Bell Tests in Physical Systems

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Bell tests in physical systems

Seung-Woo Lee

St. Hugh’s College, Oxford

A thesis submitted to the Mathematical and Physical Sciences Division for the degree of Doctor of Philosophy in the University of Oxford

Trinity Term, 2009

Atomic and Laser Physics, University of Oxford

Bell tests in physical systems

Seung-Woo Lee, St. Hugh’s College, Oxford

Trinity Term, 2009

Abstract

Quantum non-locality and entanglement in realistic physical systems have been of great interest due to their importance, both for gaining a better understanding of quantum physical principles and for applications in quantum information processing. Both quantum non-locality and entanglement can be effectively detected by testing Bell inequalities. Thus, finding Bell inequalities applicable to realistic physical systems has been an important issue in recent years. However, there have been several conceptual difficulties in the generalisation of Bell inequalities from bipartite 2-dimensional cases to more complex cases, which give rise to many fundamental questions about the nature of quantum non-locality and entanglement. In this thesis, we contribute to answering several fundamental questions by formulating new types of Bell inequalities and also by proposing a practical entanglement detection scheme that is applicable to any physical system.

To start with, we formulate a generalised structure of Bell inequalities for bipartite arbitrary dimensional systems. The generalised structure can be represented either by correlation functions or by joint probabilities. We show that all previously known Bell inequalities can be written in the form of the generalised structure. Moreover, the generalised structure allows us to construct new Bell inequalities in a convenient way.

Subsequently, based on this generalised structure, we derive a Bell inequality that fulﬁlls two desirable properties for the study of high-dimensional quantum non-locality. The ﬁrst property is the maximal violation of Bell inequalities by maximal entanglement which agrees with the intuition of “maximal violation of local

i ii realism by maximal entanglement”. The second property is that the Bell inequality written in correlation space should exactly represent a boundary between quantum mechanics and local realism. In contrast to any previously known Bell inequality, the derived Bell inequality is shown to satisfy both conditions. We apply this Bell inequality to continuous variable systems and demonstrate maximal violation by the maximally entangled state associated with position and momentum.

We then formulate a generalised Bell inequality in terms of arbitrary quasi- probability functions in phase space formalism. This includes previous types of Bell inequalities formulated using the Q and Wigner functions as limiting cases. We show that the non-locality of a quantum system is not directly related to the negativity of its quasi-probability distribution beyond the previously known fact for the case of the Wigner function. We also show that the Bell inequality formulated using the Q-function permits the lowest detector efficiencies out of the quasi-probability distributions considered. Finally, we present a general approach for witnessing entanglement in phase space by significantly inefficient detectors. Its implementation does not require any additional process for correcting errors in contrast to previous proposals. Moreover, it allows detection of entanglement without full a priori knowledge of detection efficiency. We show that entanglement in single photon entangled and two-mode squeezed vacuum states is detectable by means of tomography with detector efficiency as low as 40%. This approach enhances the possibility of witnessing entanglement in various physical systems using current detection technologies. Contents

Abstract i

Chapter 1. Introduction 1

Chapter 2. Basic Concepts 9 2.1 High-dimensional quantum systems ...... 9 2.1.1 Quantum states in high-dimensional Hilbert space . . . . 9

2.1.2 High-dimensional physical systems ...... 12 2.2 Bell’s theorem ...... 14

2.2.1 Bell’s theorem and Bell inequalities ...... 14 2.2.2 Bell test experiments ...... 18 2.2.3 Loopholes in Bell tests ...... 20

2.2.4 Polytope representation of Bell’s theorem ...... 23 2.3 Quantum entanglement ...... 24

2.3.1 Entanglement ...... 24 2.3.2 Genuine entanglement ...... 26

2.3.3 Entanglement Witness ...... 27

2.4 Phase space representations ...... 29 2.4.1 Generalised quasi-probability functions ...... 29

2.4.2 Bell inequalities in phase space ...... 30

iii iv Contents

Chapter 3. Generalised structure of Bell inequalities for arbitrary- dimensional systems 33

3.1 Introduction ...... 33

3.2 Generalised arbitrary dimensional Bell inequality ...... 35

3.3 Violation by Quantum Mechanics ...... 40

3.4 Tightness of Bell inequalities ...... 46

3.5 Remarks ...... 48

Chapter 4. Maximal violation of tight Bell inequalities for maximal

entanglement 51

4.1 Introduction ...... 51

4.2 Optimal Bell inequalities ...... 53

4.3 Extension to continuous variable systems ...... 60

4.4 Conclusions ...... 63

Chapter 5. Testing quantum non-locality by generalised quasi-probability functions 65

5.1 Introduction ...... 65

5.2 Generalised Bell inequalities of quasi-probability functions . . . . . 68

5.3 Testing Quantum non-locality ...... 70

5.4 Violation by single photon entangled states ...... 74

5.5 Violation by two-mode squeezed states ...... 76

5.6 Discussion and Conclusions ...... 79

Chapter 6. Witnessing entanglement in phase space using ineﬃcient

detectors 83

6.1 Introduction ...... 83

6.2 Observable associated with eﬃciency ...... 85 Contents v

6.3 Entanglement witness in phase space ...... 86 6.4 Testing single photon entangled states ...... 88

6.5 Testing two-mode squeezed states ...... 90

6.6 Testing with a priori estimated eﬃciency ...... 91

6.7 Conclusions ...... 93

Chapter 7. Conclusion 95

Bibliography 99

Chapter 1

Introduction

Bell’s theorem [1] has been called “the most profound discovery of science” [2] and also “one of the greatest discoveries of modern science” [3]. These words highlight the importance of the quantum features i.e. quantum non-locality and entanglement that were then found based on Bell’s theorem. Quantum non-locality and entanglement show striking properties that can not be understood in the context of classical physics, and thus shift the paradigm of understanding fundamental principles in physical systems. Moreover, these quantum features are a promising resource for quantum information processing that is a revolutionary technology superior to classical counterpart in various ways [4]. In these senses observing quantum non-locality and entanglement in physical systems can be seen as a surprising discovery. Historically, the concepts of quantum non-locality and entanglement appeared for the ﬁrst time in the famous Einstein-Podolski-Rosen (EPR) paradox [5]1. This paradox was formulated with the theory of local realism in mind to prove the incompleteness of quantum mechanics. Later Bell showed in his theorem that local realism leads to constraints on correlations between measurement results carried out on two separated systems [1]. These constraints, known as Bell inequalities, provide a possible method for testing the validity of quantum mechanics against local realism. It

1The concept of entanglement was also introduced by E. Schr¨odinger[6, 7].

1 2 Introduction was theoretically shown that Bell inequalities are violated by the quantitative pre- dictions of quantum mechanics in the case of entangled states [1, 8]2. Subsequently, experiments have conﬁrmed the validity of quantum mechanics by demonstrating the violation of Bell inequalities [9, 10]. In general, quantum non-locality and entanglement manifest themselves in such a counterintuitive way, precisely through the violation of the local realism [9, 10] and by permitting stronger correlations beyond the level allowed in classical physics [11].

Quantum non-locality and entanglement in simple models such as bipartite or tripartite 2-dimensional systems are currently well understood [11]. However, little is known about them in realistic physical systems composed of many particles with many degrees of freedom, which we will call ‘complex systems’ throughout this thesis. In fact most physical systems in nature are complex systems, and thus generalisations of Bell’s theorem to complex systems have been regarded as one of the most important challenges in quantum mechanics [12, 13, 14, 15, 16, 17, 18, 19, 20, 11, 21, 22].

There are several motivations for studying Bell’s theorem in complex systems. Firstly, generalising Bell’s theorem to complex systems would provide a way to observe quantum features in the macroscopic world. Macroscopic systems are complex systems that are generally governed by classical physics, and thus observing quantum properties in those systems seems to be diﬃcult. However, recent studies on e.g. the relation of entanglement with macroscopic observables [21] oﬀer possibili- ties to investigate quantum properties in macroscopic systems [23]. For example, Bell type inequalities constructed with macroscopic observables would allow one to detect quantum non-locality and entanglement in macroscopic physical systems.

Secondly, Bell inequalities can help us to investigate quantum phenomena arising in realistic physical systems. For example, studying the role of entanglement in a

2Note that not all entangled states violate Bell inequalities. 3 quantum phase transition is one of the most interesting issues in recent relevant research [24]. In fact, quantum phase transition can be regarded as a change of the dominant degrees of freedom of a system. Thus Bell inequalities defined in both degrees of freedom that the system traverse would be an effective tool for the study of entanglement in the vicinity of a quantum phase transition. This can be also used for studying, for example, entanglement transfer between different physical degrees of freedom.

Thirdly, studying quantum non-locality and entanglement in realistic physical systems is essential for applications in quantum information processing. Quantum non-local properties and entanglement have been considered as essential resources for various quantum information processing protocols [25, 11, 20]. All candidates of quantum information processing are complex systems [20, 21, 26, 27]. Although a specific degree of freedom is chosen as the basic unit of quantum information processing, i.e. a qubit, the other degrees of freedom can still affect the qubit and should be considered in realistic implementations. Thus investigation of a qubit system in high-dimensional formalism might be useful to understand the behaviour of qubits in realistic implementations. Furthermore, implementations of quantum information processing in complex systems can provide practical advantages. For example, high-dimensional quantum cryptography can be more secure than 2-dimensional cases [28, 29, 30]. Quantum teleportation, quantum computation, and quantum cryptography can be efficiently implemented by optical continuous variable systems [20, 31].

In spite of these motivations, realisation of Bell tests in complex physical systems still suffers from both conceptual and technical difficulties, and there are numerous relevant open questions [16, 15]. Yet we do not have a full picture of all possible manifestations of quantum phenomena in complex systems. For example, there have been no known Bell inequalities for high-dimensional systems that are maximally vi- 4 Introduction olated by maximally entangled states without bias at the degree of violation [32, 33]. We still do not have a clear picture of quantum non-locality in phase space formalism and its relations to other quantum properties. Moreover, there is no general method for quantifying entanglement in complex systems. Due to several effects arising in complex systems, it is difficult to observe quantum behaviour clearly. For example, since complex physical systems are generally macroscopic and interact strongly with their environment, decoherence can influence entanglement properties in those systems. Moreover, with increasing size or dimensionality, a precise measurement of a system becomes difficult, especially if one wants to preserve its quantum properties. This is often due to detector imperfections.

In this thesis we focus on several topics related to Bell tests in realistic physical systems, which shall be described as follows. The first topic studied in this thesis is the extension of Bell inequalities to arbitrary dimensional bipartite systems. We consider two desirable conditions of Bell inequalities in order to investigate quantum non-locality properly in high-dimensional quantum systems. For 2-dimensional systems the Clauser-Horne-Shimony-Holt (CHSH) inequality [8] has the desirable property of only being maximally violated for a maximally entangled state. In addition, the CHSH inequality is a tight Bell inequality i.e. a facet of the polytope defining the boundary between local realism and quantum mechanics. This means that any violation of local realism on this particular facet is indicated by the CHSH inequality [33]. Note that tightness is a desirable property since only sets of tight Bell inequalities can provide necessary and sufficient conditions for the detection of pure state entanglement. For d-dimensional systems with d > 2 there has been no known Bell inequality to satisfy both desirable properties so far. For example, the Bell inequality proposed by Collins et al. [13] is maximally violated by non- maximal entanglement [32] and the Bell inequality in the case of Son et al. [14] was shown to be non-tight [17]. Therefore, it would be ideal to find a Bell inequality 5 satisfying both conditions for arbitrary d-dimensional systems. In addition, such a Bell inequality may provide practical advantages in e.g. the preparation of an ideal channel for higher-dimensional quantum teleportation [34] or cryptography [25]. We will address this topic in chapter 3 and 4, where we formulate a generalised structure of Bell inequalities for bipartite arbitrary d-dimensional systems and derive a Bell inequality that fulfills two desirable properties: maximal violation by maximally entangled states and tightness.

Another topic of this thesis is to study quantum non-locality in phase space formalism. Phase space representations are a convenient tool for investigating quantum states as they provide insights into the boundaries between quantum and classical physics. Any quantum stateρ ˆ can be fully characterised by the quasi-probability function [35, 36]. The negativity of the quasi-probability function has been regarded as a non-classical feature of quantum states, and is thus believed to have a fundamental relation to quantum non-locality.

Bell argued [37] that the original EPR state [5] will not exhibit non-locality since its Wigner-function is positive everywhere and hence serves as a classical probability distribution for hidden variables. However, later Banaszek and W´odkiewiczshowed how to demonstrate quantum non-locality using the Q- and Wigner-functions [38, 19, 39]. Remarkably, this showed that there is no direct relation between the negativity of the Wigner function and quantum non-locality. Since then, in spite of various eﬀorts to explain more precisely the relation between quantum non-locality and the negativity of quasi-probability functions [40], a clear answer has been missing. This is mainly because we still do not have a general method to describe quantum non-locality in phase space formalism. In particular, a Bell inequality formulated by generalised quasi-probability functions would be necessary, by which one could demonstrate how non-locality changes the extent of the negativity. We will propose such a method in chapter 5 to test Bell inequalities using arbitrary quasi-probability 6 Introduction functions and study the relation between quantum non-locality and the negativity of quasi-probability functions.

Our final topic is to find an efficient detection scheme for entanglement in phase space formalism. Entanglement detection is one of the primary tasks both for investigating fundamental aspects of quantum systems and for applications in quantum information processing [11]. However, in most cases, experimental realisations of testing entanglement suffer from detector imperfections since measurement errors wash out quantum correlations. This difficulty becomes more significant as the size or dimensionality of the systems increases. This is unfortunate as entanglement in larger systems is gaining more attention [20, 21]. For example, the violation of a Bell type inequality for continuous variable systems e.g. two-mode squeezed states

[19, 41] requires almost perfect photo-detection eﬃciency [42]. Several schemes have been proposed to overcome this problem such as e.g. numerical inversion of measured data [43] and iterative reconstruction methods [18, 44], but require a large number of calculation or iteration steps. Therefore, an entanglement detection scheme that is practically usable in the presence of noise is necessary. Furthermore, a general entanglement criterion is required, which is applicable independently of the particular physical systems in phase space.

Such an entanglement criterion would also be essential in entanglement based quantum information protocols [20]. For example, preparing an entangled channel is a necessary precondition for any secure quantum key distribution protocol [45]. In chapter 6 we will propose a detection scheme of entanglement using ineﬃcient detectors.

As motivated from these three topics, we investigate quantum non-locality and entanglement in physical systems. Our research aims to provide answers the questions arising in those topics. Detailed descriptions of each chapter are presented as follows. 7

• In chapter 2, we will review basic concepts that are necessary for describing most of the content in this thesis. We start by introducing high-dimensional quantum systems. Then, the basic concepts of quantum non-locality will be explained in the context of the original Bell theorem. We consider loophole

problems that occur in experimental Bell tests. We also introduce the polytope representation of Bell’s theorem that is useful to extend Bell inequalities to complex systems. Then, we compare entanglement witnesses, which discriminate entangled states from separable states, to Bell inequalities. Finally, we introduce the phase space representation of quantum states and a Bell inequality represented in this formalism.

• In chapter 3 we will formulate a generalised structure of Bell inequalities for bipartite arbitrary d-dimensional systems, which can be represented either in terms of correlation functions or joint probabilities. The two known high-

dimensional Bell inequalities proposed by Collins et al. [13] and by Son et al. [14], will be considered in the framework of the generalised structure. We then investigate the properties of these Bell inequalities with respect to the degree of entanglement.

• In chapter 4 we will derive a Bell inequality for even d-dimensional bipartite quantum systems that fulﬁlls two desirable properties: maximal violation by maximally entangled states and tightness. These properties are essential to investigate quantum non-locality properly in higher-dimensional systems. Then we apply this Bell inequality to continuous variable systems and investigate its violations. We also discuss which local measurements lead to maximal

violations of Bell inequalities in continuous variable Hilbert space.

• In chapter 5 we will investigate quantum non-locality in phase space formalism.

We ﬁrst formulate a generalised Bell inequality in terms of the s-parameterised 8 Introduction

quasi-probability function. We then demonstrate quantum non-locality by the single-photon entangled and two-mode squeezed states as varying the parameter s and detection eﬃciencies. We ﬁnally discuss the relation between quantum non-locality and the negativity of quasi-probability functions.

• In chapter 6 we propose an entanglement detection scheme using significantly inefficient detectors. This is applicable to arbitrary quantum states described in phase space formalism. We formulate an entanglement witness in the form of a Bell-like inequality using directly measured Wigner functions. For this, we include the effects of detector efficiency into possible measurement outcomes. Using this entanglement witness, we detect entanglement in the single photon entangled and two-mode squeezed states with varying detection efficiency. We finally discuss the effects of a priori knowledge of detection efficiency on the

capability of our scheme.

• We conclude this thesis in chapter 7 and give an outlook on the directions of further research.

Apart from the research presented in this thesis, I have also contributed to propose entanglement puriﬁcation protocols which are applicable to multipartite high-dimensional systems [46]. Chapter 2

Basic Concepts

In this chapter, we present basic ideas for the study of Bell’s theorem in physical systems. We review a high-dimensional description of quantum states, and deﬁne high-dimensional physical systems. Basic concepts of Bell’s theorem and its experimental implementations are presented. As considering implementations we introduce loophole problems arising in realistic Bell tests. We also introduce a polytope representation of Bell inequalities that is useful to study Bell’s theorem in complex systems. For the extension of Bell’s theorem to the phase space formalism we review Bell inequalities formulated by quasi-probability functions. In addition, we also consider entanglement witnesses as comparing to Bell inequalities.

2.1 High-dimensional quantum systems

2.1.1 Quantum states in high-dimensional Hilbert space

Pure quantum states of a system are represented by vectors in Hilbert space. A system completely described in d-dimensional Hilbert space, Hd, is called a d- dimensional quantum system (or equivalently d-level quantum system). For the simplest example, a 2-dimensional quantum system is described in the 2-dimensional

Hilbert space H2 spanned by two orthonormal basis vectors, | ψ0i and | ψ1i. An ar-

9 10 Basic Concepts bitrary superposition of two basis vectors | ψi = α | ψ0i + β | ψ1i is also a possible state of the system, where their amplitudes α and β are arbitrary complex numbers. The normalisation condition for the state | ψi i.e. |α|2 + |β|2 = 1, leads us to interpret |α|2 and |β|2 as the probabilities that the system is measured to be in states | ψ0i and | ψ1i respectively. This state can be regarded as a basic unit of quantum information processing i.e. a qubit. Then, the two computational basis vectors are labeled by | ψ0i = | 0i and | ψ1i = | 1i. The computational basis can be arbitrarily chosen by transformation in 2-dimensional Hilbert space. For example, √ √ | +i = (| 0i + | 1i)/ 2 and | −i = (| 0i − | 1i)/ 2 constitute another orthonormal basis of qubits. In general, the unitary transformations in 2-dimensional Hilbert space, SU(2), provide an inﬁnite number of possible basis sets.

While a pure quantum state is described by a single vector as described above, P a mixed state is given as a statistical ensemble of pure statesρ ˆ = i pi | ψii hψi | where pi is the probability that the system in the state | ψii. The concept of a mixed state comes up when the state of a system is not exactly known but given as a mixture of diﬀerent states. The most general description of a 2-dimensional

1 system is given as a density matrixρ ˆ = 2 (1 + ~a · ~σ) where ~a = (a1, a2, a3) is a real vector and ~σ = (ˆσx, σˆy, σˆz) is a vector of pauli operators. This can be visualised by the Bloch sphere which represents all possible states of a single qubit [4]. Pure quantum states correspond to the surface of the sphere, that is |~a|2 = 1, while mixed states correspond to the interior region of the sphere.

A high-dimensional system is deﬁned as a system which should be described in more than 2-dimensional Hilbert space. Its mathematical description is straightfor- wardly extended from the 2-dimensional formalism. We consider a d-dimensional (d > 2) Hilbert space spanned by d orthonormal basis vectors, {| 0i , | 1i , ..., | d − 1i}.

All pure states in d-dimensional Hilbert space can be represented using the basis 2.1. High-dimensional quantum systems 11 vectors

Xd−1 | ψi = αk | ki , (2.1) k=0

P 2 2 where the complex numbers αk satisfy the condition k |αk| = 1, and |αk| is the probability that the system is found to be in the state | ki. This can be regarded as the quantum version of a d-dimensional computational basic unit i.e. a qudit. Like in the 2-dimensional case, the qudit basis can be freely chosen by an arbitrary unitary transformation in d-dimensional Hilbert space, SU(d).

