Eqs.(5) and ordering, where this ordering is determined with respect (6) are both violated then λ is 2-way signalling. to a fixed background frame. If Alice performs X before So far, this discussion has followed many treatments of Bob performs Y , the probabilities are given by quantum nonlocality, in that no attention has been given P A