Hidden Nonlocality

Flavien Hirsch

September 2012 - June 2013

University of Geneva

Supervisor : Prof. Nicolas Brunner Abstract

In this essay we study the hidden nonlocality which can be revealed by local filtering. We present the idea and the physical motivations, and we recapitulate all the known cases. We then construct some new examples in lower dimension and discuss the local models of the Werner states. For the bipartite qubits case, we propose some improvements on these models that allow us to simulate a larger class of states, and we construct a protocol that, from a projective-local state, gives a POVM-local one. Finally, we prove the existence of genuine hidden nonlocality with a few concrete examples, including maximal and genuine hidden nonlocality.

2 Contents

Contents 3

1 Introduction 5

2 Concepts and motivations 6 2.1 Nonlocality and Bell’s game ...... 6 2.1.1 The scenario ...... 6 2.1.2 Classification of the probability distribution ...... 7 2.1.3 Bell’s game ...... 9 2.2 Entanglement ...... 11 2.2.1 Quantum formalism ...... 11 2.2.2 Characterization of quantum states ...... 12 2.3 SLOCC and hidden nonlocality ...... 14 2.4 Notations ...... 16 2.5 Classes of measurement ...... 17 2.5.1 Quantum observable ...... 18

3 Local states 19 3.1 Local models ...... 19 3.1.1 Simulate POVMs ...... 20 3.2 Werner state ...... 20 3.2.1 Werner’s model ...... 21 3.2.2 Barrett’s model ...... 22 3.2.3 Interpretation of Werner’s and Barrett’s model ..... 22 3.3 New classes of local states ...... 23 s 3.3.1 ra ...... 23 3.3.2 The model ...... 24 s 3.3.3 ra,b ...... 26 s 3.3.4 ra,b,g ...... 28

4 Examples of filterable states 30 4.1 General filtering and qubits filtering ...... 30 4.2 Werner state ...... 32 4.3 Gisin’s state ...... 33 4.4 Collins-Gisin’s state ...... 34 4.5 Hidden nonlocality for a two-qubits state ...... 35

3 CONTENTS 4

5 Maximal and genuine hidden nonlocality 36 5.1 From projective measurements to POVMs ...... 36 5.2 Genuine hidden nonlocality ...... 38 5.3 Maximal hidden nonlocality ...... 39 5.4 Maximal and genuine hidden nonlocality ...... 42 5.5 Generalization of rGHNL and of the POVM protocol ...... 44 5.5.1 The protocol for a qudit ...... 44 a,b,g,d 5.5.2 rGHNL ...... 44 6 Open questions 46 6.1 Sequential filtering ...... 46 6.2 POVM versus projectors ...... 48 6.3 Genuine locality ...... 48

7 Conclusion 49

8 Acknowledgments 50

Bibliography 51 Chapter 1

Introduction

For more than a century Quantum Mechanic has been studied and tested every day by the physicists. Thanks to the progress of the science and in particular those of the experimental apparatus, the precision with which one is able to verify the theories has become better everyday. So, for more than a century Quantum Mechanic has been put to the proof more and more rigor- ously, and to this day has never been faulted. The precision of the quantum theory is spectacular and it therefore seems to describe perfectly the observed phenomena; in all cases better than any other theory.

Yet, however good are its predictions, Quantum Mechanic continues to defy the human intuition. Indeed, three properties of the theory are very difficult to accept: intrinsic randomness, collapse of the wavefunction and nonlocality. While the two first ones are fundamental properties of the theory and have given rise to fruitful discussions about independence of Nature and about determinism, the third one was discovered later and was considered first as an anomaly of the theory. Nowadays, we have tested and accepted nonlocality as a characteristic of Nature. It means that two systems can be well-separated in space (and in particular space-like separated) but act as if they were one single system. We strongly think that no realistic theories are able to predict the future.

Knowing this, it is not astonishing that nonlocality has been vastly studied in the past thirty years. What seems less normal is the actual theoretical disorder about nonlocality. Multipartite scenarios, Bell’s inequalities, link between entanglement and local models: despite a lot of nice results and the work of brilliant people, these tasks are very partially understood, and we still lack an overview, a general formalism, an analytical method to solve the problems we have.

Genuine hidden nonlocality is one of these big questions: can a fully local state becomes nonlocal after local operations? Even if some partial results are known (for projective measurements), no definitive answer has been yet given. In this work we will prove that the answer to the above question is yes.

5 Chapter 2

Concepts and motivations

The aim of this first chapter is to introduce all the concepts that will be used later in this work and to motivate them physically.

2.1 Nonlocality and Bell’s game

Since quantum mechanics gives probability to events, i.e. is not deterministic, we will focus ourselves on probability theory, define nonlocality and charac- terize the quantum probability distributions. Finally we will explain what we call a Bell’s game.

2.1.1 The scenario The standard situation we will focus on is the one with two parties, sharing some resources. Let’s call Alice the "left" one and Bob the "right" one for convenience.

Figure 1: the general situation we consider

Now Alice and Bob can perform a measurement on their respective re- sources, and obtain a result. We do not care about the exact mechanism of what they do, but our assumptions are the following ones: they can freely choose between a finite set of possible measurements and the set of the results they can obtain after the measurement is finite as well.

If Alice has NA possible measurements we symbolically define MA = A , A , ..., A as the set of measurements and we call the integers num- { 1 2 NA } bers that label the measurements the inputs of Alice. We define RA = RA , RA , ..., RA as the set of the n results she can obtain and we call the { 1 2 na } A integers numbers that label the results the outputs. Similarly, the Bk represent

6 CHAPTER 2. CONCEPTS AND MOTIVATIONS 7

Bob’s measurements and the RBk his results (with set of size NB and nB, respectively).

Note that it is a very general way to study two physical systems; the assumption about the finite number of inputs and outputs is used just for convenience (one can extend to countable and uncountable infinite number of possibility, in particular the quantum formalism allow such case) and we did not mention anything about the physical system nor what the measurement is.

Since we do not focus on what really happens when a measurement is applied on a part, we can represent our scenario in the following way: both parties have a box, with a finite number of buttons on the top which can be pushed individually and a finite number of lights on the bottom which can light individually. What Alice and Bob can do is push one of the buttons -which therefore represent the inputs- and note which light is lit -i.e. which output they obtain. This approach is often called the device-independent approach.

Figure 2: Each party has a box with possible inputs and outputs

All the information about this bipartite system is contained in the prob- ability of obtaining a pair of outputs given a pair of measurements. In other terms we want to know

p(a, b A , B ) (2.1) | x y where a 1, 2, ..., N , b 1, 2, ..., N That is (2.1) is the so-called joint 2{ A} 2{ B} probability that Alice obtains the result RAa given the measurement Ax and Bob obtains the result RBb given the measurement By.

2.1.2 Classification of the probability distribution From here we will rewrite (2.1) in the following form: p(a, b A, B), for con- | venience. Please keep in mind that A, respectively B, can be any measurement chosen in the set MA, respectively MB. Definition 1 (Determinism). A joint probability distribution p(a, b A, B) is de- | terministic if

p(a, b A, B)=d d (2.2) | a,a0 b,b0 CHAPTER 2. CONCEPTS AND MOTIVATIONS 8

where a0 (b0) is the result that Alice (Bob) obtains after measurement A (B), and di,j = 1 if i = j, 0 otherwise. That is, a measurement determines precisely its result. Obviously this will never be the case with a quantum system.

Definition 2 (Independence). A joint probability distribution p(a, b A, B) is inde- | pendent if it can be written as

p(a, b A, B)=p(a A)p(b B) (2.3) | | | i.e. if the probability distribution factorizes.

This corresponds to the trivial case where Alice and Bob are completely uncorrelated; there is no particular interest to study them together rather than separately in this situation.

Definition 3 (Locality). A joint probability distribution p(a, b A, B) is local if it | can be written as

p(a, b A, B)= p(a A, l)p(b B, l)r(l)dl (2.4) | ZL | | where L is a set, l L and r : L [0, 1] is a probability density. 2 ! The interpretation of this decomposition of the probability distribution is that Alice and Bob share some local resources, described by the variable l. The variable is said to be local because we assume that physically it is created somewhere (according to the probability distribution r(l)) and sent to Alice and Bob. The origin of the randomness on this variable can be quantum or classical but the point is that the correlations of Alice and Bob are completely explained by this shared variable and what they do locally.

Definition 4 (Nonlocality). A joint probability distribution p(a, b A, B) is nonlocal | if it cannot be written as (2.4).

Definition 5 (Non-signalling). A joint probability distribution p(a, b A, B) is | nonsignalling if

 p(a, b A, B)= p(a, b A, B0) (2.5) b | b |

 p(a, b A, B)= p(a, b A, B0) (2.6) a | a |

A, A , B, B 8 0 0 It means that the measurement of Bob does not influence the result of Alice and vice versa. We notice that a local probability distribution is always non-signalling. CHAPTER 2. CONCEPTS AND MOTIVATIONS 9

Definition 6 (Marginals). We call p(a A, B)=Â p(a, b A, B) the marginals of | b | Alice. Bob’s marginals are defined similarly. Note that if a probability distribution is non-signalling we have p(a A, B)= | Â p(a, b A, B)=Â p(a, b A, B )=p(a A, B ), B, B . Therefore p(a AB)= b | b | 0 | 0 8 0 | p(a A) in the non-signalling case. | To summarise, non-signalling is the weakest condition we can put on a joint probability distribution, it strictly includes the local probability distributions, which strictly includes the independent ones, which strictly includes the deterministic. The two relevant sets for this essay will be the local and non-signalling ones and the nonlocal and non-signalling ones.

2.1.3 Bell’s game From now on, we will add the condition that Alice and Bob are space-like separated during the time between the choice of their measurements and their appliance. So, according to relativity, without faster-than-light communication what one party does locally cannot give any information to the other one. Accordingly we are in the non-signalling scenario, since if Alice’s marginals depended on the measurement that Bob would choose, it would be possible, in principal, to extract locally some informations on Bob’s choice whatever the distance between the two parties is, and that would violates the relativity. However, there still is an interesting property that the probability distribution may have: the nonlocality. As we defined above, a probability distribution is nonlocal if it cannot be written in the form (2.4), means it cannot be deduced from a common variable that Alice and Bob have, and of what they do locally.