In general an arbitrary quantum state in d-dimensional Hilbert space can be described as

2 1 dX−1 ρˆ = (1 + a λˆ ), (2.2) d i i i=0

ˆ ~ ˆ ˆ where ai = Trλiρˆ and a0 = 1. Here λ = (λ0, ..., λd2−1) is a generalised pauli operator

2 in d-dimensional Hilbert space and the vector ~a = (a0, ..., ad2−1) is a d -dimensional real vector. This is the so called generalised Bloch representation of d-dimensional ˆ quantum states [47]. The generalised Pauli operators, λi, in d-dimensional Hilbert space, Hd, are given by [48, 49]

ˆ a ˆ b (Xd) (Zd) , a, b ∈ 0, 1, ...d − 1, (2.3)

ˆ ˆ where Xd and Yd are deﬁned as

µ ¶ Xd−1 2πi Xˆ = | k + 1i hk | , Zˆ = exp Nˆ . (2.4) d d d k=0

ˆ P ˆ ˆ Here N = k k | ki hk | is the number operator, and Xd and Yd transform the d- 12 Basic Concepts dimensional computational basis by

µ ¶ 2πi Xˆ | ki = | k + 1i , Zˆ | ki = exp k | ki . (2.5) d d d

ˆ ˆ ˆ ˆ ˆ ˆ Note that Xd and Zd do not commute as ZdXd = exp(2πi/d)XdZd. The generalised Pauli operators in Eq. (2.3) provide a basis for arbitrary unitary operations in d- dimensional Hilbert space, which we will use to change the measurement basis in high-dimensional Bell tests.

2.1.2 High-dimensional physical systems

General physical systems are composed of many particles with many degrees of freedom. In a given physical system one may be interested in a specific degree of freedom which we will call the target degree of freedom, or specific subsystems which we will call target subsystems. Based on these concepts we define the following cases as high-dimensional physical systems.

First, most degrees of freedom in physical systems are represented by superposed states with more than 2 possible outcomes i.e. high-dimensional. For example, the position and momentum of a free particle are continuous variables in inﬁnite- dimensional Hilbert space. The angular momentum of an electron in atoms is ﬁnite high-dimensional. Therefore, if the target degree of freedom that we are interested in is high-dimensional, we regard the corresponding system as a high-dimensional system.

Second, most physical systems have many degrees of freedom. In realistic implementations, it is very hard to single out a specific degree of freedom properly so that any removed degrees of freedom do not affect considerably the degree of freedom being singled out. One might consider the effects of other degrees of freedom as noises caused by complexity. However, in order to understand properly several 2.1. High-dimensional quantum systems 13 quantum phenomena in realistic systems, it is necessary to consider more than two degrees of freedom simultaneously and their influences on each other. For example, a quantum phase transition can be generally understood as an abrupt change of the dominant degree of freedom by varying an external parameter in physical systems

[50] as shown in the case that cold atoms in optical lattices traverse two states, Mott-insulator and superﬂuid [51]. Therefore, if we consider multiple degrees of freedom simultaneously in a given system, their properties should be described in the framework of a high-dimensional formalism. Thus the corresponding system is high-dimensional.

Third, most physical systems are composed of many particles. Thus it is also very hard to single out the target subsystems properly so that the removed rest subsystems do not affect them considerably. In certain cases the target subsystems can be selected as open systems and the effects from rest subsystems can be regarded as effects of the environment. In most cases, in order to investigate effectively many body systems, it is necessary to choose target subsystems as collective bodies of many particles which should be described in high-dimensional Hilbert spaces. For example, a subsystem composed of N qubits can be considered in the d = 2N dimensional Hilbert space [52].

Therefore, we can consider the systems

• with a higher-dimensional target degree of freedom,

• with multiple target degrees of freedom,

• having target subsystems composed of many particles as high-dimensional systems. Several quantum features in such systems, which might not arise in e.g. 2-dimensional bipartite systems, can be eﬀectively investigated in the framework of the high-dimensional formalism. For example, entanglement of 14 Basic Concepts multiple target degrees of freedom, which is called as the hyper entanglement, has been realised and investigated based on the high-dimensional formalism [53, 54]. Polarisation, time-bin and spatial degree of freedom have been used to create high- dimensional systems with d = 3 [55], d = 4 [56], and d = 8 [57]. In addition, high- dimensional systems are applicable to quantum information processing and provide some advantages e.g. a robust quantum key distribution [28, 29, 30], superdense coding [58], fast high ﬁdelity quantum computation [59, 60, 61].

To summarise, physical systems existing in nature are high-dimensional systems, and several quantum phenomena in such systems can be eﬀectively investigated in the framework of high-dimensional Hilbert space. This may lead to obtaining fundamental insight into the properties of complex quantum systems. Moreover, their properties are also applicable to quantum information processing.

2.2 Bell’s theorem

2.2.1 Bell’s theorem and Bell inequalities

Bell introduced a theorem about quantum non-locality in his seminal paper entitled “On the Einstein-Podolsky-Rosen paradox” [1]. In his theorem he discussed the famous paradox presented by Einstein, Podolsky, and Rosen (EPR) [5], which was intended to prove the incompleteness of quantum mechanics. Bell’s argument begins with two assumptions as asserted in the EPR paper:

• Reality - the measurable quantity must have a deﬁnite value before the measurement takes place.

• Locality - the physical quantities within reality would not inﬂuence each other at a large distance. 2.2. Bell’s theorem 15

Based on these assumptions together, called local realism, Bell derived a constraint in the form of an inequality which limits the correlational expectation values of measurement outcomes for two spatially separated parties. This is called the Bell inequality. Any violation of this inequality by the quantitative prediction of quantum mechanics implies that at least one of the two assumptions, reality or locality, must be abandoned. The most considerable achievement of Bell’s argument is to provide a possible method for testing the validity of quantum mechanics against local realism. Thus, it was thought that Bell’s theorem may put an end to the debate between local realism and quantum mechanics. However, it took more time to realise a test of Bell inequality due to several technical diﬃculties. With the progress of quantum control techniques, ﬁnally the violation of Bell inequalities has been demonstrated [9] as clear evidence of existing quantum non-local properties which defeats local realism.

Ever since the EPR argument and Bell’s theorem, many versions of Bell inequalities have been derived similar to Bell’s original inequality. The most famous version was proposed by Clauser, Horne, Shimony and Holt (CHSH) [8]. Suppose that two spatially separated parties named Alice and Bob perform measurements independently. Each observable can be chosen from two possible settings denoted by A1, A2 for Alice and B1, B2 for Bob. There is no inﬂuence on the measurement selection between two parties. We here assume that all observables have two possible outcomes ±1, i.e. 2-dimensional outcomes. Then we can consider a combination of all possible correlations, A1B1, A1B2, A2B1, and A2B2, as

A1B1 + A1B2 + A2B1 − A2B2 = ±2, (2.6) which can take either the deterministic value +2 or −2 depending on measurement outcomes of each observable. Let us deﬁne the joint probability P (a1, a2, b1, b2) 16 Basic Concepts which indicates that the system is in a state where A1 = a1, A2 = a2, B1 = b1, and

B2 = b2 before the measurement. Then the average of the combination in Eq. (2.6) is written by

E(A1B1 + A1B2 + A2B1 − A2B2) X = P (a1, a2, b1, b2)(a1b1 + a1b2 + a2b1 − a2b2) a ,a ,b ,b 1 X2 1 2 = P (a1, a2, b1, b2)a1b1 + P (a1, a2, b1, b2)a1b2

a1,a2,b1,b2

+P (a1, a2, b1, b2)a2b1 − P (a1, a2, b1, b2)a2b2

= E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2). (2.7)

From Eq. (2.6) and Eq. (2.7), we can obtain an inequality

−2 ≤ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) ≤ 2, (2.8) which is called the CHSH inequality. Note that the upper and lower bound of the CHSH inequality are given as the deterministic maximum and minimum value of the Eq. (2.6).

Let us now assume that two parties share a quantum state

1 | ψi = √ (| 0i | 1i − | 1i | 0i). (2.9) 2

~ ~ Each party can choose observables A1 = SA · ~a1 and A2 = SA · ~a2 for Alice, and ~ ~ ~ ~ ~ B1 = SB · b1 and B2 = SB · b2 for Bob as varying the unit vectors of ~a1, ~a2, b1, ~ ~ ~ ~ and b2. Here SA and SB are the spin operator deﬁned as S = (Sx,Sy,Sz) which ~ is proportional to the pauli operators. If the chosen observables are SA · ~a1 = Sz, √ √ ~ ~ ~ ~ ~ SA ·~a2 = Sx, SB · b1 = −(Sz + Sx)/ 2, and SB · b1 = (Sz − Sx)/ 2, the expectation 2.2. Bell’s theorem 17 value for the quantum state in Eq. (2.9) is given by

√ hA1B1i + hA1B2i + hA2B1i − hA2B2i = 2 2, (2.10) which violates the CHSH inequality (2.8). This is a remarkable result which shows the existence of the non-local correlations in the state Eq. (2.9). It is also inevitably shown that local realism should be abandoned.1

Let us then consider another version of Bell inequality proposed by Clauser and Horne (CH) in 1974 [65] as

−1 ≤ P (X1,Y1) + P (X1,Y2) + P (X2,Y1) − P (X2,Y2) − P (X1) − P (Y1) ≤ 0,(2.11)

where P (X1,Y1) is the joint probability for the local measurement X1 and Y1, and likewise for others. It can be derived from the CHSH combination given in Eq. (2.7)

−2 ≤ a1b1 + a1b2 + a2b1 − a2b2 ≤ 2, (2.12)

with outcome variables deﬁned by x1 = (1 + a1)/2, x2 = (1 + a2)/2, y1 = (1 + b1)/2 and y2 = (1 + b2)/2, x1, x2, y1, y2 ∈ {0, 1}. We can then obtain another form of inequality from Eq. (2.12) as

−4 ≤ a1b1 + a1b2 + a2b1 − a2b2 − 2

= 4(x1y1 + x1y2 + x2y1 − x2y2 − x1 − y1) ≤ 0. (2.13)

Finally, we ﬁnd that the statistical average of this combination gives the CH inequality given in Eq. (2.11).

1There are several other interpretations which explain this result e.g. the non-local hidden variable model by David Bohm [62, 63] and many world interpretations [64]. Details of these interpretations are beyond the scope of this thesis. 18 Basic Concepts

2.2.2 Bell test experiments

The test of the CHSH inequality can be implemented by various 2-dimensional systems with measurements of 2-dimensional outcomes. For example, one can consider the Stern-Gerlach measurement for entangled spin pairs or the polarisation measurement for entangled photon pairs. Let us here consider a Bell test performed by an optical setup to measure polarisations of entangled photon pairs [9]. Suppose that Alice and Bob are spatially separated and share entangled photon pairs which are generated from an optical source. Each photon goes through a polarising beam splitter whose orientation can be freely chosen by each party as shown in Fig. 2.1. Photons are detected at two output channels of the polarising beam splitter. We assume that measured data is recorded only for coincident detections at both parties. Possible outcomes for each are denoted by + or −, and thus there are four possible compound data for a single trial: ++, +−, −+, and −−. After many trials of experiments with a speciﬁed measurement setting we can obtain the joint probability by statistical average as

N P (+ + |ab) = ++ , (2.14) N

where N++ is the number of detections with outcomes ++ and likewise for others, and N = N++ + N+− + N−+ + N−−. Here a and b denote the measurement setting changed by rotating the polarisation at the beam splitter. We then deﬁne the correlation function between measurement results of a and b as

Ea,b = P (+ + |ab) − P (+ − |ab) − P (− + |ab) + P (− − |ab). (2.15) 2.2. Bell’s theorem 19

Figure 2.1. Optical setup for the Bell test (CHSH inequality). Entangled photons generated from the source are distributed between two separate parties. Photons go through a beam splitter and are measured at the detector with coincidence counting.

After measurements for all a, b = 1, 2, we can obtain E1,1, E1,2, E2,1, and E2,2. Finally the statistical average of the CHSH combination can be obtained as

|E1,1 + E1,2 + E2,1 − E2,2| ≤ 2, (2.16) which should be bounded by the value 2 in the local realistic theories. Any statistical average exceeding this bound guarantees that the shared states have non-local properties.

Let us consider the case when the shared photon pairs are in the state

1 | ψi = √ (| +i | +i + | −i | −i). (2.17) 2

2 expectation value as P (+ + |ab) = | hψ | | θa, +i | φb, +i | and likewise for others. 20 Basic Concepts

The correlation function in Eq. (2.15) can be then obtained as

Ea,b = cos 2(θa − φb), a, b = 1, 2. (2.18)

If the measurement basis are chosen as θ1 = 0, θ2 = π/4 for Alice and φ1 = π/8 and

φ2 = −π/8 for Bob, the expectation value of the CHSH combination can exceed √ the local realistic bound 2 (it reaches 2 2 in the case of perfect measurements). Therefore, one can observe the violation of the CHSH inequality experimentally.

Since Bell’s theorem, a great number of Bell test experiments have been performed and confirmed quantum mechanics against local realistic theories. The first test of Bell’s theorem was performed in 1972 by Freedman and Clauser [66]. They demonstrated violations of a CH-type inequality by polarisation correlations of photons emitted by calcium atoms. Later, a test of the CHSH inequality was implemented by Aspect et al. in 1982 [9].2 Tittel et al. [69] and Weihs et al. [70] conducted Bell tests using photon pairs that were space-like separated. Progresses of quantum technologies have led to significant improvement in both efficiencies and variety of Bell tests. However, in realistic implementations of Bell tests, there exist several conceptual difficulties as we will explain in the following subsection.

2.2.3 Loopholes in Bell tests

Beyond technical diﬃculties in the realisation of Bell tests, there exist also some conceptual diﬃculties. Indeed, so far there have been no experimental demonstra- tions of quantum non-locality without supplementary assumptions. Thus, there are still many scientists who point out the fact that the violation of Bell inequalities can be explained as faults of experimental setup in local realistic theories. This is the

2Aspect et al. also conducted other versions of Bell type inequalities [67, 68] such as the type proposed by Clauser and Horne [65] and original Bell inequality [1]. 2.2. Bell’s theorem 21 so called loophole problem of the Bell test. Let us here consider two main loophole problems as follows:

• locality loophole - This arises from the difficulty in separating two local particles. When the distance between two local measurements is small it conflicts with the assumption of no communication between one observer’s measurements to the other. Therefore, to prevent such a loophole problem the two parties should be space-like separated. However, it is also technically difficult

to separate two sub-systems without losing their quantum properties. It seems that the most promising candidate for closing this loophole problem is optical photons [69] as other massive particles are diﬃcult to separate suﬃciently so

that space-like measurements can be performed. [10]. In 1998, Weihs et al. conducted, for the ﬁrst time, a Bell test experiment that closed the locality loophole using photons. In their scheme, the choice of local polarisation measurement was ensured to be random in order to avoid any connection between separated measurements [70].

• detection loophole - All experiments suﬀer from the imperfection of realistic detectors. This causes the so called detection loophole problem. The ineﬃ- ciency of detectors makes a violation of Bell inequalities compatible with local realism. Let us consider the Bell test scheme described in the previous section

2.2.2 with detection efficiency less than 100%. In this scheme measured data is recorded when particles are detected coincidentally. However, a coincident measurement is affected by the detection efficiency. Thus the effect changes the CHSH inequality into [71]

4 |Ecoinc + Ecoinc + Ecoinc − Ecoinc| ≤ − 2, (2.19) 1,1 1,2 2,1 2,2 η 22 Basic Concepts

where η is the detection eﬃciency. For the quantum states given in Eq. (2.9), √ the left-hand side still can reach the value 2 2. Therefore, the violation of √ Eq. (2.19) can be demonstrated only if η > 2( 2 − 1) ' 0.83. This means that in order to demonstrate quantum non-locality without any supplemen-

tary assumption highly eﬃcient detectors (η > 0.83) are required. Scientists usually assume that the sample of detected pairs is representative of the pairs generated at the sources, i.e. the fair sampling assumption that reduces the right-hand side in Eq. (2.19) to 2. However, it is a supplementary assumption in addition to local realism which prevents us observing genuine non-local

properties. The minimum required eﬃciency for a non-locality test has been investigated for various measurement setups and systems [72, 73, 74, 75]. Nu- merous eﬀorts have attempted to close the detection loophole problem, which

may lead to the so called loophole-free Bell test [76, 77]. For example, Rowe et al. [10] performed Bell tests using massive ions instead of photons with 100% eﬃciency, though it has a locality loophole problem.

These loophole problems have been also taken seriously in applications to quantum information processing. For example, the detection loophole affects the security of quantum key distribution protocols [25]. In the presence of detection loophole several attacks are possible in the protocols of quantum key distribution [78, 79]. The threshold of efficiency for secure protocols has been studied, for instance, it has been shown that the overall efficiency taking into account the channel loss and the detection efficiency together is required to be higher than 50% for secure standard quantum key distribution [78], which is not feasible yet with current technologies. 2.2. Bell’s theorem 23

Figure 2.2. Schematic diagram of the ploytope in the joint probability space or alternatively in the correlation function space. The inside region of the polytope represents the accessible region of local-realistic (LR) theories and outside region contains the region of quantum mechanics (QM). Each facet of the polytope corresponds to a tight Bell inequality. A non-tight Bell inequality including bias at the boundary is represented by a line deviating from the boundary.

2.2.4 Polytope representation of Bell’s theorem

Bell’s theorem can be represented in the space constructed by the vectors of measurement outcomes. The basis vectors correspond to either the joint probabilities or correlation functions for possible outcomes. In both spaces of joint probabilities and correlations functions the set of possible outcomes for a given measurement setting of

Bell tests constitutes a convex polytope as schematically plotted in Fig. 2.2 [33, 80]. Each generator of the polytope, being an extremal point of the polytope, represents the predetermined measurement outcome called a local-realistic conﬁguration. All interior points of the polytope are given by the convex combination of generators and they represent the accessible region of local-realistic (LR) theories associated with the probabilistic expectations of measurement outcomes. Therefore, every facet of the polytope is a boundary of halfspace characterised by a linear inequality, which we call a tight Bell inequality. There are non-tight Bell inequalities which contain the polytope in its halfspace. As a non-tight Bell inequality has interior bias at the boundary between local realistic and quantum correlations, one might call it a worse detector of non-local properties. We note that the polytope representation is 24 Basic Concepts useful to study Bell’s theorem in complex systems, since it provides complete geometric boundary between quantum mechanics and local realism in principle for all dimensional cases.

2.3 Quantum entanglement

2.3.1 Entanglement

Entanglement is associated with the quantum correlation between two or more subsystems of a composite body. Let us consider a state that is in a Hilbert space of two qubits Ha ⊗ Hb,

1 | Ψi = √ (| 0i | 1i + | 1i | 0i ). (2.20) ab 2 a b a b

It is impossible to determine whether the ﬁrst qubit carries the value 0 or 1, and likewise for the second qubit. Only after one qubit is measured, the other qubit can be assumed to be measured in a certain state. A pure entangled state cannot be represented by a direct products of two arbitrary states as

0 | Ψiab = | ψia | ψ ib , (2.21)

0 where | ψia and | ψ ib are the states deﬁned in the Hilbert space Ha and Hb respectively. More generally, including mixed states, an entangled quantum state is deﬁned as a state which cannot be represented by the probability sum of direct product of

i i density operatorsρ ˆ1 andρ ˆ2:

X X i i ρˆab = piρˆa ⊗ ρˆb, pi = 1 (2.22) i i 2.3. Quantum entanglement 25 which is called a separable state. The separable state can always be prepared in terms of local operations and classical communications (LOCC) between two separated parties, while an entangled state cannot be prepared from a separable state by LOCC. Note that entanglement decreases under LOCC and is always invariant under the local unitary transformations.