Now an important question is the following one: if we have a certain probability distribution in hand, how can we check whether it is nonlocal or not? To show that it is local, one can give explicitly the functions p(a A, l), | p(b B, l) and r(l) which satisfy the condition (2.4). But if it is not, one has to | show that for all p(a A, l), p(b B, l) and r(l) this condition is never fulfilled, | | that is, there exists no such decomposition of the probability distribution. This seems to be a very tedious task. Fortunately, we have a powerful tool: the Bell’s inequalities. Definition 7 (Bell’s inequality). Let M = A , A , ..., A and M = B , B , ..., B A { 1 2 NA } B { 1 2 NB } be two sets of measurements with nA and nB their number of outputs, respectively. Let p(a, b A, B) be a probability distribution define for all A M and B M . | 2 A 2 B Let f be a function on the following form:

f (MA, MB, p)= Â aa,b,x,y p(a, b Ax, By) (2.7) a,b,x,y |

where the aa,b,x,y are some real numbers.

We call a Bell’s inequality all pair of f , L with f on the form (2.7) and L a real { } number such that the condition f (MA, MB, p) > L is a sufficient condition for the nonlocality of the probability distribution p. CHAPTER 2. CONCEPTS AND MOTIVATIONS 10

The first example of such an inequality was discovered by Clauser, Horne, Shimony and Holt [1]. Let’s consider a set of two possible measurements on Alice’s side M = MA , MA and similarly for Bob M = MB , MB . A { 1 2} B { 1 2} The so-called CHSH inequality is define in the following way:

CHSH(MA, MB, p)= p(00 MA , MB )+p(11 MA , MB ) p(01 MA , MB ) p(10 MA , MB ) | 1 1 | 1 1 | 1 1 | 1 1 + p(00 MA , MB )+p(11 MA , MB ) p(01 MA , MB ) p(10 MA , MB ) | 2 1 | 2 1 | 2 1 | 2 1 + p(00 MA , MB )+p(11 MA , MB ) p(01 MA , MB ) p(10 MA , MB ) | 1 2 | 1 2 | 1 2 | 1 2 p(00 MA , MB ) p(11 MA , MB )+p(01 MA , MB )+p(10 MA , MB ) | 2 2 | 2 2 | 2 2 | 2 2 (2.8)

Defining the correlator of MAi and MAj as

MA MB = p(i = j MA , MB ) p(i = j MA , MB ) (2.9) i j | i j 6 | i j we can rewriting⌦ CHSH↵ in a more convenient form:

CHSH(M , M , p)= MA MB + MA MB + MA MB MA MB A B h 1 1i h 1 2i h 2 1i h 2 2i (2.10)

For any local probability distribution p we have 2 CHSH(M , M , p)  A B  2. With this tool, answering the question of nonlocality of a probability distribution p is quite simple: one has to find the existence of two sets MA and M such that CHSH(M , M , p) > 2. That is, optimise the CHSH B | A B | function over all possible MA, MB. At this point, we have everything we need to give the definition of a Bell’s game. The goal of this kind of game is to check concretely the violation of a Bells inequality. If one has the analytical expression of a probability distribution, the problem is entirely mathematical and can be solved, at least in principal. Consequently the aim of the game is to obtain the probability distribution from the actual situation.

The game will imply a third party that we will call the "referee", whose role is to verify that Alice and Bob do not "cheat". The game is the following: Alice and Bob share some resources. In the first round, each of them receives a measurement from the referee, from their respective measurements set, and they have to give their results to the referee. The referee fixes a time during which they have to give their results, and he ensures that they are space-like separated during the interval that separates the receipt of the measurement and the answer they give. The second round is played according to same rules with the possibility of others measurement. At each round, the referee collect Alice’s and Bob’s outputs. After a certain number of rounds, the game is stopped by the referee and he can estimate the probability distribution for the measurements he gave. CHAPTER 2. CONCEPTS AND MOTIVATIONS 11

Since the number of rounds is finite, this will not be the exact probabilities, but one can have arbitrarily close values by increasing the number of rounds. Then the referee can compute the value of a suitable Bell’s inequality (note that the number of measurement involves in a Bell’s inequality is fixed, but for each number of measurements there exists in principal a Bell’s inequality) and determine if the probability distribution is nonlocal or not. Why do we need a external person to calculate these quantities? If Alice and Bob simply choose some measurements that they then apply to their system to evaluate the probability distribution, where is the problem? The answer is that we did not make any assumption on Alice and Bob and we do not trust them. If we did, it would be indeed true that the above game is useless, Alice and Bob would be able to play without any referee and to obtain the right answer. But it could be the case that they fool us, willingly or not. Although we call them with human names, in general they can represent any entity. And for instance, if these entity are causally connected, they can tune their outputs to optimise the value of a Bell’s inequality. Even if they are space-like separated during the whole game, they can choose in advance the inputs that they will use at each round and the related outputs -which is totally equivalent to choose completely the probability distribution. In other terms, we have to require the no-knowledge on the other part’s inputs, and that’s what this game does.

2.2 Entanglement

Bipartite quantum states are introduced in this section as well as the notion of entanglement and its link with nonlocality.

2.2.1 Quantum formalism Let d be an n-dimensional Hilbert space on C. We will define the quantum H quantities we need to treat according to what we have already defined above ; we present everything for the bipartite case -since it is the one in which we are interested- but one can trivially extend all these definitions to the n-partite case.

Definition 8 (Quantum state). A quantum state of dimension d is a positive semi-definite operator r that acts on d and satisfies tr(r)=1. H This is the very general definition of a quantum state, also called a qudit (the "d" holds for the dimension). A bipartite state is a particular case of this formalism:

Definition 9 (Bipartite quantum state). A bipartite quantum state is a positive semi-definite operator r that acts on d1 d2 and satisfies tr(r)=1. H ⌦H This is the formalism we need to describe a quantum resource shared by Alice and Bob, corresponding to the fact that they each receive one qudit. Now we need to define precisely what is a measurement on a quantum state. CHAPTER 2. CONCEPTS AND MOTIVATIONS 12

Definition 10 (Quantum measurement). A measurement on a quantum state r is a set of N linear operators that act on d,M= M , M , ..., M which satisfy H { 1 2 N}

† Â Mk Mk = 1 (2.11) k

r M† After the measurement on the state r the final state is Mk k with probability tr( r M†) Mk k tr( rM†). Mk k One can check that for all r that are quantum states the sum over all the probabilities is indeed 1, due to condition 2.11. From a measurement M one can obtain the related POVM defined by { k} the set of M M† . It is quite convenient to work with POVMs, rather than { k k } measurement:

Definition 11 (POVM). A POVM (Positive Operator Valued Measurement) on a bipartite state r is a set of N linear positive semi-definite operators that acts on d, H E = E , E , ..., E which satisfy { 1 2 N}

 Ek = 1 (2.12) k

After the measurement on the state r the output is k with probability tr(Ekr). Since from the POVM we cannot extract the final states but only the output, we will use that formalism when we do not care about the quantum state after a measurement but only about its statistics. Finally, we will define what is a composite measurement :

Definition 12 (Composite measurement). Let r be a bipartite quantum state. If Alice performs a measurement A and Bob a measurement B on that state the composite measurement M which describes the global measurement on the state is A B. ⌦ Explicitly the measurement is the set M = A B , A B , ..., A B , A { 1 ⌦ 1 1 ⌦ 2 1 ⌦ N 2 ⌦ B , A B , ..., A B , ..., A B , A B , ..., A B 1 2 ⌦ 2 2 ⌦ N N ⌦ 1 N ⌦ 2 N ⌦ N} 2.2.2 Characterization of quantum states At this point we have everything we need to express the probability distri- bution that comes from a bipartite quantum state shared by Alice and Bob. Indeed, if Alice locally performs a POVM A and Bob a POVM B the composite POVM A B describes the global measurement applied on the system and ⌦ according to definition 11 we have

p(a, b A, B)=tr(A B r) (2.13) | a ⌦ b First we remark that this probability distribution is always non-signalling, by using the definition 6 of the marginals: CHAPTER 2. CONCEPTS AND MOTIVATIONS 13

p(a A, B)= p(a, b A, B) | b | =  tr(Aa Bbr) b ⌦ = tr( Aa Bbr) b ⌦ = tr(Aa  Bbr) ⌦ b = tr(A 1r) a ⌦ = p(a A) | Which is the definition (5) of non-signalling. Therefore quantum mechanics is actually non-signalling. For a quantum state an interesting feature is the entanglement. First we need the notion of separability:

Definition 13 (Separability). A bipartite quantum state r of dimension d d is ⇥ separable if it can be written on this form:

k k r = Â pk rA rB (2.14) k ⌦

k k where the rA and rB are quantum states of dimension d and the pk are positive and real numbers which sum to one.

Why do we call such states separable? Let’s assume that we have a k k separable state, which means we can find some rA, rB and pk that decompose our state in the form (2.14). The corresponding probability distribution will be

p(a, b A, B)=tr(A B r) | a ⌦ b k k = tr(Aa Bb  pkrA rB) ⌦ k ⌦ k k =  pk tr(Aa BbrA rB) k ⌦ ⌦ k k =  pk tr(AarA) tr(BbrB) k k k =  pk pA(a A)pB(b B) k | | We obtain a convex combination of separated probabilities, in the sense that they factorise. Which corresponds to classical correlations with local probability distribution pk (a A) and pk (b B) for Alice and Bob. On the A | B | contrary, an entangled state cannot be describe in that way.

Definition 14 (Entanglement). A bipartite quantum state r of dimension d d is ⇥ entangled if it is not separable. CHAPTER 2. CONCEPTS AND MOTIVATIONS 14

We have said that all the information about a physical state is contained on the probability distribution. Due to the degree of freedom involved each complete collection of quantum probability distributions corresponds to a unique quantum state, i.e. there is no ambiguity or redundancy in the quantum description. Knowing this we will define quantum nonlocality in a natural way:

Definition 15 (Quantum nonlocality). A bipartite quantum state r is local if its probability distribution is local, nonlocal otherwise.