Entanglement of simple models such as bipartite or tripartite 2-dimensional systems is already well understood. The entanglement criterion in a simple model is clearly defined. A quantum state with density matrixρ ˆ is entangled if and only if its partial transpose has at least one negative eigenvalue [81]. This is called the Peres-Horodecki criterion [81, 82]. It was shown that this entanglement criterion is valid for 2 ⊗ 2 and 3 ⊗ 3 cases as a necessary and sufficient condition. However, it is not true for high-dimensional systems since there exist some entangled quantum states which can have positive partial transpose [82, 83]. Many theoretical proposals of entanglement criteria for various systems have been suggested and investigated [11], but we here will not go into the detail of them. Quantifying entanglement is one of the primal tasks for applications to quantum information processing. In all cases, maximally entangled states can provide ideal resources in protocols of quantum information processing. For example, a maximally entangled state is an ideal channel allowing a transfer of arbitrary quantum states with 100% fidelity in the quantum teleportation protocol [34]. Based on such a property entanglement can also be used for quantum cryptography. For example, the Ekert protocol of quantum key distribution (EK91) uses entangled pairs as the distribution channels [25], which can detect any eavesdropping attack using loophole free Bell tests. 26 Basic Concepts

Figure 2.3. (a) Genuine and (b) non-genuine 4-dimensional entanglement model.

2.3.2 Genuine entanglement

We introduce the genuine entanglement of high-dimensional or multipartite systems.

Genuine d-dimensional entanglement refers to a state which is not decomposable into any sub-dimensional states. Thus a genuine d-dimensional entangled state

ρˆd can not be represented by a direct sum of any lower dimensional states, i.e. L the density matrixρ ˆd cannot be written as a block-diagonal matrix,ρ ˆd 6= i Ciρˆi P where i Ci = 1. We note that even a non-genuine entangled state in a d × d- dimensional Hilbert space can be projected onto a maximally entangled state in a lower-dimensional Hilbert space [84]. For example, a maximally entangled state

0 0 | ψ4i = | 00i + | 11i + | 22i + | 33i and a mixed state ρ = | ψ2i hψ2 | ⊕ | ψ2i hψ2 |

0 where | ψ2i = | 00i + | 11i and | ψ2i = | 22i + | 33i (without normalisation factor), are diﬀerent entangled states in 4 × 4-dimensional Hilbert space (see Fig. 2.3), but both can be mapped onto a maximally entangled state in 2 × 2-dimensional Hilbert space. This shows that a projection of a system to a lower-dimensional model may not preserve the whole quantum nature of the system. In other words, one can not obtain unambiguous results in the case of investigating entanglement in lower- dimensional measurements (lower than the target degrees of freedom).

Similarly, genuine N-partite entanglement refers to the state in which none of the parties can be separated from any other party. Thus the genuine N-partite entangled stateρ ˆN can not be represented by a product state of any lower-partite 2.3. Quantum entanglement 27

P i i P states, i.e.ρ ˆN 6= i CiρˆN1 ⊗ ρˆN2 where N = N1 + N2 and i Ci = 1. For example, the Greenberger-Horne-Zeilinger (GHZ) state and the cluster state [85] are genuine multipartite entangled states. We note that the discrimination of genuine entanglement is essential to obtain proper characteristics of entanglement when increasing the size or dimensionality of physical systems. For example, one can investigate d- dimensional genuine entanglement as increasing the dimensionality, d, and compare its result with the case of N collective pairs of 2-dimensional entangled particles when d = 2N . Note that genuine d-dimensional and collective d-dimensional entanglement can play diﬀerent roles in quantum information processing.

2.3.3 Entanglement Witness

An entanglement witness (EW) is an observable which reacts diﬀerently to entangled and separable states, and thus it can be used to determine whether a state is entangled or not [86]. Its expectation value can show the diﬀerence between entangled and separable states. For example, let us consider a Hermitian operator Wˆ whose average expectation value by separable states is bounded by a maximal value

Wmax:

ˆ |hWi| ≤ Wmax. (2.23)

ˆ ψ If an expectation value of a quantum state ψ exceeds this bound |hWi | > Wmax, we can conclude that the quantum state ψ is entangled. Note that if there is a ˆ ˆ separable state satisfying |hWi| = Wmax, the witness operator W is called an optimal entanglement witness.

The role of entanglement witnesses is very crucial for the study of entanglement since it is known that for every entangled state there always exists a witness operator detecting it. Moreover, it can provide a useful tool for the implementation of 28 Basic Concepts

Figure 2.4. Schematic diagram for an entanglement witness (EW) and a Bell inequality (BI). EW detect entangled states including some states which do not violate the local realistic (LR) theories, while BI detect non-local states which violate the LR theories.

detecting entanglement. For the usage of entanglement witness, the ﬁrst step is to ﬁnd an appropriate Hermitian operator that discriminates separable and entangled states. This can be described geometrically as follows. Consider a geometrical space corresponding to the set of all possible density matrices. Then, the set of all separable states corresponds to a single convex space represented as the shaded region in Fig. 2.4. This is because all separable states satisfy the convexity condition i.e. a

AB AB AB AB AB linear combinationσ ˆ = pρˆ1 + (1 − p)ˆρ2 of two separable states,ρ ˆ1 andρ ˆ2 is also a separable state. An entanglement witness is a Hermitian operator which draws a boundary discriminating separable states and entangled states.

A Bell inequality can be seen as an entanglement witness since any violation of Bell inequalities guarantees that corresponding state is entangled. However, the assumption that all entangled states violate a Bell inequality is not true. This is because non-locality and entanglement are not the same features of quantum mechanics. For example, there can exist some entangled states which do not violate local-realistic (LR) theories [87]. Because of this fact, Bell inequalities are regarded as non-optimal entanglement witnesses. A schematic diagram for the diﬀerence between the optimal entanglement witness (EW) and Bell inequalities (BIs) is shown 2.4. Phase space representations 29 in Fig. 2.4. The details of the relation between entanglement witnesses and Bell inequalities can be found in Ref. [88].

Entanglement witnesses can be used for investigating entanglement in various physical systems. Recently, entanglement in macroscopic systems has been wit- nessed [23], which might open up the possibility to observe entanglement in various realistic physical systems around us. Moreover, based on witnessing entanglement we can obtain deeper insights into several phenomena arising in physical systems such as quantum phase transitions [24]. However, there exist several difficulties for practical applications of entanglement witnesses. Firstly, in spite of its clear defini- tion, a measurable entanglement witness is in general very hard to find, especially for higher-dimensional systems. Secondly, in most realistic cases experimental tests of entanglement witnesses suffer from imperfections of detectors which tend to neglect the effects of quantum correlations. In chapter 6 we will propose an entanglement witness which can detect continuous variable entanglement even using significantly inefficient detectors.

2.4 Phase space representations

2.4.1 Generalised quasi-probability functions

Phase space representations are a convenient tool for the analysis of continuous variable states as they provide insights into the boundaries between quantum and classical physics. Any quantum state can be fully characterised by the quasi-probability function deﬁned in phase space [35, 36].

The generalised quasi-probability functions can be written in terms of one real 30 Basic Concepts parameter, s, as [35, 36, 89]

2 W (α; s) = Tr[ˆρΠ(ˆ α; s)], (2.24) π(1 − s) where

µ ¶ X∞ s + 1 n Π(ˆ α; s) = | α, ni hα, n | , (2.25) s − 1 n=0 and | α, ni is the number state displaced by the complex variable α in phase space.

It is produced by applying the Glauber displacement operator Dˆ(α) to the number state |ni. We call W (α; s) the s-parameterised quasi-probability function which becomes the P-function, the Wigner-function, and the Q-function when setting s = 1, 0, −1 [89], respectively. For non-positive s, the s-parameterised quasi-probability function can be written as a convolution of the Wigner-function and a Gaussian weight

Z µ ¶ 2 2|α − β|2 W (α; s) = d2β W (β) exp − . (2.26) π|s| |s|

This can be identiﬁed with a smoothed Wigner-function aﬀected by noise which is modeled by Gaussian smoothing [90, 91, 92].

2.4.2 Bell inequalities in phase space

Banaszek et al. introduced Bell inequalities formulated by the Q- and Wigner functions [19]. The Q-function Bell inequality can be written in the Clauser-Horne (CH) version of Bell inequality [65] by

−1 ≤ π2[Q(α, β) + Q(α, β0) + Q(α0, β) − Q(α0, β0)] − π[Q(α) + Q(β)] ≤ 0, (2.27) 2.4. Phase space representations 31

BW-type Hr=0.2L HaL r=0.2 2 1.5 1 - + 0.5 Π2 1 W Β 0 4 + + 2 -0.5 0 -1 0 Β -2 -1.5 0 Α -2 2 -1.5-1-0.5 0 0.5 1 1.5 2 Α BW-type Hr=1L HbL r=1.0 2 1.5 1 0.5 - + Π2 1 W Β 0 4 + + 2 -0.5 0 -1 0 Β -2 -1.5 0 Α -2 2 -1.5-1-0.5 0 0.5 1 1.5 2 Α

Figure 2.5. Plot of the two mode Wigner function of the TMSS with diﬀerent squeezing parameters (a) r = 0.2 and (b) r = 1 for real value of α and β. The combination of points for maximal expectation value of the BW-type is given in the contour plots for each squeezing rate. 32 Basic Concepts where Q(α, β) = (1/π2)hα, β|ρˆ|α, βi is the two-mode Q-function for a quantum state ρˆ and Q(α) is the marginal distribution. Here α, α0, β, and β0 are the displacement parameters in phase space. The Q-function can be measured by the on-oﬀ photo detection method (i.e. the discrimination of zero versus non-zero photons) [91, 92].

Let us consider the Bell inequality formulated by the two mode Wigner function [19] in its generalised form proposed in [41]

π2 |W (α, β) + W (α, β0) + W (α0, β) − W (α0, β0)| ≤ 2, (2.28) 4 where α, α0, β, and β0 are locally independently chosen displacement parameters in phase space. The two mode Wigner function is used as a correlation function due to the fact that it is proportional to the expectation value of a two mode displaced parity operator. Note that the parity operator has two alternative outcome values, +1 or −1. Therefore, in this formalism, Eq. (2.28) can be seen as the CHSH type Bell inequality [8] satisfying the maximal expectation value 2 under the local- realistic theories. The maximal expectation value for the regularised EPR state, i.e. the two-mode squeezed vacuum states (TMSS), was shown to reach about 2.32 p at α = −α0 = β/2 = (ln 3)/16 cosh 2r → 0 and β = 0 where r is the squeezing parameter [19, 41]. The Wigner function for the two-mode squeezed vacuum states of r = 0.2 and r = 1 are shown in Fig. 2.5 for real α and β. The combination of points in phase space which leads to a maximal expectation value is also indicated in the ﬁgure. We will extend the Bell inequalities formulated in phase space formalism to a more general form and investigate its properties in chapter 5. Chapter 3

Generalised structure of Bell inequalities for arbitrary-dimensional systems

3.1 Introduction

Local-realistic theories impose constraints on any correlations obtained from measurement between two separated systems [1, 8, 37]. It was shown that these constraints, known as Bell inequalities, are incompatible with the quantitative predic- tions by quantum mechanics in case of entangled states. For example, the original Bell inequality is violated by a singlet state of two spin-1/2 particles [1]. The Clauser- Horne-Shimony-Holt (CHSH) inequality is another common form of Bell inequality, allowing more ﬂexibility in local measurement conﬁgurations [8]. These constraints are of great importance for understanding the conceptual features of quantum mechanics and draw the boundary between local-realistic and quantum correlations.

One may doubt if there is any well-deﬁned constraint for many high-dimensional subsystems which would eventually simulate a classical system as increasing its dimensionality to inﬁnity [37]. Therefore, constraints for more complex systems such

33 34 Generalised structure of Bell inequalities for arbitrary-dimensional systems as multi-partite or high-dimensional systems have been proposed and investigated intensively [16, 13, 93, 94, 33, 29, 95, 32, 96, 97, 14, 14, 98, 99, 80, 100].

For bipartite high-dimensional systems, Collins et al. suggested a local-realistic constraint, called CGLMP inequality [13]. It is violated by quantum mechanics and its characteristics of violation are consistent with the numerical results provided by Kaszlikowski et al. [93, 94]. Further, Masanes showed that the CGLMP inequality is tight [33], which implies that the inequality has no interior bias as a local-realistic constraint. However, Acin et al. found that the CGLMP inequality shows maximal violation by non-maximally entangled state [32]. Zohren and Gill found the similar results when they applied CGLMP inequality to inﬁnite dimensional systems [97]. Recently, Son et al. [14] suggested a generic Bell inequality and its variant for arbitrary high-dimensional systems. The variant will be called SLK inequality throughout this paper. They showed that the SLK inequality is maximally violated by maximally entangled state. Very recently, the CGLMP inequality was recasted in the structure of the SLK inequality by choosing appropriate coeﬃcients [14].

In this paper, we propose a generalised structure of Bell inequalities for bipartite arbitrary d-dimensional systems, which includes various types of Bell inequalities proposed previously. A Bell inequality in the given generalised structure can be represented either in the correlation function space or joint probability space. We show that a Bell inequality in one space can be mapped into the other space by Fourier transformation. The two types of high-dimensional Bell inequalities, CGLMP and SLK, are represented in terms of the generalised structure with appropriate coef- ﬁcients in both spaces (Sec. 5.2). We investigate the violation of Bell inequalities by quantum mechanics. The expectations of local-realistic theories and quantum mechanics are determined by the coeﬃcients of correlation functions or joint probabilities. The CGLMP inequality is maximally violated by non-maximally entangled state while the SLK is by maximally entangled state (Sec. 3.3). We also investigate 3.2. Generalised arbitrary dimensional Bell inequality 35 the tightness of Bell inequalities which represents whether they contain an interior bias or not at the boundary between local-realistic and quantum correlations. Then we show that the SLK is a non-tight Bell inequality while the CGLMP is tight (Sec. 3.4).

3.2 Generalised arbitrary dimensional Bell inequal-

ity

We generalise a Bell inequality for bipartite arbitrary d-dimensional systems. Sup- pose that each observer independently choose one of two observables denoted by A1 or A2 for Alice, and B1 or B2 for Bob. Here we associate a hermitian observables H to a unitary operator U by the simple correspondence, U = exp(iH), and call U a unitary observable [29, 98, 96, 14]. We note that unitary observable representation induces mathematical simpliﬁcations without altering physical results 1. Each outcome takes the value of an element in the set of order d, V = {1, ω, ..., ωd−1}, where ω = exp(2πi/d). The assumption of local-realistic theories implies that the outcomes of observables are predetermined before measurements and the role of the measurements is just to reveal the values. The values are determined only by local hidden variables λ, i.e., Aa(λ) and Bb(λ) for a, b = 1, 2.

We denote a correlation between speciﬁc measurements taken by two observers,

∗ as Aa(λ)Bb (λ). Based on the local hidden-variable description, the correlation function is the average over many trials of the experiment as

Z ∗ Cab = dλ ρ(λ)Aa(λ)Bb (λ), (3.1)

1Instead of the complex eigenvalues, one may consider real eigenvalues, but then one has to employ a rather complicated form of the correlation weights to obtain Bell inequalities equivalent to the ones derived in the present paper. 36 Generalised structure of Bell inequalities for arbitrary-dimensional systems where ρ(λ) is the statistical distribution of the hidden variables λ with the properties R of ρ(λ) ≥ 0 and dλρ(λ) = 1. The correlation function can be expanded in terms of joint probability functions over all possible outcome pairs (k, l) with complex-valued weight as

Xd−1 k−l Cab = ω P (Aa = k, Bb = l), (3.2) k,l=0

k−l where ω is called a correlation weight and P (Aa = k, Bb = l) is a joint probability of Alice and Bob obtaining outcomes ωk and ωl respectively. Here we use the powers k and l of the outcomes ωk and ωl for the arguments of the joint probability as there is one-to-one correspondence.

We assume in general a correlation weight µk,l to satisfy certain conditions [99].

C.1 - The correlation expectation vanishes for a bipartite system with a locally unpolarised subsystem:

X X µk,l = 0, ∀l and µk,l = 0. ∀k k l

C.2 - The correlation weight is unbiased over possible outcomes of each subsystem (translational symmetry within modulo d):

µk,l = µk+γ,l+γ, ∀γ

C.3 - The correlation weight is uniformly distributed modulo d:

|µk+1,l − µk,l| = |µk,l+1 − µk,l|. ∀k, l

The correlation weight in Eq. (3.2) ωk−l satisﬁes all the conditions, can be written 3.2. Generalised arbitrary dimensional Bell inequality 37

α P α as ω where α ≡ k − l ∈ {0, 1, ..., d − 1} and it obeys α ω = 0.

Let us now consider higher-order(n) correlations following also the local hidden- variable description. The n-th order correlation function averaged over many trials of the experiment corresponds to the n-th power of 1-st order correlation as

Z (n) ∗ n Cab = dλ ρ(λ)(Aa(λ)Bb (λ)) Xd−1 n(k−l) = ω P (Aa = k, Bb = l) k,l=0 Xd−1 nα . = ω P (Aa = Bb + α), (3.3) α=0 where the n-th order correlation weight ωnα also satisﬁes the above conditions, C.1, . C.2 and C.3, and P (Aa = Bb + α) is the joint probability of local measurement outcomes diﬀering by a positive residue α modulo d. The dot-equal implies that left- hand side is the same as the right-hand side modulo d, i.e. Aa ≡ Bb + α (mod d). Here we note that the higher-order correlations Eq. (3.3) show the periodicity of

(d+n) (n) Cab = Cab and they have the Fourier relation with the joint probabilities as given in Eq. (3.3).

We present a generalised Bell function for arbitrary d-dimensional system using higher-order correlation functions as

X Xd−1 (n) B = fab(n)Cab , (3.4) a,b n=0

where coeﬃcients fab(n) are functions of the correlation order n and the measurement conﬁgurations a, b. They determine the constraint of local-realistic theories with a certain upper bound and its violation by quantum mechanics will be investigated in Sec. 3.3. The zero-th order correlation has no meaning as it simply shift P the value of B by a constant and is chosen to vanish, i.e., a,b fab(0) = 0. The Bell 38 Generalised structure of Bell inequalities for arbitrary-dimensional systems function in Eq. (4.1) is rewritten in terms of the joint probabilities given in Eq. (3.3), as

d−1 X X . B = ²ab(α)P (Aa = Bb + α), (3.5) a,b α=0

. where ²ab(α) are coeﬃcients of the joint probabilities P (Aa = Bb + α).

We note that the coeﬃcients ²ab(α) are obtained by the Fourier transformation of fab(n) based on the kernel of a given correlation weight as

Xd−1 nα ²ab(α) = fab(n)ω , (3.6) n=0 1 Xd−1 f (n) = ² (α)ω−nα. (3.7) ab d ab α=0

It is remarkable that one can represent a given Bell function either in the correlation function space or joint probability space by using the Fourier transformation of the coeﬃcients between them. This is the generalisation of the Fourier transformation in 2-dimensional Bell inequalities provided by Werner et al. [80].

Different Bell inequalities can be represented by altering coefficients of the generalised structure, including previously proposed Bell inequalities in bipartite systems. In the case of d = 2, CHSH-type inequalities can be obtained with coefficients as

α f(1) = (1, 1, −1, 1) and ²ab(α) = fab(1)(−1) where α ∈ {0, 1}. For arbitrary d- dimensional systems, the two types of Bell inequalities, CGLMP and SLK, are represented in terms of the generalised structure with appropriate coeﬃcients obtained as follows.

CGLMP inequality - As it was originally proposed in terms of joint probabilities

[13], the Bell function of the CGLMP inequality is in the form of (3.5) and its 3.2. Generalised arbitrary dimensional Bell inequality 39 coeﬃcients are given as

2α 2(α −˙ 1) ² (α) = 1 − , ² (α) = −1 + , 11 d − 1 12 d − 1 2α 2α ² (α) = −1 + , ² (α) = 1 − , (3.8) 21 d − 1 22 d − 1 where the dot implies the positive residue modulo d. By using the inverse Fourier transformation in Eq. (3.7) the coeﬃcients for the correlation function representation are obtained as

µ ¶ 2 1 f (n 6= 0) = , 11 d − 1 1 − ω−n µ ¶ 2 1 f (n 6= 0) = , 12 d − 1 1 − ωn µ ¶ 2 1 f (n 6= 0) = , 21 d − 1 ω−n − 1 µ ¶ 2 1 f (n 6= 0) = , 22 d − 1 1 − ω−n

fab(n = 0) = 0 ∀a, b, (3.9) where the sum of the 0-th order coeﬃcients vanishes and does not aﬀect the characteristics of the Bell inequality.