As we already noticed above, the separable states have probability distribu- tions which can be written as a convex combination of factorised probabilities, thus they are local since we directly have the decomposition (2.4). All separ- able states are local. But is the reciprocal true: are all entangled states nonlocal? Intuitively we can think that it is actually the case since we said that quantum states and probability distributions are in one-to-one correspondence and since the definition of separability seems to be the equivalence to locality for quantum states.

However the answer is negative. Even if it is true for the particular class of pure states [2], there exist some quantum states that are entangled but local [3]. Therefore the situation is a bit more complex for quantum states than for probability distributions: while these are classified in two classes (local or nonlocal), those are classified in three classes (local, entangled or nonlocal).

This point is a crucial point of quantum theory and nonlocality. First we can ask ourselves if it is a "bug" of the formalism: are the definitions good? There is no simple answer to this question and the rest of this essay will add some new information to discuss this question.

2.3 SLOCC and hidden nonlocality

At this point entanglement is only a mathematical property of a quantum state, corresponding to the fact that there is no decomposition like (2.14), since the nonlocal-local classification refers to something physical. The nonlocality is the relevant concept so what has an entangled but local state that a separable state has not? One possible answer is: hidden nonlocality. Indeed one can "reveal" the nonlocality of some entangled and local states.

For instance in [4] Palazuelos show that one can obtain some violation of a Bell’s inequality by tensorizing a local state with itself. That is, many copies of local states can lead to a nonlocal one (in higher dimension). This phenomenon is called supractivation and answer partially the above question. Moreover, in [5] Masanes, Liang, and Doherty show that for each entangled state s, there exists a state r not violating CHSH, such that r s violates ⌦ CHSH. Which proves that entanglement "contains" necessarly some nonlocal- CHAPTER 2. CONCEPTS AND MOTIVATIONS 15 ity but we sometimes need non-trivial protocol to reveal it.

These two cases both use the idea of tensorizing a local state with some- thing to reveal nonlocality. Contrary, we will use some operations which preserve the dimension of the quantum state. In this section we introduce the concept of SLOCC (Stochastic Local Operation with Classical Communica- tion). First, we present a less powerful operation than SLOCC: the local linear operation.

Definition 16 (Local linear operation). A local linear operation is an operation that transforms a bipartite quantum state r in

† † r0 = Â Ak BlrAk Bl (2.15) k,l ⌦ ⌦

d † with the Ak and the Bl some linear operators acting on that respect Âk Ak Ak = 1 † 1 H and Âl Bl Bl = . One can check that it is the most general linear and separable operation over the constraint of preserving the trace and the positivity, which means that the final state is still a bipartite quantum state if the initial state was one. Barret proves that this operation cannot map a local state into a nonlocal one [6]. Hence, we need a more powerful kind of transformation to reveal nonlocality. Note that the transformations on which we focus are local, since with nonlocal linear operation, one can transform a quantum state in any other possible quantum states. But in that case, one needs nonlocal resources and there is nothing astonishing to obtain nonlocality from nonlocality. In fact the trick is to use a sequence of measurements. After the first (local) measurement, the quantum state has changed, so can we use this changing to map a local state into a nonlocal one? This is the idea of local filtering:

Definition 17 (Local filtering). A local filtering is a four-outcomes measurement that comes from the local two-outcomes measurements A = F , F¯ for Alice and { A A} B = F , F¯ for Bob. { B B} Explicitly it corresponds to the global measurement

F = F F , F F¯ , F¯ F , F¯ F¯ (2.16) { A ⌦ B A ⌦ B A ⌦ B A ⌦ B} Note that we need F F† 1 (this condition means that every eigenvalues A A  of F F† are smaller than one) since F F† + F¯ F¯ † = 1 and F F† 1 for the A A A A A A B B  same reason (the general solution of this matrix equation is F¯ = 1 F F† U A A A where U is a unitary matrix). q Thus after a local filtering we have four possibles state. Now we add the possibility of classical communication for Alice and Bob, that is, they have a channel with which they can send classical bits. This allows them CHAPTER 2. CONCEPTS AND MOTIVATIONS 16 to post-select the state they want: they will keep their quantum state only F F r F† F† when they obtain A⌦ B A⌦ B after the measurement. This happens with tr(F F r F† F†) A⌦ B A⌦ B probability tr(F F r F† F†). And it is the definition of SLOCC. A ⌦ B A ⌦ B Definition 18 (SLOCC). A stochastic local operation with classical communication (SLOCC) transforms a bipartite quantum state r in

† † FA FB r FA FB r0 = ⌦ ⌦ (2.17) tr(F F r F† F†) A ⌦ B A ⌦ B where F and F are linear operators that acts on d. A B H By its structure, this operation preserve the positivity. The difference with the local linear operation (definition 16) is that it does not preserve the trace, but the trace of the final state is equal to one, because we "renormalize". It is really a post-selection: Alice and Bob keep their state only when the initial state "passes" the filters F F . A ⌦ B

Then if the final r0 is nonlocal, while the initial one was local (over a cer- tain class of measurements), we will say that the initial state contains hidden nonlocality.

Several interpretations of hidden nonlocality are possible. One can high- light that well-chosen local filtering allows preparing nonlocal states from local ones, i.e. a selection on many copies of local states can lead to states that all violate a Bell’s inequality. If Alice and Bob can communicate, they can win a Bell’s game with a local state that has hidden nonlocality: when the filter does not pass they saturate the local bound, and when it does, they reach a larger value. Even if they cannot communicate local filtering increases their probability of winning the game. But as demonstrated in [7], hidden nonlocality mainly shows the failure of the local model for sequence of measurements. Are these kinds of states really local, if a local model predicts everything correctly for one measurement, but not for more than one?

2.4 Notations

Let d be a d-dimensional Hilbert space. We will note y a general vector H | i in this space and y his dual. We define 0 , 1 , 2 , ..., d the canonical h | {| i | i | i | i} basis. Hence a general vector y is defined by its decomposition in this basis: | i y = Âd c k = Âd k y k . | i i=k k| i i=kh | i| i We will sometimes use the pure state notation:

Definition 19 (Pure states). A quantum state r that acts in d is pure if it is a H rank-one projector, i.e. r = y y . | ih | CHAPTER 2. CONCEPTS AND MOTIVATIONS 17

When a state is pure we will then denote it by y instead of y y . | i | ih | Finally we define the three Pauli matrices:

01 0 i 10 s = s = s = x 10 y i 0 z 0 1 ✓ ◆ ✓ ◆ ✓ ◆ And ~s =(sx, sy, sz)

2.5 Classes of measurement

We will distinguish two classes of measurement: projective measurements and POVMs. The second one has already been defined (definition11) and is just a rewriting of the general measurement formalism. The first one is a smaller class:

Definition 20 (Projective measurements). A projective measurement (also called Von Neumann measurement or ideal measurement) on a quantum state of dimension d is a measurement P = P , P , .., P where the elements are orthogonal rank-one { 1 2 d} projectors. That is

PiPj = di,jPi (2.18)

i, j = 1...d 8 This class of measurement is in a sense the canonical one since every posit- ive matrices (i.e. POVM elements) can be decomposed in a linear combination of projectors. Moreover it has a natural quantum mechanical interpretation: each element projects on a pure state, P = y y , and all together they k | kih k| span entirely d. After the projective measurement the final state is the pure H state y with probability tr( y y r)= y r y . In particular if the initial | ki | kih k| h k| | ki state is f (i.e. if it is pure) this probability becomes y f 2, that is the | i |h k| i| norm squared of the Hilbertian scalar product.

In the 2-dimensional case a rank-one projector can be written as:

1 +~v ~s P = · (2.19) 2 where ~v is a vector on the Bloch sphere (unit sphere where the three direc- tions are the "spin" directions: sx, sy, sZ). And therefore a rank-one projector in dimension two is uniquely defined by a vector on the Bloch sphere.

We will speak about "hidden nonlocality" when a state is local for projective measurements but nonlocal after filtering, while we will reserve the term "genuine hidden nonlocality" for a state fully local (i.e. local for all POVMs) and nonlocal after filtering. CHAPTER 2. CONCEPTS AND MOTIVATIONS 18

2.5.1 Quantum observable It is sometimes more convenient to work with observables rather than meas- urements. A quantum observable is a hermitian matrix constructed from a POVM E = E , E , ..., E as follows : { 1 2 N}

O = Â akEk (2.20) k

where the real number ak represents the value we assign to the POVM element Ek. In the projective case the ak are the eigenvalues of the observable since the projectors are orthogonal. If the quantum state is a qubit it is common to choose +1 and 1 as eigenvalues and in that case we can write the observables in terms of the Pauli matrices:

O = ~a ~s (2.21) · where ~a is a vector on the two-dimensional sphere. One can check that this is a hermitian matrix with eigenvalues +1 and 1. Chapter 3

Local states

In this chapter, we will present some entangled states that are local over certain classes of measurement. We will then prove that some of them exhibit hidden nonlocality.

3.1 Local models

In order to show that a bipartite quantum state is local, we need to prove the existence of the decomposition (2.4) for its probability distribution. A sufficient condition is therefore to exhibit a set L, a probability density r(l) on this set and the functions p(a A, l) and p(b B, l), that we will call the | | response functions. Since the locality means that the probability distribution has only classical correlations, having this three functions allows us to simulate a quantum state, that is having the same probability distribution as the quantum predictions with only classical and local resources. The protocol that explains what Alice and Bob have to do to simulate a quantum state is called a local hidden variable model(LHVM).

Definition 21 (Local hidden variable model). Alice and Bob share a random variable l L (sometimes called the "hidden variable" for historical reasons) with 2 probability density r(l). Then they receive inputs A and B and output fA(A, l) and fB(B, l), respectively.

Figure 3: the scenario of a local model

In the definition above, we have explained the protocol in terms of "outputs functions" rather than response functions but one can always go from one to

19 CHAPTER 3. LOCAL STATES 20 another: the output function is obtained by simulating locally the probabilities given by the response function (on the other side the other way round can be hard).

Although one is sufficient in the local models we will present, we often give both response and output functions because we think it leads to a gain of intuition on the model. More precisely said, the output function is more convenient to understand the model while the response function appears in the proof that the model works (condition (2.4) fulfilled).

3.1.1 Simulate POVMs As pointed out by Barrett[6] to simulate POVMs on a quantum state we just need to focus on the ones with all elements proportional to rank-one projectors. Indeed each element can be decomposed as a summation of rank- one projectors, and hence using this decomposition allow us to simulate in reality something more subtle. Generally we therefore have POVMs on the form P = x P , x P , ..., x P where the real numbers x respect: 0 x 1 { 1 1 2 2 N N} i  i  and Âi xi = d, and the Pi are rank-one projectors. In the qubit case one can therefore represent such a POVM as three vectors on the Bloch sphere (that respresent the projectors) with norms smaller than one (that represent the coefficients in front of the projectors).

Figure 4: a three-outcomes POVM represented by three vectors inside the Bloch sphere that satisfy the normalization conditions

3.2 Werner state

The Werner state is a qudit-qudit state defined originally by Werner in [3] as follows:

1 W = ((d F)1 +(dF 1)V) (3.1) d3 d where 1 F 1 and V is the "flip" operator: V( y f )= f y .   | i ⌦ | i | i ⌦ | i This state is the most general d-dimensional U U invariant state (where U ⌦ CHAPTER 3. LOCAL STATES 21 is any unitary matrix).

Werner showed that this state is entangled if F < 0. However he showed that this state can be simulated for projective measurements for F = 1 + d+1 d2 [3]

3.2.1 Werner’s model Alice and Bob share a complex vector w W, where W is the unit sphere, 2 that is W = w Cd w = 1 . Its probability density is uniform. They { 2 |k k } receive projective measurements A = P , P , ..., P and B = Q , Q , ..., Q , { 1 2 d} { 1 2 d} respectively. Alice’s response function is:

p (k A, w)= w P w (3.2) A | h | k| i Bob’s response function is:

1 if w Qk w < w Ql w for all l = k pB(k B, w)= h | | i h | | i 6 (3.3) | (0 if w Q w > w Q w for some l = k h | k| i h | l| i 6 The interpretation of this response function is that Bob outputs determin- istically the k that satisfies w Q w w Q w k. 0 h | k0 | i  h | k| i 8 That is Bob chooses the k that minimizes w Q w . h | k| i Therefore, one can show that

pA(k A, w)pB(k B, w)r(w)dw = Tr(Pk QkW) (3.4) ZW | | ⌦ if F = 1 + d+1 . d2

To link the Werner state with the state rW we will present in 4.2 we will rewrite the Werner state using the fact that the flip operator can be expressed as V = 1 1 2 Â S S . With that an other way to write down a ⌦ i

2 Âi 1+d and the Werner’s d 1 model holds for a = d . CHAPTER 3. LOCAL STATES 22

3.2.2 Barrett’s model Starting again from the Werner state in form (3.5) Barrett showed that this state is local even for POVMs, but for a smaller value of a.

Alice and Bob have to simulate POVMs A = A = a P and B = { k} { k k} B = b Q , respectively. The model uses the same set W with the same { k} { k k} probability density. Alice has the following response function :

a p (k A, w)= k (1 w P w ) (3.6) A | d 1 h | k| i and Bob:

bk pB(k B, w)= w Bk w Q( w Qk w 1/d)+ 1 Â w Bl w Q( w Ql w 1/d) | h | | i h | | i l h | | i h | | i ! d (3.7)

where Q is the Heaviside function These two response functions together with the uniform probability density 3d 1 on the unit sphere reproduce the Werner state correlations for a = d+1 (d d 1 d 1) d . At the opposite of the Werner’s model a goes to 0 for very large dimension. And at the opposite of Werner’s range it is not possible to filter the state in that local range, at least not with the Popescu’s filter we will present in 4.2.

3.2.3 Interpretation of Werner’s and Barrett’s model We can try to understand better what these two models do and why they work. First, note that the choice of a uniform probability density over the unit sphere is natural since the Werner state is U U invariant. Then the ⌦ response functions must have this invariance as well and one can check that the marginals are trivially the ones of the identity because the density is uniform and there is no privilege for particular measurement elements in average. But the difficult part is of course to reproduce the correlations, not the marginals. In both cases, the shared variable w can be interpreted as a quantum state. Hence in Werner’s model Alice’s response function has a natural interpreta- tion: it corresponds to compute the probability of outputting k with respect to the quantum state w. Concretely with its response function, the most a projector is close to w w the most likely it will be outputted. We see that | ih | the entangled part of the Werner state is the projection over the antisymmetric subspace and consequently Alice and Bob are anti-correlated. Therefore Bob has do something "opposite" to Alice, and that’s what Werner’s model does: Bob chooses the element which is the farther away from w w . And all these | ih | conditions together: the symmetries, the use of w as a quantum state, and the "anti-correlation" lead to the simulation of a Werner state. CHAPTER 3. LOCAL STATES 23

The idea is more or less the same for the Barrett’s model. Let us first describe Bob’s response function as a scenario: Bob receives his POVM B = B = b Q . He selects the elements B that respect w Q w > { k} { k k} { l} h | l| i 1/d. In this subset he outputs i with probability w B w . But note that h | i| i  w B w < 1 in general (since not all elements pass the first condition). ih | i| i That means there is the possibility of not outputting anything after this step. If it is the case, he goes back to the complete set and outputs k with probability bk/d. One can check that this protocol is actually the one represented by the response function (3.7). We see that with this process the probability of passing the first test increases when the projector Q is close to w w . Moreover when an element l | ih | passes the test, its probability to be directly outputted is proportional to w Q w . That is, oppositely to the Werner’s model, Bob promotes the h | l| i elements close to w. Then Alice has to do the opposite and that’s why Barrett chose a response function proportional to (1 w P w ). h | k| i 3.3 New classes of local states

In this section we will present a few new local states in dimension two. It will lead to the first example of hidden nonlocality for qubits.

s 3.3.1 ra We define the one-parameter family of states rs = a y y +(1 a)s 1 a | ih | ⌦ 2 Where 0 a 1 and s is an arbitrary quantum state of dimension 2. Note 1  that for s = 2 this is the two-dimensional Werner state. We will prove that this state is local for projective measurement if a 1 .  2 The model is based on the Degorre’s protocol[12] with a modification on Alice’s response function. Setting a = 1 and using the Pauli matrices notation (A = ~a ~s and 2 · B =~b ~s) for Alice’s and Bob’s observables we have the following correlator · and marginals:

~a ~b AB = · (3.8) h i 2 tr(As) A = (3.9) h i 2 B = 0(3.10) h i If a local model can reproduce these correlations, it means the state is local since projective measurements on qubits allow two outcomes and there- fore everything is contained in those three quantities plus the normalisation condition (we just need to check that the response functions are normalized probability distributions). The hard bit is to reproduce the correlator. First, as pointed out by N.Gisin and B.Gisin [13], note that if we allow the following CHAPTER 3. LOCAL STATES 24

"biased" probability density:

~a ~l r(~l)=| · | (3.11) 2p (where ~l S2 ) and if Alice and Bob output f (~a,~l)= sign(~a ~l) and 2 A · f (~b,~l)=sign(~b ~l), we obtain for the correlator: B ·

~ ~ ~ ~ ~ AB = r(l) fA(~a, l) fB(b, l)dl 2 h i ZS 1 ~a ~l = | · | sign(~a ~l)sign(~b ~l)d~l p 2 p 2 ZS 2 · · 1 = (~a ~l)sign(~b ~l)d~l p 2 2 ZS · · = ~a ~b · So if they shared a variable with such a probability density, Alice and Bob would be able to simulate the perfect singlet. Of course, this is not possible since the probability density cannot depend on Alice’s input in the Bell’s game, because it would correspond to knowing in advance what the referee will choose for her. However we can use the fact that the dependence is only on Alice’s side. Once Alice receives her measurement she actually has the information to transform the shared variable she received at the same time, or, more precisely, changes its probability density. That’s what the following model does.

3.3.2 The model Alice and Bobs share a vector ~l 3 uniformly distributed on the two- ~ 21R dimensional sphere. That is r(l)= 4p .

Alice receives ~l and a vector in the Bloch sphere ~a (describing the meas- urement ~a ~s). First she "tests" her ~l. With probability ~a ~l she accepts ~l and · | · | outputs sign(~a ~l), with probability (1 ~a ~l ) she rejects ~l and outputs · 1+tr(As) | · | 1 tr(As) +1 with probability and 1 with probability . 2 2 Bob outputs sign(~b ~l). · This model reproduces the correlations (3.8),(3.9) and (3.10).

Proof First note that Bob outputs +1 if ~l is in the same half-sphere as~b, 1 if it is in the other half-sphere. Since ~l is uniformly distributed over the sphere there is no "privileged" direction for it and we have B = 0. h i The same observation (inverting +1 and 1) is true when Alice accepts ~l. 1+tr(As) 1 tr(As) When she rejects it, she outputs in average (+1) +( 1) = · 2 · 2 tr(As). CHAPTER 3. LOCAL STATES 25

~ ~ 1 The initial probability density of l is rin(l)= 4p and its probability to pass Alice’s test is ~a ~l . Thus if ~l pass the test its new probability density | · | is r (~l)=r (~l) ~a ~l 1 where N is a normalization factor that we fin in ·| · |· N ~ ~ can evaluate with the normalization condition S2 r fin(l)dl = 1. We obtain ~ ~a ~l ~ r fin(l)= |2·p | . Consequently when l pass theR test its probability density is (3.11) and if Alice output sign(~a ~l) and Bob sign(~b ~l) they simulate the · · perfect singlet. In average ~l pass the test with probability

p = r(~l) ~a ~l d~l 2 ZS | · | 1 = 2

When ~l does not pass the test, Alice and Bob are completely independent since Alice rejected the shared variable. Accordingly there is no correlations between them, therefore

1 1 AB = ( ~a ~b)+ A B h i 2 · 2 h ih i 1 = ( ~a ~b) 2 · Finally when ~l does not pass, Alice outputs tr(As) in average and it 1 happens with probability 2 then

tr(As) A = h i 2 s We see that the model we described simulates actually ra for any quantum state s. Let us now find out the response functions of that model.