SLK inequality - It was introduced in terms of correlation functions [14], and the coeﬃcients are given by

nδ (n−d)δ f11(n 6= 0) = (ω + ω )/4,

n(δ+η1) (n−d)(δ+η1) f12(n 6= 0) = (ω + ω )/4,

n(δ+η2) (n−d)(δ+η2) f21(n 6= 0) = (ω + ω )/4,

n(δ+η1+η2) (n−d)(δ+η1+η2) f22(n 6= 0) = (ω + ω )/4,

fab(n = 0) = 0 ∀a, b, (3.10) 40 Generalised structure of Bell inequalities for arbitrary-dimensional systems where δ is a real number, called a variant factor, and η1,2 ∈ {+1/2, −1/2}. By varying δ and η1,2, one can have many variants of SLK inequalities. For all the variants the coeﬃcients in the joint probability picture are obtained as

²11(α) = S(δ + α),

²12(α) = S(δ + η1 + α),

²21(α) = S(δ + η2 + α),

²22(α) = S(δ + η1 + η2 + α), (3.11) where

1 π S(x 6= 0) = (cot x sin 2πx − cos 2πx − 1), 4 d 1 S(x = 0) = (d − 1). (3.12) 2

We have shown that those two types of high-dimensional inequalities have different coefficients but the same generalised structure. In the frame work of the generalised structure we will now study how the coefficients determine the characteristics of Bell inequalities such as the degree of violation and tightness.

3.3 Violation by Quantum Mechanics

In order to see the violation of Bell inequalities by quantum mechanics we need to know the upper bound by local hidden variable theories. We note that a probabilistic expectation of a Bell function is given by the convex combination of all possible deterministic values of the Bell function and the local-realistic upper bound is decided . by the maximal deterministic value. Let αab = α such that P (Aa = Bb +α) = 1. The assumption of local-realistic theories implies that the values αab are predetermined. 3.3. Violation by Quantum Mechanics 41

For a Bell function in the form of Eq. (4.1) or (3.5), they obey the constraint,

. α11 + α22 = α12 + α21, (3.13)

because of the identity, A1 − B1 + A2 − B2 = A1 − B2 + A2 − B1. The local-realistic upper bound of the Bell function is therefore given by

X max . BLR = max[ ²ab(αab)|α11 + α22 = α12 + α21]. (3.14) αab a,b

The quantum expectation value for arbitrary quantum stateρ ˆ is written by

ˆ BQM(ˆρ) = Tr(Bρˆ) X Xd−1 ˆ(n) = fab(n)Tr(Cab ρˆ), (3.15) a,b n=0 where Bˆ is the Bell operator deﬁned by replacing the correlation function in Eq. (4.1) with correlation operator,

X ˆ(n) n(k−l) ˆ ˆ Cab = ω Pa ⊗ Pb (3.16) k,l

ˆ ˆ where Pa,Pb are projectors onto the measurement basis denoted by a, b. If an expec-

max tation value of any quantum state exceeds the local realistic bound BLR , i.e., the Bell inequality is violated by quantum mechanics, the composite system is entangled and shows nonlocal quantum correlations. The maximal quantum expectation

max is called quantum maximum BQM and corresponds to the maximal eigenvalue of the Bell operator. In the case of d = 2, with the coeﬃcients f(1) = (1, 1, −1, 1) we √ max can obtain the quantum maximum, BQM = 2 2, which is in agreement with the 42 Generalised structure of Bell inequalities for arbitrary-dimensional systems

Cirel’son bound [101].

In the presence of white noise, a maximally entangled state |ψmi becomes

1 ρˆ = p|ψ ihψ | + (1 − p) (3.17) m m d2 where p is the probability that the state is unaﬀected by noise. Then, the minimal

min max probability for the violation is p = BLR /BQM(|ψmi). We now investigate the violation of two types of Bell inequalities, CGLMP and SLK, and compare them as follows.

max CGLMP inequality - The local-realistic upper bound, BLR = 2, can be obtained as Eq. (4.2). The quantum expectation can also be obtained as Eq. (3.15) and it is consistent with the result in Ref. [13]. Acin et al. found, however, that the CGLMP inequality shows maximal violation for non-maximally entangled states [32]. For 3-

max dimensional system, the quantum maximum is BQM ' 2.9149 for the non-maximally entangled state,

1 √ (|00i + γ|11i + |22i), (3.18) n where γ ' 0.7923 and n = 2 + γ2. It is higher than the expectation by maximally entangled state, B(|ψmi) ' 2.8729. The expectation of the CGLMP is shown in Fig. 3.1 against the entanglement degree γ, once the local measurements are chosen such that they maximise the Bell function for the maximally entangled state. Further, we also note that the minimal violation probability(pmin) of the CGLMP decreases as the dimension d increases.

SLK inequality - Many variants of the SLK Bell inequality are obtained by varying δ and η1,2. All the variants of the SLK have the same quantum maximum d − 1

max for a maximally entangled state |ψmi, BQM = BQM(|ψmi), as we prove as follows. 3.3. Violation by Quantum Mechanics 43

3 2.2 HaL HbL 2.8 2 2.6 1.8

QM 2.4 QM 1.6 B B 2.2 1.4 2 1.2

0 0.5 1 1.5 2 0 0.5 1 1.5 2 Γ Γ

Figure 3.1. The expectation value of (a) the CGLMP and (b) the optimal √ SLK for d = 3 as varying the value γ for the quantum state, (1/ n)(|00i + γ|11i+|22i) where n = 2+γ2. The SLK takes the maximum 2 when the state is maximally entangled (γ = 1), whereas the CGLMP takes the maximum 2.9149 for a partially entangled state (γ ' 0.7923). The dashed lines indicate the local-realistic upper bounds.

The Bell operator of the SLK variants can be written as

1 Xd−1 Bˆ = α · β˜, (3.19) S 2 n=1

ˆ†n ˆ†n T ˜ ˆn ˆn T where α = (A1 , A2 ) and β = Uβ with β = (B1 , B2 ) and U is a 2 × 2 unitary matrix with elements,

nδ (n−d)δ U11 = (ω + ω )/2,

n(δ+η1) (n−d)(δ+η1) U12 = (ω + ω )/2,

n(δ+η2) (n−d)(δ+η2) U21 = (ω + ω )/2,

n(δ+η1+η2) (n−d)(δ+η1+η2) U22 = (ω + ω )/2, (3.20)

where η1,2 ∈ {1/2, −1/2}. 44 Generalised structure of Bell inequalities for arbitrary-dimensional systems

The expectation of the Bell operator is given by ¯ ¯ ¯ d−1 ¯ d−1 ¯ ¯ 1 ¯X ¯ 1 X ¯ ¯ ¯ hψ | α · β˜ | ψi¯ ≤ ¯hψ | α · β˜ | ψi¯ 2 ¯ ¯ 2 n=1 n=1 d−1 ³¯ ¯ ¯ ¯´ 1 X ¯ ¯ ¯ ¯ ≤ ¯hψ | α ⊗ β˜ | ψi¯ + ¯hψ | α ⊗ β˜ | ψi¯ 2 1 1 2 2 n=1 d−1 r X ¯ ¯2 ¯ ¯2 1 ¯ ˜ ¯ ¯ ˜ ¯ ≤ √ ¯hψ | α1 ⊗ β1 | ψi¯ + ¯hψ | α2 ⊗ β2 | ψi¯ 2 n=1 v d−1 u 2 X uX ¯ ¯2 1 t ¯ ˜ ¯ = √ ¯hψ | αi ⊗ βi | ψi¯ , (3.21) 2 n=1 i=1 where we consecutively used the triangle inequality and the arithmetic-geometric means inequality, 2|a||b| ≤ |a|2 + |b|2. Note that

2 2 X ¯ ¯2 X ³ ´ ³ ´ ¯ ˜ ¯ † ˜† ˜ ¯hψ | αi ⊗ βi | ψi¯ ≤ hψ | αi ⊗ βi αi ⊗ βi | ψi i=1 i=1 X2 ˜† ˜ = hψ | 1 ⊗ βi βi | ψi , (3.22) i=1 where we used that αi is unitary. Here the above inequality is obtained by reasoning that Qˆ ≡ 1 − | ψi hψ | is a positive operator as hφ | Qˆ | φi = 1 − | hφ |ψ| |i2 ≥ 0 for any | φi, and | hψ | Cˆ | ψi |2 = hψ | Cˆ† | ψi hψ | Cˆ | ψi = hψ | Cˆ†(1 − Qˆ)Cˆ | ψi = P ˆ† ˆ ˆ ˆ† ˆ ˆ ˜ ˜† ˜ hψ | C C | ψi − hψ | Q | ψi ≤ hψ | C C | ψi, where C ≡ αi ⊗ βi. Since i βi βi =

P P ∗ † P † P † jk i UijUikβj βk = jk δjkβj βk = i βi βi = 21, it is clear that

2 2 X ¯ ¯2 X ¯ ˜ ¯ † ¯hψ | αi ⊗ βi | ψi¯ ≤ hψ | 1 ⊗ βi βi | ψi = 2. (3.23) i=1 i=1

Hence the upper bound for all variants of the SLK is

¯ ¯ ¯ ˆ ¯ ¯hψ | BS | ψi¯ ≤ d − 1. (3.24) 3.3. Violation by Quantum Mechanics 45

Since all SLK Bell operators have the eigenvalue d − 1 for maximally entangled states, the upper bound is reachable. Therefore, d − 1 is the quantum maximum for all variants of the SLK inequality.

On the other hand, the local-realistic upper bounds depend on the variants. The

max local-realistic upper bound BLR is a function of the variant factor δ. It shows a

max max periodicity, BLR (δ) = BLR (δ + 1/2), and without loss of generality it suﬃces to consider 0 ≤ δ < 1/2. If δ = 0, the local-realistic upper bound is the same as the quantum maximum d − 1, and thus the corresponding Bell inequality is not violated by quantum mechanics. When δ = 1/4, we have the lowest local-realistic upper bound as

max 1 π 3π min[BLR (δ)] = (3 cot − cot ) − 1, (3.25) δ 4 4d 4d

max and for other cases the bound values are symmetric at δ = 1/4, i.e., BLR (1/4 +

max ²) = BLR (1/4 − ²) for 0 < ² ≤ 1/4. Therefore, we will call the variant of δ = 1/4, which gives the maximal diﬀerence between quantum maximum and local- realistic upper bound, as the optimal SLK inequality and use it for comparing to the CGLMP. In Fig. 3.1, we present the quantum expectation of the SLK for 3- dimensional systems against the degree γ, where the local measurements are chosen such that they maximise the Bell function for the maximally entangled state. Note that the SLK inequality shows the maximal violation by maximally entangled states and the minimal violation probability pmin increases as the dimension d increases.

By investigating the violation of two inequalities, CGLMP and SLK, based on the generalised structure of Bell inequalities, we showed that those two types have very diﬀerent characteristics. The SLK inequality is maximally violated by maximally entangled states as being consistent with our intuition whereas the CGLMP is maximally violated by non-maximally entangled states. We remark that the coef- 46 Generalised structure of Bell inequalities for arbitrary-dimensional systems

ﬁcients of the given generalised structure determine the characteristics of quantum violations.

3.4 Tightness of Bell inequalities

The set of possible outcomes for a given measurement setting forms a convex polytope in the joint probability space or alternatively in the correlation function space [100, 80, 33, 95]. A convex polytope is deﬁned either as a convex set of points in space or as a intersection of half-spaces [102]. The extreme points of the polytope are called generators. For a h-dimensional polytope, h − 1-dimensional faces of the polytope are called facets. Each generator of the polytope represents the predetermined measurement outcome called local-realistic conﬁguration. All interior points of the polytope are given by the convex combination of generators and they represent the accessible region of local-realistic theories associated with the probabilistic expectations of measurement outcomes. Therefore, every facet of the polytope is a boundary of halfspace characterised by a linear inequality, which we call tight Bell inequality. There are non-tight Bell inequalities which contain the polytope in its halfspace. As the non-tight Bell inequality has interior bias at the boundary between local-realistic and quantum correlations, one might say it to be the worse detector of the nonlocal test [80, 33, 95].

The Bell polytope is lying in the joint probability space of dimension h, the degrees of freedom for the measurement raw data. For a bipartite system, two observables per party and d-dimensional outcomes, the joint probability, P (Aa =

2 k, Bb = l) where k, l = 0, 1, ..., d − 1 and a, b = 1, 2, can be arranged in a 4d - dimensional vector space. However, the joint probabilities have two constraints, i.e., normalisation and no-signaling constraints, which reduce dimension by 4d [33]. The generators in the h-dimensional space can be written, following the notations in 3.4. Tightness of Bell inequalities 47

Ref. [33], as

G = |A1,B1i ⊕ |A1,B2i ⊕ |A2,B1i ⊕ |A2,B2i (3.26)

where |ni stands for |n mod di and is the d-dimensional vector with a 1 in the n-th component and 0s in the rest. In order to examine the tightness of a given generalised Bell inequality, in general one considers the following conditions that every tight Bell inequality fulﬁlls [33].

(T.1) All the generators must belong to the half space of a given facet.

(T.2) Among the generators on the facet, there must be h which are linearly independent. Here the dimension of polytope in joint probability space is given as h = 4d(d − 1) due to the no-signaling and normalisation conditions of joint probabilities [33]. Note that hyperplane of dimension h−1 is completely char- acterized by h independent vectors. Therefore, a facet of polytope requires h independent generators. More detail proof of this condition can be found in Ref. [33].

First, it is straightforward that all generators fulﬁll the inequality as the Bell inequality derived to do. As the local-realistic upper bound is the maximum among expectation values of local-realistic conﬁgurations, all generators are located below

max the local-realistic upper bound, BLR . Thus the ﬁrst condition T.1 is fulﬁlled. Sec- ond, we examine whether there are h linearly independent generators which give the

max value of the local-realistic bound, BLR . By the predetermined local-realistic values

αab, the generators (3.26) become

|A, A − α11i ⊕ |A, A − α12i ⊕ |A − α12 + α22,A − α11i

⊕|A − α11 + α21,A − α12i, (3.27)

where A ∈ {0, 1, ..., d − 1} and the number of linearly independent generators is 48 Generalised structure of Bell inequalities for arbitrary-dimensional systems determined by the number of sets {αab} that give the local-realistic upper bound. If the number of linear independent generators is not smaller than h = 4d(d − 1), the corresponding Bell inequality is tight.

CGLMP inequality - For the CGLMP inequality the local-realistic upper bound ˙ is achieved when α11 + α22 − (α12 − 1) − α21 + d − 1 = 0. The condition allows the suﬃcient number of linearly independent generators and the CGLMP inequality is tight [33].

SLK inequality - For the optimal SLK inequality, the upper bound is obtained in the case that {α11, α12, α21, α22} is equal to one of four sets; {0, 0, d − 1, d − 1}, {0, 0, 0, 0},{0, 1, d − 1, 0},{d − 1, 0, d − 1, 0}. Thus there are four types of generators as

|A, Ai ⊕ |A, Ai ⊕ |A − 1,Ai ⊕ |A − 1,Ai

|A, Ai ⊕ |A, Ai ⊕ |A, Ai ⊕ |A, Ai

|A, Ai ⊕ |A, A − 1i ⊕ |A − 1,Ai ⊕ |A − 1,A − 1i

|A, A + 1i ⊕ |A, Ai ⊕ |A, A + 1i ⊕ |A, Ai (3.28) which are linearly independent with A ∈ {0, 1, ..., d − 1}. There are only 4d linearly independent generators which are smaller than h = 4d(d−1), the tightness condition T.2. Thus the optimal SLK inequality is non-tight. On the other hand, the SLK inequality for δ = 0 is tight but it is not violated by quantum mechanics.

3.5 Remarks

In summary, we presented a generalised structure of the Bell inequalities for arbitrary d-dimensional bipartite systems by considering the correlation function specified by a well-defined complex-valued correlation weight. The coefficients of a given Bell 3.5. Remarks 49 inequality in the correlation function space and the joint probability space were shown to be in the Fourier relation. Two known types of high-dimensional Bell inequalities, CGLMP and SLK, were shown to have the generalised structure in common and we found their coefficients in both spaces.

Based on the generalised structure, we investigated characteristics of the Bell inequalities such as quantum violation and tightness. We found that the CGLMP and SLK inequalities show different characteristics. For instance, the SLK inequality is maximally violated by maximally entangled states, which is consistent with the intuition “the larger entanglement, the stronger violation against local-realistic theories,” whereas the CGLMP inequality is maximally violated by the non-maximally entangled state as previously shown by Acin et al. [32]. On the other hand, in ana- lyzing the tightness of the inequalities, the CGLMP is tight but the SLK inequality is found to be non-tight for δ 6= 0, implying that the SLK inequality has interior bias at the boundary between local-realistic and quantum correlations. The correlation coefficients of Bell inequalities play a crucial role in determin- ing their characteristics of quantum violation and tightness. This implies that by altering the coefficients in the generalised structure one can construct other Bell inequalities. The present work opens a possibility of finding a new Bell inequality that fulfills both conditions of the maximal violation by maximal entanglement and the tightness.

Chapter 4

Maximal violation of tight Bell inequalities for maximal entanglement

4.1 Introduction

The incompatibility of quantum non-locality with local-realistic theories is one of the most remarkable aspects of quantum theory. Local-realistic theories impose constraints on the correlations between measurement outcomes on two separated systems which are described by Bell inequalities (BIs) [1]. It was shown that Bell inequalities are violated by quantum mechanics in the case of entangled states. Therefore Bell inequalities are of great importance for understanding the conceptual foundations of quantum theory and also for investigating quantum entanglement. Since the ﬁrst discussion of quantum non-locality by Einstein-Podolski-Rosen (EPR) a great amount of relevant work has been done and numerous versions of Bell inequalities have been proposed [1, 8, 13, 32, 14, 17, 33, 103, 104, 41, 38, 19, 39, 20].

For bipartite 2-dimensional systems the CHSH Bell-type inequality [8] has the desirable property of only being maximally violated for a maximally entangled state. The CHSH inequality divides the space of correlations between measurement outcomes by deﬁning a hyperplane. Since a facet of the polytope deﬁning the region of

51 52 Maximal violation of tight Bell inequalities for maximal entanglement local-realistic correlations lies in this hyperplane the CHSH inequality is tight. This means that any violation of local-realistic theories occurring on this particular facet is indicated by the CHSH inequality [33]. Tightness is a desirable property since only sets of tight Bell inequalities can provide necessary and suﬃcient conditions for the detection of pure state entanglement. There are still many open questions regarding the generalisation of Bell inequalities to complex quantum systems [15]. For example, Bell inequalities for bipartite high-dimensional systems as e.g. that proposed by Collins et al. [13] are either not maximally violated by maximal entanglement [32] or as in the case of Son et al. [14] were shown to be non-tight [17].

In the case of continuous variable systems there is so far no known Bell inequality formulated in phase space which is maximally violated by the EPR state – the maximally entangled state associated with position and momentum [20]. Although Banaszek and Wodkievicz (BW) showed how to demonstrate non-locality in phase space [38, 19, 39] their Bell inequality is not maximally violated by the EPR state

[41]. Another approach using pseudospin operators was shown to yield maximal violation for the EPR state [104]. However, ﬁnding measurable local observables to realise this approach is challenging. Due to the lack of any known Bell inequality providing answers to these questions we still have no clear understanding of nonlocal properties of high-dimensional systems and their relation to quantum entanglement.