2 1+~l ~s First we define the vector l C as: l l := · , i.e. the projector | i 2 | ih | 2 on l is represented by the vector ~l in the Bloch sphere. With that definition | i we have:

1 +~l ~s ~l ~b = tr(~b ~s · )=tr(B l l )= l B l · · 2 | ih | h | | i In the model Bob outputs deterministically sign(~b ~l), means he outputs · +1 if~b ~l > 0, 1 if~b ~l < 0. Since~b ~l = l B l the response function of · · · h | | i Bob is:

p ( B,~l)=Q( l B l ) (3.12) B ±| ±h | | i which coincides with Werner’s model. CHAPTER 3. LOCAL STATES 26

Similarly we can evaluate Alice’s response function: when ~l pass the test Alice outputs sign(~a ~l) and when it does not she simulates the state s. · Since the probability of passing the test is ~a ~l we obtain: | · | 1 A p ( A,~l)= ~a ~l Q( ~a ~l)+(1 ~a ~l ) tr( ± s) A ±| | · | ⌥ · | · | 2 1 A = l A l Q( l A l )+(1 l A l ) tr( ± s) |h | | i| ⌥h | | i |h | | i| 2 (3.13) 1 As expected, we obtain exactly the Werner’s model for s = 2 since one can check that in this case one has:

~ 1 A pA( A, l)= l A l Q( l A l )+(1 l A l ) tr( ± )= l A l ±| |h | | i| ⌥h | | i |h | | i| 2 h | ±| i (3.14)

1 A where A = ± . ± 2 We therefore established that rs is projective-local for a 1 . a  2 s 3.3.3 ra,b s s The state ra,b is a little extension of ra:

s 1 1 r = a y y +(1 a)(bs +(1 b) ) (3.15) a,b 2 ⌦ 2 Where 0 a 1, 0 ↵⌦ b 1 and s is an arbitrary quantum state of     dimension 2. Instead of having a free quantum state, the separable part of Alice is a convex combination between a free state and the white noise. It can be showed s that this state is less entangled than ra for b < 1, and if b is not sufficiently close to 1 (the "sufficiently" depends of the value of a but the exact relation is not known) the state is no longer filterable.

The interesting part is that this state can be shown to be POVM-local for b = 3/7 and a = 5/12.

The model is a small adaptation of the Barrett’s model: Alice and Bob have to simulate POVMs A = A = a P and B = { k} { k k} B = b Q , respectively. The model uses the same set W with the same { k} { k k} probability density. Let us recall the share variable l to keep in mind that it | i is now a complex vector in C2. Alice has the following new response function:

pA(k A, l )= l Ak l Q( l Pk l 1/2)+ 1 Â l Al l Q( l Pl l 1/2) tr(Aks) | | i h | | i h | | i l h | | i h | | i ! (3.16) CHAPTER 3. LOCAL STATES 27

that is Bob’s response function in Barrett’s model with a little change on the "rejection" part, while Bob becomes Alice of Barrett’s model:

b p (k B, l )= k (1 l Q l ) (3.17) B | | i d 1 h | k| i Why does this model work? Let us recall that Bob’s response function in Barrett’s model can be understood as an "acceptance" and a "rejection" part. Indeed Bob focuses first on a subset of the POVM depending of l and then | i gives a probability to each of element in this subset to outputted directly. If none of these elements are chosen, the next step corresponds to a rejection, in the sense that there is no longer use of l . Therefore what is done in this | i part is independent of the other party and we can use that fact to simulate a free state s.

Figure 5: a schematic representation of the adapted Barret’s response function

We argue without proving that the probability of being rejected is 1/4 in average. Hence Alice will simulate locally 1/4 of the state s and 3/4 of the original marginals: the identity. Since the Barrett model in dimension 2 works for a = 5/12 the condition over b is:

1/4 =(1 a)b = 12/7 b which leads to the announced result b = 3/7.

Increasing the value of b is possible if we allow POVMs only on one side, and projective measurements on the other one. Indeed if we choose:

1 A p ( A,~l)= l A l Q( l A l )+(1 l A l ) tr( ± s) (3.18) A ±| |h | | i| ⌥h | | i |h | | i| 2

for Alice and CHAPTER 3. LOCAL STATES 28

pB(k B, l )= l Bk l Q( l Qk l 1/2)+ 1 Â l Bl l Q( l Ql l 1/2) bk/2 | | i h | | i h | | i l h | | i h | | i ! (3.19)

for Bob.

That is, Alice simulates a projective measurement with the response func- s tion of ra and Bob is Barrett’s Bob. We use again the fact that the rejection part on Alice leads to a term independent of Bob (since there is no use of the shared variable) and that she can therefore use this fact to simulate a free state when rejection occurs. We already showed that in average the Degorre’s test has a probability 1/2 of success. The condition over b is therefore:

1/2 =(1 a)b = 12/7 b

b = 6/7 (3.20)

We recapitulate all the states we can simulate in the following table. The notation (i, j) means that we can simulate this state for i outcomes for Alice and j for Bob. The symbol "Q" stands for a quantum response function (cor- related or anti-correlated), "q" for the heaviside response function, "Bar." for Bob’s response function in Barrett’s model, "s-Bar" for the modified one and "s-deg" for the adapted version of Degorre’s model.

Figure 6: a table that summarize the states we are able to simulate

s 3.3.4 ra,b,g A last extension is to put together the s-deg and s-Bar response functions, which allows us to simulate the quantum state CHAPTER 3. LOCAL STATES 29

s 1 1 r = a y y +(1 a)(bs +(1 b) ) (gs +(1 g) ) (3.21) a,b,g 2 ⌦ 2 ↵⌦ for a = 5/12 , b = 6/7 and g = 3/7. This state is entangled but not filterable, as far as we know. Chapter 4

Examples of filterable states

We call a state filterable if after filtering it violates at least one Bells inequality which was not violated initially.

4.1 General filtering and qubits filtering

Local filters are made of two matrices FA and FB that correspond to the operations performed locally by Alice and Bob, respectively. For the research of nonlocality after filtering, the only thing that matters is the final state:

† † FA FB r FA FB r0 = ⌦ ⌦ (4.1) tr(F F r F† F†) A ⌦ B A ⌦ B Consequently we do not need to work with general F , F Cd d but we A B 2 ⇥ can focus on a more restricted class of matrix. First note that the local filters F , F and c F , c F (where c and { A B} { A · A B · B} A cB are complex numbers) are equivalent since:

c F c F r (c F )† (c F )† = A A ⌦ B B A A ⌦ B B tr(c F c F r (c F )† (c F )†) A A ⌦ B B A A ⌦ B B c 2 c 2F F r F† F† = | A| | B| A ⌦ B A ⌦ B c 2 c 2 tr(F F r F† F†) | A| | B| A ⌦ B A ⌦ B F F r F† F† = A ⌦ B A ⌦ B tr(F F r F† F†) A ⌦ B A ⌦ B † where we used the facts that cAFA cBFB =(cAcB)(FA FB), (cAFA) = † ⌦ ⌦ c⇤AFA and tr(cM)=c tr(M) for any scalar c.

With that, note that for the research of nonlocality, the condition F F† 1 A A  does not need to be fulfilled since for every FA there exists a complex number c such that (cF )(cF )† 1 and this valid filter is equivalent to F (in the A A  A sense that they give the same final state after filtering).

30 CHAPTER 4. EXAMPLES OF FILTERABLE STATES 31

Moreover, using the singular value decomposition, we can write FA = UADAVA and FB = UBDBVB where DA, DB are diagonal and real and UA, UB, VA and VB are unitary. We obtain

† † FA FB r FA FB rF = ⌦ ⌦ tr(F F r F† F†) A ⌦ B A ⌦ B U D V U D V r V† D U† V†D U† = A A A ⌦ B B B A A A ⌦ B B B tr(U D V U D V r V† D U† V†D U† ) A A A ⌦ B B B A A A ⌦ B B B † † DAVA DBVB r VADA VB DB † † =(UA UB) ⌦ ⌦ (U U ) ⌦ tr(D V D V r V† D V†D ) A ⌦ B A A ⌦ B B A A ⌦ B B where we use the facts that (A B )(A B )=A A B B and 1 ⌦ 1 2 ⌦ 2 1 2 ⌦ 1 2 tr(UMU†)=tr(MUU†)=tr(M) for any U unitary.

We therefore see that applying the local filter U D V U D V corres- A A A ⌦ B B B ponds to applying the local filters U D V U D V plus the local unitary A A A ⌦ B B B U U . Since a local unitary transformation cannot transform a local state A ⌦ B into a nonlocal one (because it corresponds only to a local change of basis) the most general local filters we have to consider are on the form FA = DAVA, FB = DBVB. If we apply this result to the qubit case, it leads to the following most general filters:

a 0 cosq eifsinq F = A 01 eifsinq cosq ✓ ◆✓ ◆ where a, q and f are real parameters. We used the fact that we can multiply by a constant and obtain a equivalent filter to argue that the diagonal matrix has only one free parameter.

Although theoretically there is no reason to exclude the unitary matrix in FA and FB (which corresponds to a local unitary before the filter) all the known useful filters are diagonal. Especially for the qubits case, it seems that only the diagonal filters are relevant. CHAPTER 4. EXAMPLES OF FILTERABLE STATES 32

4.2 Werner state

As presented in 3.2 the Werner state can be written in the following form

2 Âi

d 1 Then setting a = d one obtain the following projective-local state:

1 1 1 1 rW = 2 ( ( )+2 Â Sij Sij ) (4.3) d d ⌦ i

F = 0 0 + 1 1 , F = 0 0 + 1 1 (4.4) A | ih | | ih | B | ih | | ih | Indeed after applying these local filters the final state is:

† † FA FB rW FA FB r0 = ⌦ ⌦ W tr(F F r F† F†) A ⌦ B W A ⌦ B 2d 1 = ( (1 1)+ S S ) (4.5) 2d + 4 2d ⌦ | 12ih 12| Since S is the singlet and the identity has a contribution 0 the CHSH | 12i value of this state rW0 is 2d CHSH(r0 )= 2p2(4.6) W 2d + 4 Therefore CHSH(r ) 2 for d 5. Meaning the Werner state has hidden W0 nonlocality for dimension greater than 5.