In this paper we present a Bell inequality for even d-dimensional bipartite quantum systems which, in contrast to previously known Bell inequalities, fulﬁlls the two desirable properties of being tight and being maximally violated by maximally entangled states. These properties are essential to investigate quantum non-locality appropriately and for consistency with the 2-dimensional case. We call Bell inequalities fulﬁlling these properties optimal Bell inequalities throughout this paper. Then we extend optimal Bell inequalities to continuous variable systems and demonstrate strong violations for properly chosen local measurements. 4.2. Optimal Bell inequalities 53 4.2 Optimal Bell inequalities

We begin by brieﬂy introducing the generalised formalism for deriving Bell inequalities for arbitrary d-dimensional bipartite systems [17]. Suppose that two parties, ˆ ˆ Alice and Bob, independently choose one of two observables A1 or A2 for Alice, and ˆ ˆ ˆ B1 or B2 for Bob. Possible measurement outcomes are denoted by ka for Aa and la ˆ for Bb with a, b = 1, 2, where ka, lb ∈ V ≡ {0, 1, ..., d − 1}. A general Bell function is then written as [17]

X2 Xd−1 B = ²ab(ka, lb)Pab(ka, lb), (4.1)

a,b=1 ka,lb=0

where Pab(ka, lb) is the joint probability for outcomes ka and lb, and ²ab(ka, lb) are their weighting coeﬃcients (here assumed to be real). For local-realistic (LR) systems each probabilistic expectation of B is a convex combination of all possible deterministic values. It can thus not exceed the maximal deterministic expectation value given by

½ X2 ¾ max BLR = max ²ab(ka, lb) , (4.2) C a,b=1

where C ≡ {(k1, k2, l1, l2)|k1, k2, l1, l2 ∈ V } is the set of all possible outcome con- ﬁgurations. A quantum state violates local realism if its expectation value exceeds

max the bound BLR . The flexibility in choosing the coefficients ²ab(ka, lb) allows the derivation of all previously known Bell inequalities [17] from Eq. (4.1), e.g. those proposed by Collins et al. [13] and by Son et al. [14]. Moreover, we can construct new Bell inequalities by properly choosing coefficients ²ab(ka, lb). Our aim is to find optimal Bell inequalities which fulfil the following conditions:

(C1) - The Bell inequality is tight i.e. it deﬁnes a facet of the polytope separating 54 Maximal violation of tight Bell inequalities for maximal entanglement

local-realistic from non-local quantum regions in correlation or joint probability space.

(C2) - The Bell inequality is maximally violated by a maximally entangled state. For each bipartite d-dimensional maximally entangled state there exists a basis √ max Pd−1 | ji with j = 0, ··· , d − 1 in which this state reads | ψd i = j=0 | jji / d.

As a general method, one could choose the coefficients ²ab(ka, lb) freely and examine whether the resulting Bell inequality satisfies the conditions (C1) and (C2). Here we instead propose a method which restricts this choice and is guaranteed to give tight Bell inequalities. We assume that the coefficients are products of arbitrary binning functions defined by each party as   +1 if outcome k ∈ R, ζ (k) = (4.3) R  −1 otherwise, where R is an arbitrarily chosen subset of all possible outcomes, i.e. R ⊂ V . The coefficients are then given by

²11 = ζR1 (k1)ζS1 (l1), ²12 = ζR1 (k1)ζS2 (l2),

²21 = ζR2 (k2)ζS1 (l1), ²22 = −ζR2 (k2)ζS2 (l2), (4.4)

ˆ ˆ where Ra and Sb are subsets of the outcomes of Aa and Bb, respectively. From

max Eq. (4.2) we ﬁnd the local-realistic upper bound BLR = 2.

We ﬁrst show that any Bell inequality derived by this method is tight. The extremal points of the polytope separating local-realistic and non-local quantum mechanical correlations are associated with all deterministic conﬁgurations C. They 4.2. Optimal Bell inequalities 55 are described by 4d2 dimensional linearly independent vectors

G = |k1, l1i ⊕ |k1, l2i ⊕ |k2, l1i ⊕ |k2, l2i (4.5) where |ni stands for |n mod di and is the d-dimensional vector with a 1 in the n-th component and 0s in the rest. The interior points of the polytope are given by convex combinations of these extremal points and represent the region accessible to local-realistic theories. We now only consider extremal points associated

max with configurations giving the maximal local-realistic value BLR and denote their number by M. For a polytope defined in 4d2 dimensions at least 4d(d − 1) linearly independent vectors are required to define a facet. Therefore, if M ≥ 4d(d − 1) the

max extremal points yielding BLR deﬁne a facet of the polytope distinguishing local- realistic from non-local quantum mechanical correlations [33]. We assume the number of elements in the sets R1, R2, S1, S2 to be n1, n2, m1, m2, respectively, where

max 0 ≤ n1, n2, m1, m2 ≤ d − 1. We then count the number of conﬁguration giving BLR and ﬁnd

2 2 M = d (d − d(n1 + m1) + n1(m1 + m2) + n2(m1 − m2)) ≥ 4d(d − 1). (4.6)

Therefore all Bell inequalities obtained by this method are tight, i.e. they satisfy condition (C1). Note that any loss of elements in binned subsets may cause them to become non-tight.

We now discuss the maximal violation condition (C2) by considering three different tight Bell inequalities obtained via the above method. The corresponding choices of the coeﬃcients ²ab for d = 8 are schematically shown in Fig. 4.1(a):

(T1) This is the sharp binning type which can be realised if all outcomes are iden-

tiﬁable with perfect measurement resolution. The elements of the subsets are 56 Maximal violation of tight Bell inequalities for maximal entanglement

(a) (T1) (T2) (T3)

k 3 (b) ìæ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æææææææææææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ T1 æ T2 à ìæà 2.5 T3 ì QM

B à à à à à à à à ìà à à à à à àà àà àà ààà ààà àààà à à àà àà àà à à à à à ìà à à à ìì ìì ìììì 2à ììììììììììììììììììììììììììììììììììììì

2 25 50 d

Figure 4.1. (a) The coeﬃcient distributions of Bell inequalities (T1), (T2), and (T3) for d = 8 are shown with weighting +1 (white for ²11,²12,²21 and grey for ²22) and -1 (grey for ²11,²12,²21 and white for ²22) in outcome space. (b) Quantum expectation values BQM of (T1), (T2), and (T3) are√ plotted. As d increases, the expectation value of (T1)√ reaches the bound 2 2 (solid line), while that of (T2) approaches 2.31 < 2 2 and that of (T3) decreases below the local-realistic upper bound 2 (dashed line). 4.2. Optimal Bell inequalities 57

given as the even-numbers, i.e. R1 = R2 = S1 = S2 = {0, 2, 4, ...} so that the

coeﬃcients ²ab have an alternating weight +1 or −1 when an outcome changes by one.

(T2) This is associated with unsharp binning resolution and can be used to model

imperfect measurement resolution. The subsets are chosen as R1 = R2 =

S1 = S2 = {∀k|k ≡ 0, 1(mod 4)} where k ≡ 0, 1(mod 4) indicates that k is

congruent to 0 or 1 modulo 4. The coeﬃcients ²ab alternate between +1 and −1 for every 2 outcomes.

(T3) The measurement results are classiﬁed into two divided regions by the mean outcome [d/2], where [x] denotes the integer part of x. The subsets are cho-

sen as R1 = R2 = S1 = S2 = {∀k|0 ≤ k < [d/2]}. These three types of binning correspond to diﬀerent capabilities in carrying out measurements on d-dimensional systems. Their properties will yield useful insights for testing

Bell inequalities in high dimensional and continuous variable systems.

We examine quantum violations of (T1), (T2) and (T3) by the maximally en-

max ˆ ˆ tangled state | ψd i with increasing dimension d. The measurements Aa and Bb are performed in the bases

ω = exp(2πi/d), and αa and βb are phase factors diﬀerentiating the observables of ˆ ˆ each party Aa and Bb, respectively. The expectation value of the Bell function is 58 Maximal violation of tight Bell inequalities for maximal entanglement then given by

X2 Xd−1 ²ab(k, l) BQM = 3 π . (4.8) 2d sin [ (k + l + αa + βb)] a,b=1 k,l=0 d

As shown in Fig. 4.1(b), the expectation values of (T1) for even-dimensions are √ √ 2 2, and those for odd-dimensions tend towards 2 2 with increasing d. This is the upper bound for quantum mechanical correlations which we show by deﬁning a Bell operator as

X X ˆ B = ²ab(k, l) | a, ki ha, k | ⊗ | b, li hb, l | . (4.9) a,b k,l

From Eq. (4.4),

ˆ2 ˆ ˆ ˆ ˆ B = 41d ⊗ 1d + [P1, P2] ⊗ [Q2, Q1] (4.10)

ˆ P ˆ P where Pa = k ζRa (k) | a, ki ha, k |, Qb = l ζSb (l) | b, li hb, l | and 1d is the d- ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ dimensional identity operator. Since k[P1, P2]k ≤ kP1P2k+kP2P1k ≤ 2kP1kkP2k = 2 ˆ ˆ and likewise for k[Q2, Q1]k where k·k indicates the supremum norm, we ﬁnally obtain

ˆ2 ˆ ˆ ˆ ˆ kB k = k41d ⊗ 1d + [P1, P2] ⊗ [Q2, Q1]k ˆ ˆ ˆ ˆ = 4 + k[P1, P2]kk[Q2, Q1]k ≤ 8, (4.11)

√ or kBkˆ ≤ 2 2.

We calculate the quantum mechanical expectation value of Bˆ for Bell inequality

k+l k+l (T1) by writing the coeﬃcients as ²11 = ²12 = ²21 = (−1) and ²22 = −(−1) .

Pd−1 k+l 3 π For even d we use k,l=0(−1) /2d sin [ d (k + l + αa + βb)] = cos π(αa + βb) and 4.2. Optimal Bell inequalities 59

ﬁnd the expectation value

BQM = cos π(α1 + β1) + cos π(α1 + β2)

+ cos π(α2 + β1) − cos π(α2 + β2). (4.12)

This expression also holds approximately for suﬃciently large odd d. Thus we √ obtain BQM = 2 2, i.e. the maximal quantum upper bound, for α1 = 0, α2 = 1/2,

β1 = −1/4, and β2 = 1/4. Figure 4.1(b) also shows the maximal expectation values √ of (T2) which are smaller than 2 2 and approach ≈ 2.31 with increasing d. The maximal expectation values of (T3) decrease below the local-realistic upper bound 2 with increasing d.

These results show that (T1) is an optimal Bell inequality for even d which satisﬁes conditions (C1) and (C2). The optimal correlation operator is then written P ˆ ˆ ˆ ˆ d−1 k as Eab = Πa⊗Πb. with the local measurement Πa = k=0(−1) | a, ki ha, k |. Finally, we obtain the optimal Bell inequality

B = E11 + E12 + E21 − E22 ≤ 2, (4.13)

P ˆ k+l where Eab = hEabi = k,l(−1) Pab(k, l) is the correlation function. Note that for d = 2 Eq. (4.13) is equivalent to the CHSH inequality [8]. We have thus shown that the perfect sharp binning of arbitrary even dimensional outcomes (T1) provides an optimal Bell inequality, while the other binning methods (T2) and (T3) tend to neglect quantum properties and do not show maximal violation for maximally entangled states. 60 Maximal violation of tight Bell inequalities for maximal entanglement 4.3 Extension to continuous variable systems

We extend the optimal Bell inequality (T1) to a continuous variable system and calculate its violation by a two-mode squeezed state (TMSS). This state can, for instance, be realised by non-degenerate optical parametric ampliﬁers [103] in photonic

P∞ n systems. It is written as | TMSSi = sechr n=0 tanh r | n, ni where r > 0 is the squeezing parameter and | ni are the number states of each mode. In the inﬁnite squeezing limit r → ∞, this becomes the normalised EPR state [38, 19, 39].

When directly following the procedure of the ﬁnite dimensional case two problems arise: First, we obtain the local measurement basis by applying the quantum Fourier transformation to | ni. This is equivalent to the phase states | θi =

√ P∞ (1/ 2π) n=0 exp (inθ) | ni which are not orthogonal and not eigenstates of any hermitian observable. Therefore no precise phase measurement can be carried out. Second, a naive extension of the sharp binning method to the continuous case is impossible. Note that any coarse-grained measurement tends to lose quantum properties [105] and lead to non-tight Bell inequality tests. From the above results for unsharp and regional binning we also do not expect strong violations by these methods for the continuous variable system.

Let us consider the Pegg-Barnett phase state formalism [106]. We approximate the quantum phase by an orthonormal set of phase states in a q + 1-dimensional truncated space

q 1 X | θ, ki = √ exp (inθ ) | ni (4.14) q + 1 k n=0

where θk = θ + 2πk/(q + 1) and k = 0, 1, ..., q. Note that q is a cutoﬀ parameter (assumed here to be an odd number) and in the limit q → ∞ there exists a θk arbitrarily close to any given continuous phase. The correlation operator 4.3. Extension to continuous variable systems 61 can then be written as Eˆ(θ, φ) = Π(ˆ θ) ⊗ Π(ˆ φ) using the phase parity operator P ˆ q k Π(θ) = k=0(−1) | θ, ki hθ, k |. We consider a truncated TMSS

Xq sechr n | ψqi = p tanh r | n, ni (4.15) 2q+2 1 − tanh r n=0 which tends to the q + 1-dimensional maximally entangled state for r → ∞ and to the TMSS for an inﬁnite cutoﬀ, q → ∞. The preparation of this state can for instance be achieved by the optical state truncation method [107, 108].

The expectation value of the Bell operator is given by

ˆ ˆ 0 ˆ 0 ˆ 0 0 BQM = hψq | E(θ, φ) + E(θ, φ ) + E(θ , φ) − E(θ , φ ) | ψqi q+1 √ tanh 2 r = 4 2 , (4.16) 1 + tanhq+1 r when θ = 0, θ0 = π/(q + 1), φ = −π/(2q + 2), and φ0 = π/(2q + 2). Fig. 4.2(a) shows its monotonic increase against the squeezing rate r for different cutoff parameters q. For any finite q and δ > 0 there exists a squeezing parameter r above which √ BQM ≥ 2 2 − δ. The required squeezing for this violation is

1 r ≥ ln[(1 + f(q, δ))/(1 − f(q, δ))], (4.17) 2

√ p √ √ where f(q, δ) = [(2 2 − 4 2δ − δ2)/(2 2 − δ)]2/(q+1). The shaded region in

Fig. 4.2(b) indicates the values of r for which the Bell inequality is violated BQM ≥ √ 2 and a violation better than BQM ≥ 2 2 − δ occurs for values of r above the corresponding curves for diﬀerent δ. Violations arbitrarily close to the maximum √ value 2 2 can thus be achieved by suﬃciently strongly squeezed states for any

ﬁnite value of q with r → ∞ corresponding to the EPR state. Remarkably, this is in contrast to previous types of Bell inequalities which were not able to get arbitrarily 62 Maximal violation of tight Bell inequalities for maximal entanglement

3 HaL 5 HbL ∆=0.0001 4 ∆=0.001 2 ∆=0.01 r QM 3 B 1 2

1 0 1 1 49 99 tanh HrL q

Figure 4.2. (a) Expectation values of the Bell operator for truncated TMSS with cutoﬀ parameters q = 1 (solid), q = 9 (dashed), and q = 99 (dotted). (b) The shaded region indicates the values of√r for which the Bell inequality is violated; a violation better than BQM ≥ 2 2 − δ occurs above the curves shown for δ = 0.01 (dotted), δ = 0.001 (dashed), and δ = 0.0001 (solid).

close to this bound for the EPR state. However, we should note that for large q one here again faces diﬃculties in performing precise measurements due to the indistinguishability of two local measurements as π/(q + 1) → 0 for large q.

Finally, we discuss the relation of our optimal Bell inequality with the BW inequality proposed in [38, 19, 39]. There, local measurements are performed in the basis obtained by applying a Glauber displacement operator Dˆ(α) on the number states | ni. The measurement basis is written as | α, ni = Dˆ(α) | ni with α an arbitrary complex number. The displaced number operator is deﬁned as

ˆ ˆ † nˆα ≡ D(α)ˆnD (α). Sincen ˆα | α, ni = n | α, ni, the correlation operator is given P ˆ ˆ ˆ ˆ ∞ n by E(α, β) = Π(α) ⊗ Π(β), where Π(α) = n=0(−1) | α, ni hα, n | is the displaced parity operator. Using this notation the BW inequality becomes equivalent to

Eq. (4.13), which shows that it is a tight Bell inequality for continuous variable systems. However, the maximal expectation value of the BW inequality was shown √ to be 2.32 < 2 2 [41], while our type of Bell inequality asymptotically reaches the 4.4. Conclusions 63 √ bound 2 2. This shows that the optimal measurement bases for this non-locality test are obtained by a quantum Fourier transformation on the standard bases [99], i.e. each of them is mutually unbiased to the standard basis. This may also provide a useful insight about the optimality of measuring in mutually unbiased bases for cases with more than two local measurements [109].

4.4 Conclusions

We derived, for the ﬁrst time, a Bell inequality in even d-dimensional bipartite systems which is maximally violated by maximal entanglement and is also tight. These are desirable properties for Bell inequalities in high-dimensional systems [15, 17]. Our Bell inequality is found by perfectly sharp binning of the local measurement outcomes. It can be used for testing quantum non-locality for high dimensional systems, for instance it coincides with the result for heteronuclear molecules by Milman et al. [22]. Furthermore, we extended our studies to continuous variable systems and demonstrated strong violations asymptotically reaching the maximal √ bound 2 2 for truncated TMSSs by parity measurements in the Pegg-Barnett phase basis. This provides a theoretical answer to the question of how maximal violations of Bell inequalities can be demonstrated for the EPR states in phase space formalism [20]. In the future we will investigate the susceptibility of violations of our Bell inequalities to measurement imperfections. In this context it will also be valuable to search for additional optimal Bell inequalities comparing their properties and extending optimal Bell inequalities to multipartite systems.

Chapter 5

Testing quantum non-locality by generalised quasi-probability functions

5.1 Introduction

Ever since the famous arguments of Einstein-Podolski-Rosen (EPR) [5], quantum non-locality has been a central issue for understanding the conceptual foundations of quantum mechanics. Quantum non-locality can be demonstrated by the violation of Bell inequalities (BIs) [1] which are obeyed by local-realistic (LR) theories. Realisations of Bell inequality tests are thus of great importance in testing the validity of quantum theories against local-realistic theories. In addition, Bell inequality tests play a practical role in the detection of entanglement which is one of the main resources for quantum information processing. Bell inequality tests for 2-dimensional systems have already been realised [9, 10], while Bell inequality tests in higher-dimensional and continuous variable systems remain an active area of research [15, 20]. Phase space representations are a convenient tool for the analysis of continuous variable states as they provide insights into the boundaries between quantum and

65 66 Testing quantum non-locality by generalised quasi-probability functions classical physics. Any quantum stateρ ˆ can be fully characterised by the quasi- probability function defined in phase space [35, 36]. In contrast to the probability functions in classical phase space the quasi-probability function is not always positive. For example, the Wigner-function of the single photon state has negative values in certain regions of phase space [110]. Since the negativity of the quasi- probability function inevitably reflects a non-classical feature of quantum states, the relation between negativity of quasi-probabilities and quantum non-locality has been investigated [37, 38]. Bell argued [37] that the original EPR state will not exhibit non-locality since its Wigner-function is positive everywhere and hence serves as a classical probability distribution for hidden variables. On the other hand, Ba- naszek and Wódkiewicz(BW) showed how to demonstrate quantum non-locality using the Q- and Wigner-functions [38, 19, 39]. They suggested two distinct types of Bell inequalities, one of which is formulated via the Q-function and referred to in this paper as the BW-Q inequality while the other is formulated using the Wigner-function and is referred to as the BW-W inequality. Remarkably, the BW- W inequality was shown to be violated by the EPR state [38, 39]. This indicates that there is no direct relation between the negativity of the Wigner-function and non-locality.

Quasi-probability functions can be parameterised by one real parameter s [35, 36, 89]

2 W (α; s) = Tr[ˆρΠ(ˆ α; s)], (5.1) π(1 − s)

P ˆ ∞ n where Π(α; s) = n=0((s + 1)/(s − 1)) | α, ni hα, n |, and | α, ni is the number state displaced by the complex variable α in phase space. It is produced by applying the Glauber displacement operator Dˆ(α) to the number state |ni. We call W (α; s) the s-parameterised quasi-probability function which becomes the P-function, the 5.1. Introduction 67

Wigner-function, and the Q-function when setting s = 1, 0, −1 [89], respectively. For non-positive s the function W (α; s) can be written as a convolution of the Wigner-function and a Gaussian weight

Z µ ¶ 2 2|α − β|2 W (α; s) = d2β W (β) exp − . (5.2) π|s| |s|

This can be identified with a smoothed Wigner-function affected by noise which is modeled by Gaussian smoothing [90, 91, 92]. Therefore decreasing s reduces the negativity of the Wigner function and is thus often considered to be a loss of quantumness. For example, the Q-function (s = −1), which is positive everywhere in phase space, can be identified with the Wigner function smoothed over the area of measurement uncertainty.

The purpose of this paper is to propose a method for testing quantum non- locality using the s-parameterised quasi-probability function. We will firstly formulate a generalised Bell inequality in terms of the s-parameterised quasi-probability function in Sec. 5.2. This will lead us to a s-parameterised Bell inequality which includes the BW-Q and the BW-W inequalities as limiting cases. We will then present a measurement scheme to test Bell inequalities using imperfect detectors in Sec. 5.3. The measured Bell expectation value can be written as a function of the parameter s and the overall detector efficiency η. In Sec. 5.4 violations of Bell inequalities will be demonstrated for single-photon entangled states and in Sec. 5.5 for two-mode squeezed states. We find the range of s and η which allows observing non-local properties of these two types of states. We will show that the test involving the Q-function permits the lowest detector efficiency for observing violations of local realism. We also find that the degree of violation is irrespective of the negativity of the quasi-probability function. Finally, in Sec. 5.6, we discuss the characteristics and applications of the s-parameterised Bell inequality. 68 Testing quantum non-locality by generalised quasi-probability functions 5.2 Generalised Bell inequalities of quasi-probability

functions

We begin by formulating a generalised Bell inequality in terms of quasi-probability functions. Suppose that two spatially separated parties, Alice and Bob, indepen- ˆ ˆ ˆ ˆ dently choose one of two observables, denoted by A1, A2 and B1, B2 respectively. No restriction is placed on the number of possible measurement outcomes (which may be inﬁnite). We assume that the measurement operators of the local observables ˆ ˆ ˆ ˆ A1, A2, B1, B2 can be written as

ˆ ˆ ˆ ˆ Aa = O(αa; s), Ba = O(βb; s), for a, b = 1, 2 using a Hermitian operator   (1 − s)Π(ˆ α; s) + s1 if −1 < s ≤ 0, Oˆ(α; s) = (5.3)  2Π(ˆ α; s) − 1 if s ≤ −1, parameterised by a real non-positive number s and an arbitrary complex variable α. Here 1 is the identity operator. The possible measurement outcomes of Oˆ(α; s) are given by its eigenvalues,    s+1 n (1 − s)( s−1 ) + s if −1 < s ≤ 0, λ = (5.4) n   s+1 n 2( s−1 ) − 1 if s ≤ −1, and their eigenvectors are the displaced number states. The maximum and minimum ˆ measurement outcomes of O(α; s) for any non-positive s are λmax = 1 and λmin = −1, P ˆ ˆ ∞ n respectively. For s = 0 we have O(α; 0) = Π(α; 0) = n=0(−1) | α, ni hα, n |, the displaced parity operator, while for s = −1 we ﬁnd that Oˆ(α; −1) = 2 | αi hα | − 1 5.2. Generalised Bell inequalities of quasi-probability functions 69 projects onto the coherent states.

ˆ ˆ A Bell operator can be constructed using the measurement operators Aa, Bb by way of a construction similar to the CHSH combination

ˆ ˆ ˆ ˆ ˆ B = C1,1 + C1,2 + C2,1 − C2,2, (5.5)

ˆ ˆ ˆ where Ca,b = Aa ⊗ Bb is the correlation operator. Since the expectation values ˆ ˆ of the local observables are bounded by |hAai| < 1 and |hBbi| ≤ 1 for any non- positive s, the expectation value of the Bell operator deﬁned in Eq. (6.5) is bounded by |hBi|ˆ ≡ |B| ≤ 2 in local-realistic theories. Note that the expectation value of Π(ˆ α; s) for a given density operatorρ ˆ is proportional to the s-parameterised quasi- probability function [35, 36, 89]

2 W (α; s) = Tr[ˆρΠ(ˆ α; s)] π(1 − s) µ ¶ 2 X∞ s + 1 n = hα, n | ρˆ| α, ni , (5.6) π(1 − s) s − 1 n=0 from which both the Wigner function and the Q-function can be recovered by setting s = 0 and s = −1, respectively. We do not consider the case s > 0 when the eigenvalues of Π(ˆ α; s) are not bounded. We thus obtain the following generalised 70 Testing quantum non-locality by generalised quasi-probability functions

Bell inequality

¯ ¯π2(1 − s)4 |B| = ¯ [W (α , β ; s) + W (α , β ; s) {−1

2 2 |B|{s≤−1} = |π (1 − s) [W (α1, β1; s) + W (α1, β2; s)

+ W (α2, β1; s) − W (α2, β2; s)] − 2π(1 − s)

× [W (α1; s) + W (β1; s)] + 2| ≤ 2, where W (α, β; s) = (4/π2(1−s)2)Tr[ˆρΠ(ˆ α; s)⊗Π(ˆ β; s)] is the two-mode s-parameterised quasi-probability functions, and W (α; s) and W (β; s) are its marginal distributions. We call Eq. (5.7) the s-parameterised Bell inequality for quasi-probability functions. This Bell inequality is equivalent to the BW-W inequality when s = 0 which has the form of the standard CHSH inequality [8], and the BW-Q inequality when s = −1 in the form of the Bell inequality proposed by Clauser and Horne (CH) [65]. In these cases the corresponding generalised quasi-probability function reduces to the Wigner-function W (α, β) = W (α, β; 0) and the Q-function Q(α, β) = W (α, β; −1), respectively [19].

5.3 Testing Quantum non-locality

In this section we present a scheme to test quantum non-locality using the s- parameterised Bell inequalities. For a valid quantum non-locality test the measured quantities should satisfy the local-realistic conditions which are assumed when deriving Bell inequalities. Thus we here employ the direct measurement scheme of quasi-probability functions using photon number detectors proposed in [91]. 5.3. Testing Quantum non-locality 71

Figure 5.1. The optical setup for the Bell inequality test. Each local measurement is carried out after mixing the incoming ﬁeld with a coherent state (denoted by | ξi for Alice and | δi for Bob) in a beam splitter (BS) of high transmissivity T . The photon number detectors (PNDs) have eﬃciency ηd.

A pair of entangled states generated from a source of correlated photons is distributed between Alice and Bob, each of whom make a local measurement by way of an unbalanced homodyne detection (see Fig. 5.1). Each local measurement is carried out using a photon number detector with quantum efficiency ηd preceded by a beam splitter with transmissivity T . Coherent fields | ξi and | δi enter through the other input ports of each beam splitter. For high transmissivity T → 1 and strong coherent fields ξ,δ → ∞, the beam splitters of Alice and Bob can be described by p the displacement operators Dˆ(α) and Dˆ(β) respectively, where α = ξ (1 − T )/T p nˆ P and β = δ (1 − T )/T [91]. Measurements ((s + 1)/(s − 1)) withn ˆ = n n | ni hn | the photon number operator are performed on the outgoing modes using perfect photon number detectors. Then the expectation value directly yields the value of the s-parameterised quasi-probability function at the point in phase space specified by the complex variables α and β. For example, the Wigner function can be obtained by the parity measurements (−1)nˆ (s = 0) and the Q-function by on-off (i.e. photon presence or absence) measurements (s = −1). 72 Testing quantum non-locality by generalised quasi-probability functions

Let us now consider the effects of the detector efficiencies η. If the true photon number distribution is given by P (n), then the measured distribution can be written as a function of the overall detection efficiency η = ηdT . In general, the error of measurement outcomes can be modeled by a generalised Bernoulli transformation

[111, 112]. Therefore, the photon number distributions are given by

µ ¶ X∞ n P (m) = P (n) (1 − η)n−mηm (5.8) η m n=m µ ¶ X∞ m P (n) = P (m) (η − 1)m−nη−m, (5.9) η n m=n

¡ n ¢ where m = n!/(m!(n − m)!).

For the measurement of ((s+1)/(s−1))nˆ, the measured distribution of outcomes is given by

µ ¶ µ ¶ µ ¶ X∞ s + 1 m X∞ s + 1 m X∞ n P (m) = P (n) (1 − η)n−mηm s − 1 η s − 1 m m=0 m=0 n=m µ ¶ µ ¶ X∞ X∞ s + 1 m n = (1 − η)n (1 − η)−mηm P (n) s − 1 m n=0 m=0 µ ¶ µ ¶ X∞ X∞ (s + 1)η m n = (1 − η)n P (n) (s − 1)(1 − η) m n=0 m=0 µ ¶ X∞ (s + 1)η n = (1 − η)n 1 + P (n) (s − 1)(1 − η) n=0 µ ¶ X∞ s + 1 n = 1 − η + η P (n) s − 1 n=0 µ ¶ X∞ s0 + 1 n = P (n), (5.10) s0 − 1 n=0 ¡ ¢ ¡ ¢ P∞ n n−m m P∞ n where we used (i) the equivalence n=m P (n) m (1−η) η = n=0 P (n) m (1− ¡ ¢ n−m m P∞ m n n 0 η) η , (ii) the relation m=0 α m = (1 + α) , and s ≡ −(1 − s − η)/η. 5.3. Testing Quantum non-locality 73

For α = 0 the measured quasi-probability function is thus given by

µ ¶ 2 X∞ s + 1 m W (0; s) = P (m) η π(1 − s) s − 1 η m=0 µ ¶ 2 X∞ s + 1 n = 1 − η + η P (n) π(1 − s) s − 1 ³ n=0 ´ 1−s−η 0 W 0; − η W (0; s ) = ≡ . (5.11) η η

The s-parameterised quasi-probability function measured by a detector with eﬃ- ciency η can therefore be identiﬁed with the quasi-probability function with parameter s0 = −(1 − s − η)/η. Other sources of noise (e.g. dark counts and mode mismatch) could be included into this approach but are neglected here for simplicity.

Finally, the expectation value of observable (6.1) is given as   2  π(1−s) 0 2η W (α; s ) + s if −1 < s ≤ 0, hOˆ(α; s)i = (5.12) η   π(1−s) 0 η W (α; s ) − 1 if s ≤ −1,

where h·iη represents the expectation value obtained by measurement with eﬃciency η. Note that (5.12) is the statistical average of directly measured data without postselection. The expectation value of the Bell operator (6.5) written as a function 74 Testing quantum non-locality by generalised quasi-probability functions of s and η is given by

· π2(1 − s)4 1 − s − η 1 − s − η hBˆ i = W (α , β ; − ) + W (α , β ; − ) {−1

Note that the Bell expectation values in Eq. (6.8) for s = 0 and s = −1 give the same results as tests of the BW-W and BW-Q inequalities, respectively.

5.4 Violation by single photon entangled states

We investigate violations of the s-parameterised Bell inequality (5.7) for the single photon entangled state [113, 114, 115, 116]

1 | Ψi = √ (| 0, 1i + | 1, 0i), (5.14) 2 where | n, mi is the state with n photons in Alice’s mode and m photons in Bob’s mode. This state is created by a single photon incident on a 50:50 beam splitter. 5.4. Violation by single photon entangled states 75

2.8

ÈBÈ 1.0 max2.4

2.0 0.9Η -1.5 -1.0 s -0.5 0.8 0.0

Figure 5.2. Maximum Bell expectation value |B| = |hBi|ˆ for the single photon entangled state. Only the range of parameters s and detector eﬃciencies η with |B| > 2 is shown.

Its two-mode s-parameterised quasi-probability function is given by

µ ¶ 4 1 + s 2 W (α, β; s) = − + |α + β|2 Ψ π2(1 − s)2 1 − s (1 − s)2 · ¸ 2(|α|2 + |β|2) × exp − , (5.15) 1 − s and its marginal single-mode distribution is

1 W (α; s) = (2 − 2η + 4η2|α|2) exp[−2η|α|2]. (5.16) Ψ π

Note that for 0 ≥ s > −1 Eq. (5.15) has negative values in certain regions of phase space but for s = −1 it becomes the Q-function WΨ(α, β; −1) ≥ 0.

ˆ The maximum expectation values |B|max = |hBi|max are obtained for properly chosen α1, α2, β1, β2. Figure 5.2 shows the range of parameters s and detector efficiencies η for which the Bell inequality is violated, |B|max > 2. Interestingly, the 76 Testing quantum non-locality by generalised quasi-probability functions degree of violation is not directly related to the negativity of the quasi-probability functions. The test of the Bell inequality using the Q-function (s = −1) yields strong violations and is most robust to detector inefficiencies. This is because the observable (6.1) becomes dichotomised at s = −1 corresponding to detection of none vs. some photons. For a given s, the amount of violation decreases with decreasing η. The minimum value of η indicates the required detector efficiency for a successful non-locality test [117]. For example, the minimum bound is about 83% for the Q-function (s = −1). We also find the minimum parameter s which allows demonstrating quantum non-locality for a given detector efficiency. For example, for a perfect detector (η = 1), the corresponding Bell inequality is violated when s & −1.43.

5.5 Violation by two-mode squeezed states

We consider the two-mode squeezed vacuum states (TMSSs), i.e. a continuous variable entangled state written as

X∞ | TMSSi = sech r tanhn r | n, ni , (5.17) n=0 where r > 0 is the squeezing parameter. It can be realised for instance by non- degenerate optical parametric ampliﬁers [103]. In the inﬁnite squeezing limit r → ∞, the two-mode squeezed states becomes the normalised EPR state which is the maximally entangled state associated with position and momentum [39].

For a non-positive s the quasi-probability function of the two-mode squeezed 5.5. Violation by two-mode squeezed states 77

r = 0.4

2.4 ÈBÈmax 1.0

0.9 2.0 Η -1.5 0.8 -1.0 s -0.5 0.7 0.0

r = 0.6

2.4 ÈBÈmax 1.0

0.9 2.0 Η -1.5 0.8 -1.0 s -0.5 0.7 0.0

r = 0.8

2.4 ÈBÈmax 1.0

0.9 2.0 Η -1.5 0.8 -1.0 s -0.5 0.7 0.0

Figure 5.3. Demonstration of quantum non-locality for two-mode squeezed vacuum states (TMSSs). Maximum Bell values are shown for diﬀerent squeezing r in the range of s and η where the Bell inequality is violated. 78 Testing quantum non-locality by generalised quasi-probability functions

2.4 s = 0 2.4 s = -0.5 max max

È 2.2 È 2.2 B B È È

2.0 2.0

0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 r r

2.4 s = -0.7 2.4 s = -1.0 max max

È 2.2 È 2.2 B B È È

2.0 2.0

0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 r r

Figure 5.4. Violation of the Bell inequality as a function of the squeezing r for diﬀerent s and η = 1 (solid line), η = 0.95 (dashed line), and η = 0.9 (dotted line).

states is given by

µ 4 2 W (α, β; s) = exp − {S(s)(|α|2 + |β|2) TMSS π2R(s) R(s) ¶ − sinh 2r(αβ + α∗β∗)} , (5.18) and its marginal single-mode distribution is

µ ¶ 2 2|α|2 W (α; s) = exp − , (5.19) TMSS πS(s) S(s) where R(s) = s2 − 2s cosh 2r + 1 and S(s) = cosh 2r − s. Note that these are positive everywhere in phase space. In Fig. 5.3 violations of the s-parameterised Bell inequality are shown for two-mode squeezed vacuum states. The test using the Q-function (s = −1) is most robust with respect to detector inefficiencies. The 5.6. Discussion and Conclusions 79 amount of violation shows different tendencies depending on the squeezing parameter r. In the case of low squeezing rates, i.e. when the amplitudes of small-n number states are dominant, the violation is maximal if we choose the Q-function (s = −1) as shown in Fig. 5.4. This implies that the dominant contribution to the violation comes from correlations between the vacuum and photons being present. For larger squeezing rates r & 1.2, the violation reaches a maximal value B ≈ 2.32 when we test the Wigner-function (s = 0) [41]. This indicates that the parity measurements are effective for verifying higher-order number correlations. However, the parity measurements require very high detector efficiency as shown in Fig. 5.4. The range of s within which one can demonstrate non-locality becomes narrower around s = 0 and s = −1 with increasing squeezing rate r. This is because the observable (6.1) is dichotomised at s = 0 and s = −1.

5.6 Discussion and Conclusions

We demonstrated that quantum non-locality has no direct relation to the negativity of s-parameterised quasi-probability functions. In fact the Q-function (s = −1) which never becomes negative can still be used to verify non-local properties as we showed in Fig. 5.2 and yields strong violations of the corresponding Bell inequality. This implies that the quantum properties of non-locality and negativity of the quasi- probability functions should be considered distinct features of quantum mechanics. Furthermore we showed that the Q-function test allows the lowest detector eﬃciency for demonstrating quantum non-locality. For example, it requires only η ≈ 83% for a single photon entangled state and η ≈ 75% for two-mode squeezed states with r = 0.4 to detect non-locality. This indicates that two-mode correlations between vacuum and many photons can be more robust to detector ineﬃciencies than correlations between vacuum and a single photon. 80 Testing quantum non-locality by generalised quasi-probability functions

The parameter s determines the characteristics of the detected non-local correlations. For example, if we choose s = −1 the violation of the Bell inequality exhibits only correlations between vacuum and photons. In order to test higher-order photon number correlations we need to increase s to zero, so that the factor ((s+1)/(s−1))n multiplied to the photon number probability increases in Eq. (5.6). Although parity measurements (s = 0) allow to detect higher-order correlations effectively, they also require very high detector efficiencies as shown in Fig. 5.3. If we properly choose a certain parameter −1 < s < 0, e.g. s = −0.7, we can detect higher-order correlations with a lower detector efficiency than that required for testing the Bell inequality using the Wigner function. However, we note that the violation of the Bell inequality with s = −0.7 disappears with increasing squeezing rate as shown in Fig. 5.4; this restricts the possible applications to schemes using light that contains only a few photons.

Let us ﬁnally discuss whether we can regard decoherence eﬀects as changes to s. Interactions with the environment and detection noise tend to smoothen quasi- probability functions. For example, when solving the Fokker-Planck equation for the evolution of the Wigner-function of a system interacting with a thermal environment one obtains [118]

Z µ ¶ 1 α − r(τ)β W (α, τ) = d2βW th(β)W , τ = 0 . (5.20) t(τ)2 t(τ)

√ √ Here the parameters r(τ) = 1 − e−γτ and t(τ) = e−γτ are given in terms of the energy decay rate γ, and

µ ¶ 2 2|β|2 W th(β) = exp − (5.21) π(1 + 2¯n) 1 + 2¯n is the Wigner function for the thermal state of average thermal photon numbern ¯. 5.6. Discussion and Conclusions 81

The eﬀect of the thermal environment is then identiﬁed with temporal changes of the parameter

r(τ)2 s(τ) ∼ − (1 + 2¯n) = (1 − eγτ )(1 + 2¯n). (5.22) t(τ)2

Therefore one might be tempted to consider an environment in a thermal state as giving rise to a temporal change in s in Eq. (6.1). However this idea is not applicable to tests of quantum non-locality. The s-parameterised Bell inequality is derived for observables (6.1) which contain s as a deterministic value of local-realistic theories.

Thus the local-realistic bound is no longer valid when dynamical observables are considered (even though they give the same statistical average). However, this idea might be useful for witnessing entanglement [119].

In summary, we have formulated a Bell inequality in terms of the generalised quasi-probability function. This Bell inequality is parameterised by a non-positive value s and includes previously proposed Bell inequalities such as the BW-W (s = 0) and the BW-Q (s = −1) inequalities [19]. We employed a direct measurement scheme for quasi-probability functions [91, 120] to test quantum non-locality. The violation of Bell inequalities was demonstrated for two types of entangled states, single photon entangled and two-mode squeezed vacuum states. We found the range of s and η which allow the observation of quantum non-local properties. We discussed the types of correlations and their robustness to detection ineﬃciencies for diﬀer- ent values of s. We also demonstrated that the negativity of the quasi-probability function is not directly related to the violation of Bell inequalities. The realisation of s-parameterised Bell inequality tests is expected along with the progress of photon detection technologies [121, 122] in the near future. Our investigations can readily be extended to other types of states like photon subtracted gaussian states

[123, 124, 125, 126], or optical Schr¨odingercat states [127, 128]. 82 Testing quantum non-locality by generalised quasi-probability functions Chapter 6

Witnessing entanglement in phase space using inefficient detectors

6.1 Introduction

Entanglement is one of the most remarkable features of quantum mechanics which can not be understood in the context of classical physics. It has been shown that entanglement can exist in various physical systems and play a role in quantum phenomena [21, 23]. Moreover, its properties can be used as a resource for quantum information technologies such as quantum computing, quantum cryptography, and quantum communication [4]. Therefore, detecting entanglement is one of the most essential tasks both for studying fundamental quantum properties and for applications in quantum information processing. Although various entanglement detection schemes have been proposed [11], their experimental realisation suffers from imperfections of realistic detectors since measurement errors wash out quantum correlations. This difficulty becomes more significant with increasing system dimensionality and particularly in continuous variable systems where entanglement is increasingly attracting interest [20].