We can intuitively understand the fact that the filtering works better for large dimension: for very large d the state is almost entangled for each value 1 of a since the condition a > 1+d becomes easier to satisfy. However the range in which we can simulate the state with the Werner’s model becomes larger: a = d 1 1. Then we can construct a state local for projective loc d ! measurement but with a lot of entanglement. It seems likely that there is a way to extract some nonlocality of such a state. This is why the violation of CHSH after filtering becomes closer than the quantum maximum 2p2 when the dimension increases, as we see in (4.6). CHAPTER 4. EXAMPLES OF FILTERABLE STATES 33

4.3 Gisin’s state

The Gisin’s state is [9]:

1 q r = q y y + ( 00 00 + 11 11 ) (4.7) G | qih q| 2 | ih | | ih | where y = cosq 01 + sinq 10 ,0 q p and 0 q 1 | qi | i | i   2   Gisin’s original motivation was to find a simpler and more intuitive ex- ample of hidden nonlocality that Popescu’s one (only true for dimension five or more). However, his example is not precisely a case of hidden nonlocality but only a filterable state since there is no proof of the locality of this state, even for projective measurement.

Although this case seems less powerful and significant than Popescu’s one, it has some interesting properties and moreover it is fundamentally different in the way it works.

First, let us prove that rG does not violate CHSH for certain values of q and q. For that we invoke the Horodecki criterion [10]: for a state r we construct the 3 3 matrix T with entries T = tr(s s r) where the s are ⇥ ij i ⌦ j k the Pauli matrix (in any order). Then we define H(r) as the sum of the two largest eigenvalues of the matrix TT†. The Horodecki criterion says that r violates CHSH if and only if H(r) > 1. Moreover, there is a bijection between the maximal CHSH value and the Horodecki value given by:

max CHSH(r) = 2 H(r) (4.8) { } q For the Gisin’s state we obtain:

H(r )=max (2q 1)2 + 4(qsinqcosq)2,8(qsinqcosq)2 (4.9) G { } and if we assume

2q(1 sinqcosq) > 1(4.10) the maximum in (4.9) is always the first term. Therefore if we set

q(1 + sin2qcos2q) 1(4.11)  we have H(r ) 1, i.e. no violation of CHSH. G  Next if we choose the local filters

F = ptanq 0 0 + 1 1 , F = 0 0 + ptanq 1 1 (4.12) A | ih | | ih | B | ih | | ih | we obtain CHAPTER 4. EXAMPLES OF FILTERABLE STATES 34

F 1 1 q r = [2qsinqcosq y y + ( 00 00 + 11 11 )] G 2qsinqcosq +(1 q) 2 | ih | | ih | ↵⌦ (4.13) That is, these filters leave totally invariant the separable part ( 00 00 + | ih | 11 11 ) while it transform y in a pure singlet. | ih | | qi Using again the Horodecki criterion this state is nonlocal if and only if

q[1 + sinqcosq(p2 1)] > 1(4.14) The conditions (4.10), (4.11) and (4.14) are compatible provided sinqcosq  p2 1. Therefore there are some values of the parameters for which the Gisin’s state cannot violate CHSH directly but actually violates after filtering.

Even if the criterion we tested was only relevant for CHSH, no violation of others Bell’s inequalities are known for the Gisin state, meaning that for the time being the unique way to extract nonlocality of this state is the filtering. The next result we will show is a kind of little extension of the Gisin’s state.

4.4 Collins-Gisin’s state

This state was presented initially by Collins and Gisin [11] for its property to violate I3322 but not CHSH for some value of the parameters.

The Collins-Gisin’s state is:

r = q y y +(1 q) 00 00 (4.15) CG | qih q| | ih | where y = cosq 01 + sinq 10 ,0 q p and 0 q 1 | qi | i | i   2   The idea is then to apply Gisin’s filters with a symmetric adaptation:

F = #ptanq 0 0 + 1 1 , F = # 0 0 + ptanq 1 1 (4.16) A | ih | | ih | B | ih | | ih | The aim is to "kill" the 00 component much more than the others. It leads | i to :

F 1 2 (1 q) r = (q y y + # 00 00 ) (4.17) CG N 2sinqcosq | ih | ↵⌦ 2 (1 q) where the normalization constant N = # 2sinqcosq + q.

Note that # is independent of q and q which means that we can always 2 (1 q) make the term # 2sinqcosq arbitrarily small. Therefore we can obtain a final state arbitrarily close to the perfect singlet. And this is actually the case for every value of the two parameters q and q. However, like for the Gisin’s state, no range of these parameters where the state is local are known. CHAPTER 4. EXAMPLES OF FILTERABLE STATES 35

4.5 Hidden nonlocality for a two-qubits state

Let’s define rq as follows: 1 r = q y y +(1 q) 0 0 (4.18) q | ih | ⌦ 2 ↵⌦ Since this state belongs to the the class of states we presented in 3.3.1 we know that it is local for q 2.  Using local filters

F = # 0 0 + 1 1 , F = # 0 0 + pq 1 1 (4.19) A | ih | | ih | B | ih | | ih | we obtain :

1 (1 q) rF = ( 01 01 + 10 10 pq( 01 10 + 10 01 )+#2 00 00 ) q N | ih | | ih | | ih | | ih | q | ih | (4.20) 1 q 2 where N = 2 + q #

And due to Hodorecki criterion we see that this state is nonlocal if # < q 2(1 q) . In the limit of # goes to 0 the maximal CHSH value goes to 2 1 + q. q That is, for all value of q the final state violates CHSH after filtering.p This is the first example of hidden nonlocality for qubits. Chapter 5

Maximal and genuine hidden nonlocality

This chapter contains the main results of this essay. Thanks to the results of the last chapter together with a new protocol, we will construct an example of genuine hidden nonlocality for a two qubits state, an example of maximal hidden nonlocality for a qubit-qutrit state, and finally, an example of maximal and genuine hidden nonlocality for a qutrit-qutrit state.

5.1 From projective measurements to POVMs

In this section we will show how a projective model for a state r can lead to a POVM model for a (less entangled) state r0.

Let r be a bipartite quantum state local for projective measurements and rA, rB Alice’s and Bob’s reduce states, respectively. We will show that the state

1 r0 = (r + r s + s r + s s ) (5.1) 4 A ⌦ B A ⌦ B A ⌦ B

is local for POVMs, for any quantum states sA,sB.

Let focus on the qubit case. The simulation protocol goes as follow:

Alice receives a POVM A = A , A , ..., A where each element is pro- { 1 2 N} portional to rank-one projector: Ak = akPk. First she chooses the element i ai with probability 2 . Then she uses the projective model to compute the prob- ability of obtaining Pi. With this probability she outputs i, else she outputs k with probability tr(AksA). In other words she simulates the (projective) measurement P = P , 1 P with, say, label +1 for P and 1 for 1 P ; if { i i} i i she obtains +1 she logically outputs i, if she obtains 1 she simulates locally the state sigmaA.

36 CHAPTER 5. MAXIMAL AND GENUINE HIDDEN NONLOCALITY 37

Bob does exactly the same protocol with his POVM B = B , B , ..., B = { 1 2 M} b Q , b Q , ..., b Q and s instead of s . { 1 1 2 2 M M} B A We can decompose this protocol in four cases; in figure 7 a schematic representation of this protocol is given, with the average probabilities of being in each case and the the respective probability distributions. But first, let us prove formally that why the protocol works:

Given that they chose Pi and Qj the probability of outputting directly i and j is tr(P Q r), since they use their projective model that repro- i ⌦ j duces the quantum correlations. The probability of choosing Pi and Qj is ai bi because these are independent events, which leads to a probability 2 · 2 1 tr(a P b Q r)= 1 tr(A B r) of outputting directly i and j. 4 i i ⌦ j j 4 i ⌦ j In the three other cases, the probabilities factorize since there is no use of any shared variable. Let’s consider these three cases:

Alice obtains +1 and Bob obtains 1: In this situation Alice obtained +1 she then outputs the projector she has chosen and it is i with probability ai . Bob obtained 1 which means that 2 whatever the projector he has chosen he outputs j with probability tr(BjsB). ai The probability of having i, j in that case is then 2 tr(BjsB). To compute the probability of being in this situation (+1 for Alice and 1 for Bob, given that Alice chose i) we have to sum over all possibilities of choices for Bob, since all projectors contribute in the same way in the 1 case (oppositely only the term i contribute for Alice because she obtained +1):

bk p(+1, 1 i)= tr(Pi (1 Qk)r) | k 2 ⌦ 1 = tr(Pi  bk(1 Qk)r) 2 ⌦ k 1 = tr(P (21 1)r) 2 i ⌦ 1 = tr(P 1r) 2 i ⌦ 1 = tr(P r ) 2 i A

1 ai 1 The total contribution of this case is thus 2 tr(PirA) 2 tr(BjsB)= 4 tr(AirA) tr(BjsB).

Alice obtains 1 and Bob obtains +1: Since the situation is totally symmetric the contribution of this case is: 1 4 tr(AisA) tr(BjrB).

Alice and Bob obtain 1: CHAPTER 5. MAXIMAL AND GENUINE HIDDEN NONLOCALITY 38

In this case the probability of having i, j is tr(AisA) tr(BjsB) whatever was the choice of projector for both Alice and Bob; we have consequently to sum over all the possible choices to determine the probability of being in that case:

al bk p( 1, 1)=Â tr((1 Pl) (1 Qk)r) l,k 2 2 ⌦ 1 = tr(1 1r) 4 ⌦ 1 = 4 1 Which finally give a contribution 4 tr(AisA) tr(BjsB).

Therefore we obtain:

1 p(i, j A, B)= (tr(A B r)+tr(A r ) tr(B s )+tr(A s ) tr(B r )+tr(A s ) tr(B s )) | 4 i ⌦ j i A j B i A j B i A j B (5.2)

This probability distribution is exactly the one that comes from the state (5.1).