Quantum tomography provides a method to reconstruct complete information of

83 84 Witnessing entanglement in phase space using inefficient detectors quantum states in phase space formalism [129, 130, 91, 120, 131]. The reconstructed data can be used to determine whether the state is entangled or not with the help of an entanglement witness (EW). Bell inequalities that were originally derived for discriminating quantum mechanics from local realism [1] can also be used for witnessing entanglement since their violation guarantees the existence of entanglement. Banaszek and Wódkiewicz(BW) [39, 38, 19] suggested a Bell-type inequality (referred to BW-inequality in this paper) which can be tested by way of reconstructing the Wigner function at a few specific points of phase space. However, imperfections of tomographic measurements constitute a crucial obstacle for its practical applications. Several schemes have been considered to overcome this problem [131] such as numerical inversion [43] and maximum-likelihood estimation [18, 44], but they require a great amount of calculations or iteration steps for high dimensional and continuous variable systems.

In this paper we propose an entanglement detection scheme in phase space formalism, which can be used in the presence of detection noise. We formulate an entanglement witness (EW) in the form of a Bell-like inequality using the experimentally measured Wigner function. For this, we include eﬀects of detector eﬃciency into possible measurement outcomes. Possible expectation values of the entanglement witness are bounded by the maximal expectation value when separable states are assumed. Any larger expectation value guarantees the existence of entanglement.

Our approach shows the following remarkable features: (i) in contrast to previous proposals [131, 18, 44] it does not require any additional process for correcting measurement errors; (ii) it allows us to witness entanglement e.g. in single-photon entangled and two-mode squeezed states with efficiency as low as 40%; (iii) our scheme is also valid when precise detector efficiency is not known prior to the test; (iv) finally, we note that our approach is applicable to detect any quantum state represented in phase space formalism. 6.2. Observable associated with efficiency 85 6.2 Observable associated with efficiency

We begin by introducing an observable associated with the detector eﬃciency η and an arbitrary complex variable α:    1 ˆ 1 1 η Π(α) + (1 − η )1 if 2 < η ≤ 1, Oˆ(α) = (6.1)   ˆ 1 2Π(α) − 1 if η ≤ 2 ,

Let us then consider the expectation value of observable (6.1) when the measurement is carried out with eﬃciency η. In general, measurement errors occur when not all particles are counted in the detector. Thus the real probability distribution of particles, P (n), transforms to another distribution, Pη(m), by the generalised ¡ ¢ P∞ n n−m m Bernoulli transformation [112]: Pη(m) = n=m P (n) m (1 − η) η . Thus the expectation value of the parity operator is obtained as

X∞ X∞ ˆ m n hΠ(α)iη = (−1) Pη(α, m) = (1 − 2η) P (α, n), (6.2) m=0 n=0

where h·iη implies the statistical average measured with eﬃciency η. Here Pη(α, m) and P (α, n) are the measured and real particle number distributions in the phase space displace by α, respectively.

We deﬁne the Wigner function experimentally measured with eﬃciency η as

2 W η(α) ≡ hΠ(ˆ α)i , (6.3) π η which is given as a rescaled Wigner function by Gaussian smoothing. Note that a 86 Witnessing entanglement in phase space using ineﬃcient detectors smoothed Wigner function can be identiﬁed with a s-parameterised quasi-probability

η P∞ function as W (α) = W (α; −(1−η)/η)/η where W (α; s) = (2/(π(1−s))) n=0((s+ 1)/(s + 1))nP (α, n) [35, 36]. This identiﬁcation is available both for homodyne [129, 130, 90] and number counting tomography methods [91, 120]. After series of measurements with eﬃciency η, we can obtain the expectation value of the observable (6.1) as    π η 1 1 2η W (α) + 1 − η if 2 < η ≤ 1, hOˆ(α)i = (6.4) η   η 1 πW (α) − 1 if η ≤ 2 ,

ˆ which is bounded as |hO(α)iη| ≤ 1 for all η.

6.3 Entanglement witness in phase space

Let us formulate an entanglement witness (EW) in the framework of phase space. Suppose that two separated parties, Alice and Bob, measure one of two observables, ˆ ˆ ˆ ˆ denoted by A1, A2 for Alice and B1, B2 for Bob. All observables are variations of the ˆ ˆ ˆ ˆ operator (6.1) as Aa = O(αa) and Ba = O(βb) with a, b = 1, 2. We then formulate ˆ ˆ a Hermitian operator as a combination of each local observable Aa, Bb in the form

ˆ ˆ ˆ ˆ ˆ W = C1,1 + C1,2 + C2,1 − C2,2, (6.5)

ˆ ˆ ˆ ˆ where Ca,b = Aa ⊗ Bb is the correlation operator. We call W an entanglement witness operator. Note that the operator in Eq. (6.5) can also be regarded as a Bell operator Bˆ which distinguishes non-local properties from local realism. The bound expectation value of the operator in Eq. (6.5) is determined according to whether it is regarded as Wˆ or Bˆ. In other words, the entanglement criterion given by the operator (6.5) is diﬀerent with the non-locality criterion as we will show below. 6.3. Entanglement witness in phase space 87

Let us ﬁrstly obtain the bound expectation value of the operator (6.5) as an entanglement witness by which one can discriminate entangled states and separable

P A B P states. For a separable stateρ ˆsep = i piρî ⊗ ρî where pi ≥ 0 and i pi = 1, the expectation value of the correlation operator measured with efficiency η is given by

X X∞ ˆ sep n+m A B hCa,biη = pi (1 − 2η) hα, n | ρˆi | α, ni hβ, m | ρˆi | β, mi i n,m X ˆ i ˆ i = pihAaiηhBbiη. (6.6) i

Since expectation values of all local observables with eﬃciency η are bounded as

ˆ i ˆ i |hAaiη|, |hBbiη| ≤ 1 for a, b = 1, 2, we can obtain the statistical maximal bound of the entanglement witness operator (6.5) with respect to the separable states:

¯ ¯ ¯X ¯ ˆ sep ¯ ˆ i ˆ i ˆ i ˆ i ˆ i ˆ i ˆ i ˆ i ¯ |hWiη | = ¯ pi(hA1iηhB1iη + hA1iηhB2iη + hA2iηhB1iη − hA2iηhB2iη)¯ i X sep ≤ 2 pi = 2 ≡ Wmax. (6.7) i

ˆ ψ sep Therefore, if |hWiη | > Wmax = 2 for a quantum state ψ, we can conclude that the quantum state ψ is entangled.

Let us then consider the operator (6.5) as a Bell operator. Note that the local- realistic (LR) bound of a Bell operator is given as the extremal expectation value of the Bell operator, which is associated with a deterministic conﬁguration of all possible measurement outcomes. If 1/2 < η ≤ 1, the maximal modulus outcome of (6.1) is |1 − 2/η| when the outcome of parity operator Π(ˆ α) is measured as −1.

ˆ LR Thus the expectation value of (6.5) is bounded by local realism as |hBiη| ≤ Bmax =

2 LR LR sep 2(1−2/η) . Likewise for η ≤ 1/2, we can obtain Bmax = 18. Note that Bmax ≥ Wmax

LR sep for all η, and Bmax = Wmax in the case of unit eﬃciency (η = 1). It shows that some entanglement can exist without violating local realism, and thus the Bell operator can be regarded as a non-optimal entanglement witness as pointed out already in 88 Witnessing entanglement in phase space using ineﬃcient detectors

[88]. For the purpose of this paper we will focus on the role of an entanglement witness in the following parts.

From Eq. (6.5) and Eq. (6.7), we can ﬁnally obtain an entanglement witness in the form of an inequality obeyed by any separable state:

¯ ¯ 2 ˆ ¯ π η η η η |hWiη> 1 | = [W1,1 + W1,2 + W2,1 − W2,2] 2 ¯4η2 ¯ π(η − 1) 1 ¯ + [W η + W η ] + 2(1 − )2¯ ≤ 2, η2 a=1 b=1 η ¯ (6.8)

2 η η η η η η |hWiˆ 1 | = |π [W + W + W − W ] − 2π[W + W ] + 2| ≤ 2, η≤ 2 1,1 1,2 2,1 2,2 a=1 b=1

η where Wa,b is the two-mode Wigner function measured with eﬃciency η (here we re- place the notation αa and βb in the conventional representation of two-mode Wigner

η η function W (αa, βb) with the notation a, b for simplicity), and Wa(b)=1 is its marginal single-mode distribution. Any violation of Eq. (6.8) guarantees that the measured quantum state is entangled. Remarkably, our scheme allows one to detect entanglement without correcting measurement errors. Note that in the case of unit efficiency (η = 1) the inequality in Eq. (6.8) becomes equivalent to the BW-inequality [19]. It is also notable that any violation of this inequality for η < 1 ensures the violation of the BW-inequality in the case of a unit efficiency (η = 1). Therefore the proposed entanglement witness in Eq. (6.8) can be used effectively for detecting entanglement instead of the BW-inequality in the presence of measurement noise.

6.4 Testing single photon entangled states

Let us now apply the entanglement witness in Eq. (6.8) for detecting entangled photons. We here ﬁrstly consider the single photon entangled state | Ψi = (| 0, 1i + √ | 1, 0i)/ 2 where | 0, 1i (| 1, 0i) is the state with zero (one) photons in the mode of 6.4. Testing single photon entangled states 89

2.8 2.8 HaL HbL

2.4 2.4 \È W ÈX

2.0 2.0

0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 Η Η 2.5 HcL 2.4 Η=0.5 Η=1 2.3 \È 2.2 W ÈX Η=0.99 2.1 Η=0.7 2.0

1.9 0.5 1.0 1.5 2.0 2.5 r

Figure 6.1. Maximum expectation value of the entanglement witness operator in Eq. (6.5) for an input of (a) a single photon entangled state and (b) a two-mode squeezed state with r = 0.4 (black) and r = 0.8 (grey). Entangle- sep ment exists if the expectation value exceeds the dashed line Wmax = 2. Note LR 2 that the shaded region which exceeds Bmax = 2(1 − 2/η) (for 1/2 < η ≤ 1) is the criterion of non-locality. (c) Witnessing entanglement with varying squeezing rate r of a two-mode squeezed states for detector eﬃciencies η = 1 (solid line), η = 0.99 (dashed line), η = 0.7 (dotdashed line) and η = 0.5 (dotted line). 90 Witnessing entanglement in phase space using ineﬃcient detectors

Alice and one (zero) photon in the mode of Bob [113, 114, 115, 116]. This state can be created by a single photon incident on a 50:50 beam splitter. Its two-mode Wigner function measured with eﬃciency η is

4 W η = (1 − 2η + 2η2|α + β |2) exp[−2η(|α |2 + |β |2)] (6.9) a,b π2 a b a b and its marginal single-mode distribution is

1 W η = (2 − 2η + 4η2|α |2) exp[−2η|α |2]. (6.10) a π a a

The expectation values of operator (6.5) with properly chosen αa and βb are plotted in Fig. 6.1(a) against the overall eﬃciency η. It is remarkable that entanglement can be detected even with detection eﬃciency η as low as 40%.

6.5 Testing two-mode squeezed states

Let us consider the entanglement witness in continuous variable systems e.g. two- mode squeezed states (TMSSs). This state can be generated by non-degenerate opti-

P∞ n cal parametric ampliﬁers [103], and be written as | TMSSi = sech r n=0 tanh r | n, ni where r > 0 is the squeezing parameter. The measured Wigner function with eﬃ- ciency η for a two-mode squeezed state is given by

µ 4 2 W η = exp − {S(η)(|α |2 + |β |2) a,b π2η2R(η) R(η) a b ¶ ∗ ∗ − sinh 2r(αaβb + αaβb )} , (6.11) and its marginal single-mode Winger function is

µ ¶ 2 2|α |2 W η = exp − a , (6.12) a πηS(η) S(η) 6.6. Testing with a priori estimated efficiency 91 where R(η) = 2(1 − 1/η)(1 − cosh 2r) + 1/η2 and S(η) = cosh 2r − 1 + 1/η. The expectation values of the entanglement witness operator (6.5) for two-mode squeezed states are shown in Fig. 6.1(b) with different squeezing rates r. It shows that our scheme allows one to detect some continuous variable entanglement with detector efficiency of about 40 %. As shown in Fig. 6.1(c), violations of the inequality show different tendencies depending on efficiency η with increasing the squeezing parameter r. In the case of low squeezing rates the violation is maximised when η = 0.5, while for larger squeezing rates about r ≥ 1.2 the violation is maximised when η = 1. This is because the dominant degree of freedom of entanglement detected by observable in Eq. (6.1) changes with decreasing the efficiency η. Note that in the case η = 0.5 the dominant contribution to the entanglement arises from quantum correlations between the vacuum and the photon being present, while for η = 1 it comes from higher-order correlations of photon number states.

6.6 Testing with a priori estimated eﬃciency

So far it has been assumed that the detector efficiency is known precisely prior to the tests both in our scheme presented above and in other proposals proposed previously [43, 18, 44, 131]. This can be realised e.g. by a full characterisation of detectors when doing a quantum tomography on the detectors which has been experimentally achieved [132]. However, in most cases a priori estimates of the detector efficiency (≡ ε) may not be perfect and thus can be different from the real efficiency η that affects measured data. Let us assume that we can discriminate perfectly only whether the real efficiency η > 1/2 or η ≤ 1/2. If η ≤ 1/2, we can see that the entanglement witness in Eq. (6.8) is formulated only by experimentally measured Wigner functions. Thus, in this case our EW can be tested without knowing the real efficiency. On the other hand, for the case η > 1/2, the efficiency 92 Witnessing entanglement in phase space using inefficient detectors

2.6 Η=0.55 HaL 2.6 Η=0.55 HbL

2.4 2.4 \È W ÈX 2.2 2.2

2.0 2.0

0.55 0.60 0.65 0.70 0.75 0.80 0.55 0.60 0.65 0.70 0.75 0.80 Ε Ε

Figure 6.2. Witnessing entanglement with a real eﬃciency η = 0.55 as varying the estimated eﬃciency ε for an input of (a) a single photon entangled state and (b) a two-mode squeezed state with r = 0.4 (black) and r = 0.8 (grey).

variable η is explicitly included in the entanglement witness (6.8) and should be replaced with the estimated eﬃciency ε as

¯ ¯ 2 ˆ ¯ π η η η η |hWiη> 1 | = [W1,1 + W1,2 + W2,1 − W2,2] (6.13) 2 ¯4ε2 ¯ π(ε − 1) 1 ¯ + [W η + W η ] + 2(1 − )2¯ ≤ 2. ε2 a=1 b=1 ε ¯

Note that Eq. (6.13) is valid subject to the condition

η(real eﬃciency) ≤ ε(estimated eﬃciency), since otherwise the right-hand side of inequality in Eq. (6.13) is not valid i.e. the

sep expectation values of separable states are not bounded by Wmax = 2. In this case, one can also detect entanglement even with non-perfect estimates of the efficiency as shown in Fig. 6.2. For example, even when one estimates the efficiency as ε = 0.65 for the detector with real efficiency η = 0.55, one can still detect entanglement of 6.7. Conclusions 93 the two-mode squeezed state with r = 0.4 using our scheme.

6.7 Conclusions

We remake on advantages of our scheme. The entanglement witness (6.8) can be used to test arbitrary quantum states described in phase space formalism. It can be implemented by any tomography method without additional steps for error correction. Moreover, since the required minimal detection eﬃciency for our scheme is as low as 40%, it may be realisable using current detection technologies. In addition, our entanglement witness can be used without knowing the detection eﬃciency precisely prior to the test. Finally, we note that our approach is applicable to other frameworks e.g. cavity QED or ion trap systems with the help of the direct measurement scheme of Wigner function in such systems [133].

In summary, we proposed an entanglement witness which can detect entanglement even with signiﬁcantly imperfect detectors. Its implementation requires neither an error correcting process nor a full priori knowledge of detection eﬃciency. It is generally applicable to any quantum state in phase space formalism. We expect that our scheme enhances the possibility of witnessing entanglement in complex physical systems using current photo-detection technologies.

Chapter 7

Conclusion

In this thesis, we formulated new types of Bell inequalities that are applicable to complex systems. For testing proposed Bell inequalities, we have studied quantum non-locality and entanglement in high-dimensional and continuous variable systems. We aimed to ﬁnd answers to fundamental questions on quantum non-locality and entanglement. Here, we summarise our results and conclude with the directions of further research. • In chapter 3 we proposed a generalised structure of Bell inequalities for bi-

partite arbitrary d-dimensional systems, which includes various types of Bell inequalities known previously. A Bell inequality in the given generalised structure can be represented either in correlation function space or joint probability space. Interestingly, a Bell inequality in one space can be mapped into the other by Fourier transformation. The two types of high-dimensional Bell inequalities, CGLMP and SLK, are represented in terms of the generalised structure with appropriate coeﬃcients in both spaces. We demonstrated the violation of Bell inequalities for varying degrees of entanglement, and also in-

vestigated the tightness of Bell inequalities. The generalised structure allows us to construct new types of Bell inequalities in a convenient way and to in-

vestigate their properties. We expect that this generalised structure can be

95 96 Conclusion

extended to phase space formalism due to the fact that quasiprobability functions and their characteristic functions are related by a Fourier transformation similarly to the relation between joint probabilities and correlation functions in our formalism. Thus, ﬁnding a generalised structure in phase space formalism

is expected based on our result.

• In chapter 4 we formulated a Bell inequality for even d-dimensional bipartite quantum systems which fulﬁlls two desirable properties: maximal violation by maximal entanglement and tightness. These properties are essential for in-

vestigating quantum non-locality properly in higher-dimensional systems. We called such a Bell inequality “optimal”. We found an optimal Bell inequality and applied it to continuous variable systems. Maximal violations of the optimal Bell inequality are obtained asymptotically by the EPR state, and showed that a parity measurement in the Pegg-Barnett phase basis is an optimal measurement. We believe that an optimal Bell inequality can be an eﬀective tool

for applications in quantum information processing. Speciﬁcally, it would be useful for the preparation of a maximally entangled channel in arbitrary dimensional quantum teleportation [34] and cryptography protocols [25]. It will be valuable to search for additional optimal Bell inequalities comparing their properties and extending optimal Bell inequalities to multipartite systems.

• In chapter 5 we proposed a method for testing quantum non-locality using the generalised quasi-probability function. We formulated a generalised Bell inequality in terms of the s-parameterised quasi-probability function, which includes previously proposed Bell inequalities in phase space as limiting cases.

We then presented a measurement scheme for testing Bell inequalities with a realistic setup. The measured Bell expectation value was written as a function

of the parameter s and the overall detector eﬃciency η. We demonstrated the 97

violation of proposed Bell inequalities for the single-photon entangled and two- mode squeezed states for varying s and η. We showed that there is no direct relation between the negativity of arbitrary quasi-probability functions and quantum non-locality. Furthermore, we also showed that the test involving

the Q-function permits the lowest detector eﬃciency for observing violations of local realism. The realisation of s-parameterised Bell inequality tests is expected along with the progress of photon detection technologies [121, 122] in the near future. Our investigations can readily be extended to other types of states like photon subtracted gaussian states [123, 124, 125, 126], or optical Schr¨odingercat states [127, 128].

• In chapter 6 we proposed an entanglement detection scheme in phase space which is directly applicable to any tomography method performed with inefficient detectors. An entanglement witness was formulated in the form of a Bell-like inequality using the directly measured Wigner function. The effects of detector efficiency are included into possible measurement outcomes of local observables. Possible expectation values of the entanglement witness are bounded by the maximal expectation value in the assumption of separable

states so that any larger expectation value guarantees the existence of entanglement. In contrast to previous proposals, our proposal does not require any error correction in the reconstruction process. Moreover, our scheme allows to detect entanglement even with significantly inefficient detectors e.g. as low efficiency as 40% for the single-photon entangled and two-mode squeezed states. Our scheme can also be used without knowing precisely the detection efficiency before the tests. Finally, we note that our approach is generally applicable to

various physical systems that can be described in phase space formalism. 98 Conclusion Outlook

In this thesis we have focused on improving detection methods for non-locality and entanglement. Based on the results we present some ideas for future research as follows.

(i) High-dimensional Bell inequalities would be an effective tool for testing quantum non-locality and entanglement of multiple degrees of freedom. We can thus study hyper entangled states [53, 54], i.e. the entangled states with more than two degrees of freedom, which might provide more efficient resources for quantum information processing. (ii) We are interested in the role of entanglement in quantum phenomena. For example, Bell tests on a quantum system traversing two different quantum phase may provide a tool to examine the role of entanglement in quantum phase transition. As a first step of this research, we can formulate a Bell type inequality with local measurements associated with both dominant degrees of freedom in two quantum phases. (iii) Moreover, an entanglement witness formulated in phase space formalism would be a useful tool for studying the role of entanglement in any quantum phenomena described in phase space.

Therefore, we believe that our work provides a signiﬁcant step forward in the research of quantum non-locality and entanglement in complex physical systems. Bibliography

[1] J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics 1, 195 (1964).

[2] H. P. Stapp. Bell’s theorem and world process. Nuovo Cimento Soc. Ital. Fis. B 29, 270 (1975).

[3] M. Zukowski.˙ On the paradoxical book of bell. Stud. Hist. Phil. Mod. Phys. 36, 566 (2005).

[4] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Infor- mation. Cambridge Univ Press, Cambridge, (2000).

[5] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935).

[6] E. Schr¨odinger.Discussion of probability relations between separated systems.

Proceeding of the Cambridge Philosophical Society 31, 555 (1935).

[7] E. Schr¨odinger.Probability relations between separated systems. Proceeding of the Cambridge Philosophical Society 31, 446 (1936).

[8] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969).

99 100 Bibliography

[9] A. Aspect, P. Grangier, and G. Roger. Experimental realization of einstein- podolsky-rosen-bohm gedankenexperiment: A new violation of bell’s inequalities. Phys. Rev. Lett. 49, 91 (1982).

[10] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland. Experimental violation of a bell’s inequality with eﬃcient detection. Nature 409, 791 (2001).

[11] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009).

[12] N. D. Mermin. Quantum encodings in spin systems and harmonic oscillators. Phys. Rev. Lett. 65, 1838 (1990).

[13] D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu. Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett. 88, 040404 (2002).

[14] W. Son, J. Lee, and M. S. Kim. Generic bell inequalities for multipartite arbitrary dimensional systems. Phys. Rev. Lett. 96, 060406 (2006).

[15] N. Gisin. Bell inequalities: many questions, a few answers. arXiv:quant- ph/0702021 (2007).

[16] Open Problems. http://www.imaph.tu-bs.de/qi/problems/1.html.

[17] S.-W. Lee, Y. W. Cheong, and J. Lee. Generalized structure of bell inequalities for bipartite arbitrary-dimensional systems. Phys. Rev. A 76, 032108 (2007).

[18] K. Banaszek. Maximum-likelihood estimation of photon-number distribution from homodyne statistics. Phys. Rev. A 57, 5013 (1998).

[19] K. Banaszek and K. Wodkiewicz. Testing quantum nonlocality in phase space. Phys. Rev. Lett. 82, 2009 (1999). Bibliography 101

[20] S. L. Braunstein and P. van Loock. Quantum information with continuous variables. Rev. Mod. Phys. 77, 513 (2005).

[21] L. Amico, R. Fazio, A. Osterloh, and V. Vedral. Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008).

[22] P. Milman, A. Keller, E. Charron, and O. Atabek. Bell-type inequalities for cold heteronuclear molecules. Phys. Rev. Lett. 99, 130405 (2007).

[23] V. Vedral. Quantifying entanglement in macroscopic systems. Nature 453,

1004 (2008).

[24] A. Osterloh, L. Amico, G. Falci, and R. Fazio. Scaling of entanglement close to a quantum phase transition. Nature 416, 608 (2002).

[25] Artur K. Ekert. Quantum cryptography based on bell theorem. Phys. Rev. Lett. 67, 661 (1991).

[26] D. Jaksch and P. Zoller. The cold atom Hubbard toolbox. Ann. Phys. (NY) 315, 52 (2005).

[27] D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller. Entan- glement of atoms via cold controlled collisions. Phys. Rev. Lett. 82, 1975 (1999).

[28] H. Bechmann-Pasquinucci1 and A. Peres. Quantum cryptography with 3-state systems. Phys. Rev. Lett. 85, 3313 (2000).

[29] N. J. Cerf, S. Massar, and S. Pironio. Greenberger-horne-zeilinger paradoxes for many qudits. Phys. Rev. Lett. 89, 080402 (2002).

[30] D. Bruß and C. Macchiavello. Optimal eavesdropping in cryptography with three-dimensional quantum states. Phys. Rev. Lett. 88, 127901 (2002). 102 Bibliography

[31] E. Knill, R. Laﬂamme, and G. Milburn. A scheme for eﬃcient linear optics quantum computation. Nature 409, 46 (2001).

[32] A. Ac´ın,T. Durt, N. Gisin, and J. I. Latorre. Quantum nonlocality in two three-level systems. Phys. Rev. A 65, 052325 (2002).

[33] L. Masanes. Tight bell inequality for d-outcome measurements correlations. Quant. Inf. Comp. 3, 345 (2003).

[34] Charles H. Bennett, Gilles Brassard, Claude Cr´epeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993).

[35] K. E. Cahill and R. J. Glauber. Ordered expansions in boson amplitude operators. Phys. Rev. 177, 1857 (1969).

[36] K. E. Cahill and R. J. Glauber. Density operators and quasiprobability distributions. Phys. Rev. 177, 1882 (1969).

[37] J. S. Bell. Speakable and unspeakable in quantum mechanics. Cambridge Univ Press, Cambridge, (1987).

[38] K. Banaszek and K. Wodkiewicz. Nonlocality of the einstein-podolsky-rosen state in the wigner representation. Phys. Rev. A 58, 4345 (1998).

[39] K. Banaszek and K. Wodkiewicz. Nonlocality of the einstein-podolsky-rosen

state in the phase space. Acta Phys. Slovaca 49, 491 (1999).

[40] M. Revzen, P. A. Mello, A. Mann, and L. M. Johansen. Bell’s inequality violation with non-negative wigner functions. Phys. Rev. A 71, 022103 (2005). Bibliography 103

[41] H. Jeong, W. Son, M. S. Kim, D. Ahn, and C.ˇ Brukner. Quantum nonlocality test for continuous-variable states with dichotomic observables. Phys. Rev. A 67, 012106 (2003).

[42] S.-W. Lee, H. Jeong, and D. Jaksch. Testing quantum non-locality by generalized quasi-probability functions. Phys. Rev. A 80, 022104 (2009).

[43] T. Kiss, U. Herzog, and U. Leonhardt. Compensation of losses in photodetec-

tion and in quantum-state measurements. Phys. Rev. A 52, 2433 (1995).

[44] A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homodyne tomography. J. Opt. B: Quantum Semiclass. Opt. 6, s556 (2004).

[45] M. Curty, M. Lewenstein, and N. Lutkenhaus. Entanglement as a precondition for secure quantum key distribution. Phys. Rev. Lett. 92, 217903 (2004).

[46] Y. W. Cheong, S.-W. Lee, J. Lee, and H.-W. Lee. Entanglement puriﬁcation for high dimensional multipartite systems. Phys. Rev. A 76, 042314 (2007).

[47] F. T. Hioe and J. H. Eberly. N-level coherence vector and higher conservation laws in quantum optics and quantum mechanics. Phys. Rev. Lett. 47, 838 (1981).

[48] D. Gottesman, A. Kitaev, and J. Preskill. Multipath entanglement of two photons. Phys. Rev. A 64, 012310 (2001).

[49] S. D. Bartlett, H. de Guise, and B. C. Sanders. Quantum encodings in spin

systems and harmonic oscillators. Phys. Rev. A 65, 052316 (2002).

[50] S. Sachdev. Quantum Phase Transitions. Cambridge University Press, Cam- bridge, (1999). 104 Bibliography

[51] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108 (1998).

[52] A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni. Multipath entanglement of two photons. Phys. Rev. Lett. 102, 153902 (2009).

[53] J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat. Generation of hyperentangled photon pairs. Phys. Rev. Lett. 95, 260501 (2005).

[54] C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter. Complete deterministic linear optics bell state analysis. Phys. Rev. Lett. 96, 190501 (2006).

[55] G. Vallone, E. Pomarico, F. De Martini, P. Mataloni, and M. Barbieri. Experi- mental realization of polarization qutrits from nonmaximally entangled states. Phys. Rev. A 76, 012319 (2007).

[56] E. V. Moreva, G. A. Maslennikov, S. S. Straupe, and S. P. Kulik. Realization of four-level qudits using biphotons. Phys. Rev. Lett. 97, 023602 (2006).

[57] L. Neves, G. Lima, J. G. A. G´omez,C. H. Monken, C. Saavedra, and S. P´adua. Generation of entangled states of qudits using twin photons. Phys. Rev. Lett. 94, 100501 (2005).

[58] J. T. Barreiro, T.-C. Wei, and P. G. Kwiat. Beating the channel capacity limit for linear photonic superdense coding. Nature Physics 4, 282 (2008).

[59] G. Vallone, E. Pomarico, P. Mataloni, F. De Martini, and V. Berardi. Re- alization and characterization of a two-photon four-qubit linear cluster state.

Phys. Rev. Lett. 98, 180502 (2007).

[60] K. Chen, C.-M. Li, Q. Zhang, Y.-A. Chen, A. Goebel, S. Chen, A. Mair, and

J.-W. Pan. Experimental realization of one-way quantum computing with

two-photon four-qubit cluster states. Phys. Rev. Lett. 99, 120503 (2007). Bibliography 105

[61] G. Vallone, E. Pomarico, F. De Martini, and Paolo Mataloni. Active one-way quantum computation with two-photon four-qubit cluster states. Phys. Rev. Lett. 100, 160502 (2008).

[62] D. Bohm. A suggested interpretation of the quantum theory in terms of “hidden variables” i. Phys. Rev. 85, 166 (1952).

[63] D. Bohm. A suggested interpretation of the quantum theory in terms of

“hidden variables” ii. Phys. Rev. 85, 180 (1952).

[64] H. Everett. The Many-Worlds Interpretation of Quantum Mechanics. Prince- ton University Press, (1973).

[65] J. F. Clauser and M. A. Horne. Experimental consequences of objective local theories. Phys. Rev. D 10, 526 (1974).

[66] S. J. Mermin and J. F. Clauser. Experimental test of local hidden-variable theories. Phys. Rev. Lett. 28, 938 (1972).

[67] A. Aspect, P. Grangier, and G. Roger. Experimental tests of realistic local theories via bell’s theorem. Phys. Rev. Lett. 47, 460 (1981).

[68] A. Aspect, P. Grangier, and G. Roger. Experimental test of bell’s inequalities using time- varying analyzers. Phys. Rev. Lett. 49, 1804 (1982).

[69] W. Tittel, J. Brendel, H. Zbinden, and N. Gisin. Violation of bell inequalities by photons more than 10 km apart. Phys. Rev. Lett. 81, 3563 (1998).

[70] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger. Violation

of bell’s inequality under strict einstein locality conditions. Phys. Rev. Lett.

81, 5039 (1998). 106 Bibliography

[71] A. Garg and N. D. Mermin. Detector ineﬃciencies in the einstein-podolsky- rosen experiment. Phys. Rev. D 35, 3831 (1987).

[72] S. Pironio. Violations of bell inequalities as lower bounds on the communication cost of nonlocal correlations. Phys. Rev. A 68, 062102 (2003).

[73] A. Cabello and J.-A.˚ Larsson. Minimum detection eﬃciency for a loophole-free

atom-photon bell experiment. Phys. Rev. Lett. 98, 220402 (2007).

[74] N. Brunner, N. Gisin, V. Scarani, and C. Simon. Detection loophole in asym-

metric bell experiments. Phys. Rev. Lett. 98, 220403 (2007).

[75] A. Cabello, D. Rodriguez, and I. Villanueva. Necessary and suﬃcient detection eﬃciency for the mermin inequalities. Phys. Rev. Lett. 101, 120402 (2008).

[76] S. Massar. Nonlocality, closing the detection loophole, and communication complexity. Phys. Rev. A 65, 032121 (2003).

[77] A. Cabello. Bipartite bell inequalities for hyperentangled states. Phys. Rev. Lett. 97, 140406 (2006).

[78] X. Ma, T. Moroder, and N. L¨utkenhaus. Quantum key distribution secure against the eﬃciency loophole. arXiv:0812.4301 [quant-ph] (2008).

[79] V. Scarani and C. Kurtsiefer. The black paper of quantum cryptography: real implementation problems. arXiv:0906.4547 [quant-ph] (2009).

[80] R. F. Werner and M. M. Wolf. All-multipartite bell-correlation inequalities for two dichotomic observables per site. Phys. Rev. A 64, 032112 (2001).

[81] A. Peres. Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996). Bibliography 107

[82] P. Horodecki. Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333 (1997).

[83] T. Mor P. W. Shor J. A. Smolin C. H. Bennett, D. P. DiVincenzo and B. M. Terhal. Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 5385 (1999).

[84] C.ˇ Brukner, M. S. Kim, J.-W. Pan, and A. Zeilinger. Correspondence between

continuous-variable and discrete quantum systems of arbitrary dimensions. Phys. Rev. A 68, 062105 (2003).

[85] R. Rasussendorf and Hans J. Briegel. A one-way quantum computer. Phys. Rev. Lett. 86, 5188 (2001).

[86] V. Vedral. Introduction to Quantum Information Science. Oxford University Press, (2006).

[87] R. F. Werner. Quantum states with einstein-podolsky-rosen correlations ad- mitting a hidden-variable model. Phys. Rev. A 40, 4277 (1989).

[88] P. Hyllus, O. G¨uhne,D. Bruß, and M. Lewenstein. Relations between entanglement witnesses and bell inequalities. Phys. Rev. A 72, 012321 (2005).

[89] H. Moya-Cessa and P. L. Knight. Series representation of quantum-ﬁeld quasiprobabilities. Phys. Rev. A 48, 2479 (1993).

[90] U. Leonhardt and H. Paul. Realistic optical homodyne measurements and

quasiprobability distributions. Phys. Rev. A 48, 4598 (1993).

[91] K. Banaszek and K. W´odkiewicz.Direct probing of quantum phase space by photon counting. Phys. Rev. Lett. 76, 4344 (1996). 108 Bibliography

[92] K. Banaszek, A. Dragan, K. W´odkiewicz,and C. Radzewicz. Direct measurement of optical quasidistribution functions: Multimode theory and homodyne tests of bells inequalities. Phys. Rev. A 66, 043803 (2002).

[93] D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger. Violations of local realism by two entangled n-dimensional systems are stronger than for two qubits. Phys. Rev. Lett. 85, 4418 (2000).

[94] T. Durt, D. Kaszlikowski, and M. Zukowski. Violations of local realism with quantum systems described by n-dimensional hilbert spaces up to n=16. Phys.

Rev. A 64, 024101 (2001).

[95] W. Laskowski, T. Paterek, M. Zukowski, and C.ˇ Brukner. Tight multipartite

bell’s inequalities involving many measurement settings. Phys. Rev. Lett. 93, 200401 (2004).

[96] J. Lee, S.-W. Lee, and M. S. Kim. Greenberger-horne-zeilinger nonlocality in arbitrary even dimensions. Phys. Rev. A 73, 032316 (2006).

[97] S. Zohren and R. D. Gill. Maximal violation of the cglmp inequality for inﬁnite dimensional states. Phys. Rev. Lett. 100, 120406 (2008).

[98] C.ˇ Brukner, M. Zukowski, and A. Zeilinger. Quantum communication complexity protocol with two entangled qutrits. Phys. Rev. Lett. 89, 197901 (2002).

[99] W. Son, J. Lee, and M. S. Kim. D-outcome measurement for a non-locality test. J. Phys. A 37, 11897 (2004).

[100] A. Peres. All the bell inequalities. Foundations of Physics 29, 589 (1999).

[101] B. S. Cirel’son. Quantum generalizations of bell’s inequality. Lett. Math. Phys. 4, 93 (1980). Bibliography 109

[102] G. M. Ziegler. Lectures on Polytopes. Springer-Berlag, (1995).

[103] M. D. Reid and P. D. Drummmond. Quantum correlations of phase in non- degenerate parametric oscillation. Phys. Rev. Lett. 60, 2731 (1988).

[104] Z.-B. Chen, J.-W. Pan, G. Hou, and Y.-D. Zhang. Maximal violation of bells inequalities for continuous variable systems. Phys. Rev. Lett. 88, 040406 (2002).

[105] J. Koﬂer and C.ˇ Brukner. Classical world arising out of quantum physics under

the restriction of coarse-grained measurements. Phys. Rev. Lett. 99, 180403 (2007).

[106] D. T. Pegg and S. M. Barnett. Phase properties of the quantized single-mode electromagnetic ﬁeld. Phys. Rev. A 39, 1665 (1989).

[107] D. T. Pegg, Lee S. Phillips, and S. M. Barnett. Optical state truncation by projection synthesis. Phys. Rev. Lett. 81, 1604 (1998).

[108] A. Miranowicz and W. Leonski. Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers. J. Phys. B: At. Mol. Opt. Phys. 39, 1683 (2006).

[109] Se-Wan Ji, Jinhyoung Lee, James Lim, Koji Nagata, and Hai-Woong Lee. Multisetting bell inequality for qudits. Phys. Rev. A 78, 052103 (2008).

[110] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller. Quantum state reconstruction of the single-photon fock state. Phys. Rev. Lett. 87, 050402 (2001).

[111] Rodney Loudon. The Quantum Theory of Light. Oxford University Press, Oxford, third edition, (2000). 110 Bibliography

[112] U. Leonhardt. Measuring the Quantu State of Light. Cambridge University Press, Cambridge, (1997).

[113] S. M. Tan, D. F. Walls, and M. J. Collett. Nonlocality of a single photon. Phys. Rev. Lett. 66, 252 (1991).

[114] L. Hardy. Nonlocality of a single photon revisited. Phys. Rev. Lett. 73, 2279 (1994).

[115] B. Hessmo, P. Usachev, H. Heydari, and G. Bj¨ork.Experimental demonstra-

tion of single photon nonlocality. Phys. Rev. Lett. 92, 180401 (2004).

[116] J. Dunningham and V. Vedral. Nonlocality of a single particle. Phys. Rev. Lett. 99, 180404 (2007).

[117] J.-A. Larsson. Bells inequality and detector ineﬃciency. Phys. Rev. A 57, 3304 (1998).

[118] H. Jeong, J. Lee, and M. S. Kim. Dynamics of nonlocality for a two-mode squeezed state in a thermal environment. Phys. Rev. A 61, 052101 (2000).

[119] S.-W. Lee, H. Jeong, and D. Jaksch. Witenessing entanglement in phase space using ineﬃcient detectors. arXiv:0904.1058 [quant-ph] (2009).

[120] S. Wallentowitz and W. Vogel. Unbalanced homodyning for quantum state measurements. Phys. Rev. A 53, 4528 (1996).

[121] A. Divochiy et al. Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths. Nature Photon. 2, 302 (2008).

[122] M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walm-

sley, and C. Silberhorn. Photon number statistics of multimode parametric

down-conversion. Phys. Rev. Lett. 101, 053601 (2008). Bibliography 111

[123] J. Wenger, R. Tualle-Brouri, and P. Grangier. Non-gaussian statistics from individual pulses of squeezed light. Phys. Rev. Lett. 92, 153601 (2004).

[124] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier. Generating optical schrodinger kittens for quantum information processing. Science 312, 83 (2006).

[125] K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki. Photon subtracted

squeezed states generated with periodically poled ktiopo4. Optics Express 15, 3568 (2007).

[126] H. Jeong. Testing bell inequalities with photon-subtracted gaussian states. Phys. Rev. A 78, 042101 (2008).

[127] A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier. Generation of optical ’schrodinger cats’ from photon number states. Nature 448, 784 (2007).

[128] H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki. Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction. Phys. Rev. Lett. 101, 233605 (2008).

[129] K. Vogel and H. Risken. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev.

A 40, 2847 (2008).

[130] D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani. Measurement of the wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. Phys. Rev. Lett. 70, 1244 (1993).

[131] A. I. Lvovsky and M. G. Raymer. Continuous-variable optical quantum-state tomography. Rev. Mod. Phys. 81, 299 (2009). 112 Bibliography

[132] J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, Ch. Silber- horn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley. Tomography of quantum detectors. Nature Physics 5, 27 (2009).

[133] L. G. Lutterbach and L. Davidovich. Method for direct measurement of the

wigner function in cavity qed and ion traps. Phys. Rev. Lett. 78, 2547 (1997).