Figure 7: a representation of the POVM protocol with the average probability of passing each "tree" and the final probability distributions

5.2 Genuine hidden nonlocality

0 1 1 In the section 3.3 we showed that the state r|1 i = 2 ( y y + 0 0 2 ) is 2 | ih | | ih | ⌦ local for projective measurements. Therefore the above protocol allow us to argue that the state:

sA,sB 1 0 rGHNL = (r|1 i + rA sB + sA rB + sA sB) 4 2 ⌦ ⌦ ⌦ 1 1 1 1 = ( y y + 0 0 + 0 0 s + s + 2s + 2s s ) 8 | ih | ⌦ 2 | ih | ⌦ B 2 ⌦ B A ⌦ 2 A ⌦ B ↵⌦ (5.3) CHAPTER 5. MAXIMAL AND GENUINE HIDDEN NONLOCALITY 39

is local for POVMs for any quantum states sA, sB.

Let’s choose s = s = 0 0 . We obtain A B | ih |

1 1 1 r = ( y y + 3 0 0 + 3 0 0 0 0 + 0 0 ) (5.4) GHNL 8 | ih | ⌦ 2 | ih | ⌦ | ih | 2 ⌦ | ih | ↵⌦ Using local filters given by

F = # 0 0 + 1 1 , F = #p2 0 0 + 1 1 ,(5.5) A | ih | | ih | B | ih | | ih | we obtain after filtering:

† † FA FB rGHNL FA FB rF = ⌦ ⌦ tr(F F r F† F†) A ⌦ B GHNL A ⌦ B 1 p2 = (10#2 00 00 + 01 01 + 10 10 ( 01 10 + 10 01 )) 2 + 10#2 | ih | | ih | | ih | 4 | ih | | ih | (5.6)

which is nonlocal if # 1 .  p10 5.3 Maximal hidden nonlocality

We say that a state has maximal hidden nonlocality if it is local for projective measurements and violates maximally a Bell’s inequality after filtering. Initially presented in [14] the erasure state is a qubit-qutrit defined as follows:

1 E = a y y +(1 a) 2 2 (5.7) a | ih | ⌦ 2 ↵⌦ where 0 a 1.   The trick is that Alice can decide to "ignore" the separable part, since it lives in a space orthogonal to the singlet. That is, using F = 0 0 + 1 1 A | ih | | ih | (and Bob does nothing: FB = 1) the final state is

† † FA FB Ea FA FB rF = ⌦ ⌦ tr(F F E F† F†) A ⌦ B a A ⌦ B = y y ↵⌦ This means that they can extract the perfect singlet, and thus violates maximally CHSH. The probability of passing the filter is:

tr(F F E F† F†)=a (5.8) A ⌦ B a A ⌦ B CHAPTER 5. MAXIMAL AND GENUINE HIDDEN NONLOCALITY 40

Therefore for all value of a Alice and Bob finish with the pure singlet with probability a.

We will prove that for a 1 the nonlocality is actually hidden, since the  3 state is local for projective measurements.

The proof is decomposed in two steps: first we construct a model for s dichotomic outcomes using the same idea of the one for the state ra (3.3) with a little adaptation; then we use our protocol to extend for three outcomes on Alice’s side.

1 Step 1: The erasure state for a = 2 can be simulated for dichotomic out- comes.

Proof. The state we consider is E = 1 y + 2 2 I in the case where a 2 | ih | ⌦ 2 Alice and Bob perform projective measurements⇣ with dichotomic⌘ answers, say +1 or 1. Since Bob has a qubit he can construct the usual observable, B = ~b ~s, where ~b is a vector on the Bloch’s sphere. Alice has to construct · a dichotomic observable as well but has three projectors and can assign only two eigenvalues. So if P1, P2, P3 are the three projectors one can define A = P (P + P ). This is an acceptable dichotomic quantum measure- 1 2 3 ment on a qutrit. The outputs of this measurement tells that the final state is P1rP1 if it is +1, or one of the two other states if it is 1 ( P2rP2 or P3rP3 ). tr(P1rP1) tr(P2rP2) tr(P3rP3) Since A is a hermitian matrix, one can use the following (unique) decom- position: A = b(~a ~s)+gI2 2 + d 2 2 + R where b, g, d are real numbers, · ⇥ | ih | ~a a vector on the Bloch’s sphere and R is a hermitian matrix with non-zero elements only on position (3,1),(1,3),(2,3) and (3,2). With these definitions one obtain:

~a ~b AB = b · h i 2 d g A = + h i 2 2 B = 0 h i (5.9) To simulate these correlations, one can use the Degorre’s model with an adaptation on Alice. The model is the following: Alice and Bob share a ran- dom vector ~l in 3, chosen according to a uniform probability distribution R ~ 1 on the 2-dimensional sphere. That is, r(l)= 4p . Below the description of what Alice and Bob must do with this vector:

Alice First Alice tests her ~l: with probability ~a ~l she accepts ~l. When | · | she accepts ~l there still are two possibilities: with probability b she out- puts sign(~a ~l), and with probability (1 b) she outputs +1 or 1, with 1 · probability 2 for each. CHAPTER 5. MAXIMAL AND GENUINE HIDDEN NONLOCALITY 41

~ 1+ A When she does not accept l she outputs +1 with probability 2h i and 1 A 1 with probability h i . 2 Bob Bob outputs sign(~b ~l). · For Bob it is the usual Degorre’s protocol while when Alice accepts ~l there is a probability of simulating the identity instead of using the shared variable. That is, with probability b Alice and Bob simulate the singlet when Alice 1 passes her test, which happens with probability 2 since the test and the probability density are the same as in the Degorre’s model. When Alice simulates the identity or her marginals it does not contribute to the correlator since there are no correlations with Bob in those cases and Bob simulates locally the identity. Then this model gives in average:

~a ~b AB = b · h i 2 d g A = + h i 2 4 B = 0 h i (5.10)

As we have already said, in a dichotomic game, having these three quantit- ies is equivalent to having all the probabilities, meaning that we have actually proved that we can simulate E 1 for dichotomic outcomes. 2

1 2 Step 2: If Eq is local for dichotomic measurements then r = 3 Eq + 3 2 2 1 | ih | ⌦ 2 is local for projective measurements.

Proof. Bob is already able to simulate general projective measurements with the initial model since his projective measurements are dichotomic. Alice 1 has to follow this protocol: first she chooses the projector Pi with probability 3 . Secondly Alice constructs the observable defined by A = P (P + P ) where i j k Pj and Pk are the projectors she did not choose. Then with this dichotomic ob- servable Alice uses the model and obtains an answer. If it is +1 Alice outputs i, if it is 1 Alice outputs k with probability tr(Pk 2 2 ). The probability of 3 | ih | 1 1 obtaining +1 is  3 tr(PkEq)= 3 and it that case Alice and Bob simulate Eq. i=1 In the other case Alice does not use ~l and they simulate the separable state 2 2 I and this happens with probability 2 . | ih | ⌦ 2 3

Putting everything together we have the announced result: the state E 1 = 6 1 y y + 5 2 2 1 is local for projective measurements and violates 6 | ih | 6 | ih | ⌦ 2 maximally CHSH after filtering. CHAPTER 5. MAXIMAL AND GENUINE HIDDEN NONLOCALITY 42

5.4 Maximal and genuine hidden nonlocality

To construct an example of genuine and maximal hidden nonlocality we start again with the erasure state Ea (defined in (6.1)) and use the fact that E 1 is 2 local for dichotomic outcomes.

Lets Bob have a qutrit as well and use our POVMs protocol (5.1) that is:

Alice receives a POVM A = Ak = akPk and chooses the projector ai { } { } Pi with probability 3 , then she uses her dichotomic model to simulate the measurement P , 1 P . If she obtains +1 (corresponding to P ) she outputs { i i} i i, otherwise she simulates the quantum state sA. Bob does the same with his POVM B = B = b Q and s . { k} { k k} B The differences with the protocol for a qubit-qubit state are the probabil- ai ai ities of obtaining +1 or 1 for Alice and Bob, since 2 is replaced by 3 (we ak need this because in dimension three we have Âk 3 = 1). Let’s compute this new probabilities:

ai bj Alice and Bob obtain +1 : The probability of choosing i, j is 3 3 and they both obtain +1 with probability tr(Pi QjE 1 ), the quantum probability (by ⌦ 2 assumption reproduced by the two-outcomes model). The total contribution is therefore

ai bj tr(Pi QjE 1 )=tr(Ai BjE 1 ) (5.11) 3 3 ⌦ 2 ⌦ 2 Alice obtains +1 and Bob obtains 1: ai The probability of outputting i, j in this situation is 3 tr(BjsB) To compute the probability of being in this situation, we have to sum over all the possibilities of Bob’s projector choice:

bk p(+1, 1 i)= tr(Pi (1 Qk)E 1 ) Â 2 | k 3 ⌦ 1 = tr(Pi bk(1 Qk)E 1 ) Â 2 3 ⌦ k 1 = tr(Pi (31 1)E 1 ) 3 ⌦ 2 2 = tr(Pi 1E 1 ) 3 ⌦ 2 2 = tr(P E ) 3 i A

where EA is Alice’s reduce state.

2 ai 2 The total contribution of this case is thus 3 tr(PirA) 3 tr(BjsB)= 9 tr(AirA) tr(BjsB). CHAPTER 5. MAXIMAL AND GENUINE HIDDEN NONLOCALITY 43

Alice obtains 1 and Bob obtains +1: 2 By symmetry the we have: 9 tr(AisA) tr(BjEB), with EB Bob’s reduce state.

Alice and Bob obtain 1: In this case, the probability of having i, j is tr(AisA) tr(BjsB) and we have to sum over Alice’s and Bob’s choices to determine the probability of being here:

al bk p( 1, 1)= tr((1 Pl) (1 Qk)E 1 ) Â 2 l,k 3 3 ⌦ 1 = tr((31 1) (31 1)E 1 ) 9 ⌦ 2 4 = tr(1 1E 1 ) 9 ⌦ 2 4 = 9

4 Which finally gives a contribution 9 tr(AisA) tr(BjsB).

Putting the four terms together:

1 p(i, j A, B)= (tr(Ai BjE 1 )+tr(AiEA) tr(BjsB)+tr(AisA) tr(BjEB)+tr(AisA) tr(BjsB)) | 9 ⌦ 2 (5.12)

1 which corresponds to the quantum state 9 (E 1 + 2EA sB + 2sA EB + 2 ⌦ ⌦ 4s s ). A ⌦ B Setting s = s = 2 2 we obtain : A B | ih |

2 1 1 1 E| i = ( y y + 5 2 2 + 2 2 + 5 2 2 2 2 (5.13) 1/18 18 | ih | ⌦ 2 2 ⌦ | ih | | ih | ⌦ | ih | ↵⌦ Although it is much less entangled than E 1 this state has the same property 2 of having his separable part in a orthogonal space to the singlet, at least for one of the two parties. Therefore using the projectors on the subspace 0 , 1 , {| i | i} that is

F = 0 0 + 1 1 , F = 0 0 + 1 1 (5.14) A | ih | | ih | B | ih | | ih | we have

2 † † FA FB E| i F F r = ⌦ 1/18 A ⌦ B F 2 tr(F F E| i F† F†) A ⌦ B 1/18 A ⌦ B = y y ↵⌦ CHAPTER 5. MAXIMAL AND GENUINE HIDDEN NONLOCALITY 44

5.5 Generalization of rGHNL and of the POVM protocol

5.5.1 The protocol for a qudit The procedure we made in the above section can easily be extended to the general case of dimension d. Indeed, if one has a two-outcomes model for a bipartite quantum state r acting in Cd Cd one can use this d-dimensional 1 ⌦ protocol:

Alice receives A = Ak = akPk and chooses the projector Pi with prob- ai { } { } ability d , then she uses her dichotomic model to simulate the measurement P , 1 P . If she obtains +1 (corresponding to P ) she outputs i, otherwise { i i} i she simulates the quantum state sA. Bob does the same with his POVM B = B = b Q and s . { k} { k k} B This allows us to simulate

1 r = (r +(d 1)r s +(d 1)s r +(d 1)2s s ) (5.15) 2 d2 1 A ⌦ B A ⌦ B A ⌦ B

a,b,g,d 5.5.2 rGHNL In section 5.2 we defined r = 1 ( y y + 3 0 0 1 + 3 0 0 GHNL 8 | ih | | ih | ⌦ 2 | ih | ⌦ 0 0 + 1 0 0 ) | ih | 2 ⌦ | ih | Let’s generalize this state. That is let’s define:

a,b,g,d 1 1 r = a y y + b 0 0 + g 0 0 0 0 + d 0 0 ) (5.16) GHNL | ih | ⌦ 2 | ih | ⌦ | ih | 2 ⌦ | ih | ↵⌦ where d =( 1 a b g). This state has some interesting property:

1. Violates CHSH a > 1/p2 (Horodecki criterion [10]) () 2. Violates CHSH after filtering a > 0 8 3. a 1 = local model for POVMs (section 5.2)  8 ) 4. a > 0 has singlet fraction < 1 8 2 A singlet fraction greater than 1/2 means that the state can be "super- actived" ([4]). Since it is not the case here, this state is not known to be superactivable and local filtering seems to the be the only way to reveal its hidden nonlocality.

The optimal filters for this state are CHAPTER 5. MAXIMAL AND GENUINE HIDDEN NONLOCALITY 45

a + b FA = # 0 0 + 1 1 , FB = # 0 0 + 1 1 (5.17) | ih | | ih | s a + g | ih | | ih | which leads to a maximal CHSH, value in the limit of # goes to 0,

a,b,g,d 2 max CHSH(r ) = 2 1 + a { F } (a+b)(a+g) q Note that for b = g = 0 the state is the Collins-Gisin’s state and the formula above give a maximal CHSH value of 2p2 as expected. Chapter 6

Open questions

In this chapter, we will present a few questions which we think are the natural continuation of what has been presented in this essay.

6.1 Sequential filtering

In this essay we have always been focusing on the states that pass the filters, since we optimized the filters for those states. For instance, let’s go back to the erasure state :

1 E = a y y +(1 a) 2 2 (6.1) a | ih | ⌦ 2 We explain in section 5.3 that↵⌦ the local filter F = 0 0 + 1 1 (and A | ih | | ih | identity for Bob) map the state into a pure singlet. But what if we choose rather F (q)=cosq 0 0 + sinq 1 1 ? A | ih | | ih | One can check that the state that passes the filter is:

F (q) 1E F (q)† 1 r = A ⌦ a A ⌦ F tr(F (q) 1E F (q)† 1) A ⌦ a A ⌦ = y y We are still obtaining a pure ↵⌦ singlet. But there is a difference in the state that does not pass the filter. Indeed with the initial one we had:

F¯ = 1 F F† = 2 2 A A A | ih | q = )

¯ 1 ¯† 1 FA EaFA rF¯ = ⌦ ⌦ A tr(F¯ 1E F¯ † 1) A ⌦ a A ⌦ 1 = 2 2 | ih | ⌦ 2

46 CHAPTER 6. OPEN QUESTIONS 47

i.e. the state that does not pass the filter is separable. Whereas using FA(q) we have:

F ¯(q)= 1 F F† A A A = sinq q 0 0 + cosq 1 1 + 2 2 | ih | | ih | | ih |

= )

¯ ¯ † FA(q) 1EaFA(q) 1 rF ¯(q) = ⌦ †⌦ ) A tr(F ¯(q) 1E F ¯(q) 1) A ⌦ a A ⌦ 1 = a y y +(1 a) 2 2 | qih q| | ih | ⌦ 2 where y = sinq 0 + cosq 1 . | qi | i | i This state that does not pass is filterable, for instance again with FA(q)= cosq 0 0 + sinq 1 1 . And we can show that this protocol can be continued | ih | | ih | ad infinitum. Moreover, since everything is in Alice’s hand, it seems that with an unbounded amount of time, Alice and Bob can transform a local state into a nonlocal one (in this case a pure singlet) with probability one and without communication. But actually, this is false. Indeed, the difference with these filters that do not break the filterability of the states that do not pass, is their probabilities to pass: they are necessarily lower. And for the case of the erasure state. one can check that the probability of having a nonlocal state after an infinite sequence of adapted filter is equal to a, that is, the probability that one can obtain with a single filter. Although in that example it seems completely useless to use a sequence a filter rather that a one-shot filtering, the question is still open in the general case. But we conjecture that the sequential filtering can never help the violation of any Bell’s inequality. CHAPTER 6. OPEN QUESTIONS 48

6.2 POVM versus projectors

We have seen in this essay that the POVM-locality seems really more restrictive than the projective-locality. However no Bell’s inequalities are known to be violated with the help of POVMs if they are not violated with projectors. Is that a lack in the construction of these inequalities or is that a lack in the understanding of the relation between POVM and projectors?

The discovery of genuine hidden nonlocality tells us that maybe there is no fundamental difference between POVM-local and projective-local. We now know that we can extract nonlocality even from a POVM-local state, which decreases the powerful of this concept.

A track to progress with this problem could be the states that possess hidden nonlocality but not genuine hidden nonlocality. These states are projective-local but the fact that they violate a Bell’s inequality after filtering means maybe that an adapted POVM-inequality can reveal the nonlocality with one measurement.

6.3 Genuine locality

Finally, the natural question that comes from these new results is the possibil- ity of genuine locality. Means, an entangled state that remains local after all possible filtering. Even for projective measurements no such states are known.

The Werner state in dimension two is a good candidate since the white noise is the most difficult state to filter. Moreover the partial traces of the entangled part and of the separable part are exactly the same, which means that locally there is no distinction between the noise and the "interesting" part for Alice and Bob, therefore how would they be able to select one thing from another if they cannot distinguish between these two things?

A partial result that we have is that the Werner state in dimension two cannot violate CHSH after filtering. Very likely, this results can be extended to all inequalities. Chapter 7

Conclusion

In this work we exhibited some simpler examples of hidden nonlocality, that stand for qubits-qubits. We showed some improvements in Werner’s and Barrett’s models for the qubit case as well. Mainly, we showed the existence of genuine hidden nonlocality.

The progression in the understanding of nonlocality continues: we see now that local filters can be very powerful and that even a POVM-local state can be nonlocal after an appropriate filtering. Then the natural question of genuine locality remains: can we find a local model that resists to an arbitrary sequence of measurements?

The new local models we found are hopeful: they are the first that can simulate mixed states with non-trivial noise. Moreover we constructed a POVM protocol which leads to new local models from any two-outcomes model for quantum state of any dimension. This new tool is very general and can probably be used for many other cases.

Even if proving the existence of something that we ignored it could exist is always a big step, we have the feeling to be at the very beginning of a new task. What we found opens a lot of possibilities and asks more questions than it answers. These results should be seen as an opportunity to address problems with a new enthusiasm rather than be seen as an accomplishment.

Finally, we notice that once again some results in quantum theory are far from human intuition. From a state of which all correlations can be classically explained, two parties perform locally some operations and obtain correlations that no local stories are able to tell...

49 Chapter 8

Acknowledgments

I would like to thank my supervisor, Nicolas Brunner, who was immedi- ately motivated to host me in his group. He helped me to enter in this passionating field and transmitted me a lot of enthusiasm and insight. He had the intuitions and the ideas at the origin of almost everything in this essay.

The great experience I had in this group is also due to Marco Tulio Quintino and Joseph Bowles. I would like to thank them for their kindness, their en- thusiasm, and their big contributions to my research, which became their research as well (which was a delight and a success).

I thank Nicolas Gisin who introduced me my supervisor, and Yeong- Cherng Liang for his interest and contributions to my work and for fruitful discussions.

I would like to thank my mother who read this essay and tried to correct my awful English.

Finally I thank Florian Curchod for having brought me to this fascinating field of quantum information, and for more generally maintaining and ex- panding my desire to study physics, particularly through lively and enriching discussions.

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