Bell tests in physical systems - Department of Physics at … · 2010. 7. 8. · Bell tests in physical systems Seung-Woo Lee, St. Hugh’s College, Oxford Trinity Term, 2009 Abstract

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 process-

ing. Both quantum non-locality and entanglement can be effectively detected by

testing Bell inequalities. Thus, finding Bell inequalities applicable to realistic phys-

ical 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 the-

sis, 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 bipar-

tite 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 effi-

ciency as low as 40%. This approach enhances the possibility of witnessing entan-

glement 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 entangle-

ment 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 clas-

sical 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 incom-

pleteness 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 pos-

sible method for testing the validity of quantum mechanics against local realism. It

1The concept of entanglement was also introduced by E. Schrodinger [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 entan-

glement 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 ob-

serve quantum features in the macroscopic world. Macroscopic systems are complex

systems that are generally governed by classical physics, and thus observing quan-

tum 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 pro-

cessing, 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 informa-

tion 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 sys-

tems. 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 quan-

tum 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 ad-

dition, 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 for-

malism. 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 state ρ 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 funda-

mental 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 Wodkiewicz showed

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 inves-

tigating 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 en-

tanglement 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 ques-

tions 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 poly-

tope representation of Bell’s theorem that is useful to extend Bell inequalities

to complex systems. Then, we compare entanglement witnesses, which dis-

criminate 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 in-

vestigate 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 pa-

rameter 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 phys-

ical 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 poly-

tope 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, |ψ0〉 and |ψ1〉. An ar-

9

10 Basic Concepts

bitrary superposition of two basis vectors |ψ〉 = α |ψ0〉 + β |ψ1〉 is also a possible

state of the system, where their amplitudes α and β are arbitrary complex num-

bers. The normalisation condition for the state |ψ〉 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 |ψ0〉 and |ψ1〉 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 |ψ0〉 = | 0〉 and |ψ1〉 = | 1〉. The computational basis can be

arbitrarily chosen by transformation in 2-dimensional Hilbert space. For example,

|+〉 = (| 0〉 + | 1〉)/√2 and | −〉 = (| 0〉 − | 1〉)/√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,

a mixed state is given as a statistical ensemble of pure states ρ =∑

i pi |ψi〉〈ψi |where pi is the probability that the system in the state |ψi〉. 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

system is given as a density matrix ρ = 12(1 + ~a · ~σ) where ~a = (a1, a2, a3) is a real

vector and ~σ = (σx, σy, σz) 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, {| 0〉 , | 1〉 , ..., | d− 1〉}.All pure states in d-dimensional Hilbert space can be represented using the basis

2.1. High-dimensional quantum systems 11

vectors

|ψ〉 =d−1∑

k=0

αk | k〉 , (2.1)

where the complex numbers αk satisfy the condition∑

k |αk|2 = 1, and |αk|2 is the

probability that the system is found to be in the state | k〉. 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

ρ =1

d(1 +

d2−1∑i=0

aiλi), (2.2)

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

in d-dimensional Hilbert space and the vector ~a = (a0, ..., ad2−1) is a d2-dimensional

real vector. This is the so called generalised Bloch representation of d-dimensional

quantum states [47]. The generalised Pauli operators, λi, in d-dimensional Hilbert

space, Hd, are given by [48, 49]

(Xd)a(Zd)

b, a, b ∈ 0, 1, ...d− 1, (2.3)

where Xd and Yd are defined as

Xd =d−1∑

k=0

| k + 1〉〈k | , Zd = exp

(2πi

dN

). (2.4)

Here N =∑

k k | k〉〈k | is the number operator, and Xd and Yd transform the d-

12 Basic Concepts

dimensional computational basis by

Xd | k〉 = | k + 1〉 , Zd | k〉 = exp

(2πi

dk

)| k〉 . (2.5)

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 im-

plementations, 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 free-

dom 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 mea-

surement 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 inequal-

ities 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 indepen-

dently. 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 out-

comes ±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)

=∑

a1,a2,b1,b2

P (a1, a2, b1, b2)(a1b1 + a1b2 + a2b1 − a2b2)

=∑

a1,a2,b1,b2

P (a1, a2, b1, b2)a1b1 + P (a1, a2, b1, b2)a1b2

+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

|ψ〉 =1√2(| 0〉 | 1〉 − | 1〉 | 0〉). (2.9)

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


value for the quantum state in Eq. (2.9) is given by

〈A1B1〉+ 〈A1B2〉+ 〈A2B1〉 − 〈A2B2〉 = 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 in-

equality given in Eq. (2.11).

1There are several other interpretations which explain this result e.g. the non-local hiddenvariable model by David Bohm [62, 63] and many world interpretations [64]. Details of theseinterpretations 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 sys-

tems with measurements of 2-dimensional outcomes. For example, one can consider

the Stern-Gerlach measurement for entangled spin pairs or the polarisation mea-

surement 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 par-

ties. 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

P (+ + |ab) =N++

N, (2.14)

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)


Figure 2.1. Optical setup for the Bell test (CHSH inequality). Entangledphotons generated from the source are distributed between two separateparties. Photons go through a beam splitter and are measured at the detectorwith 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 statis-

tical 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

|ψ〉 =1√2(|+〉 |+〉+ | −〉 |−〉). (2.17)

The measurement basis are varying by rotation from a fixed direction associated

with the standard basis |+〉 and | −〉 in a plane. Therefore, the measurement basis

for Alice are written as | θa, +〉 = cos θa |+〉 − sin θa | −〉 and | θa,−〉 = sin θa |+〉 +

cos θa | −〉, and for Bob as |φb, +〉 = cos φb |+〉−sin φb | −〉 and |φb, +〉 = sin φb |+〉+cos φb | −〉. From Eq. (2.14) we can measure the joint probability that has the

expectation value as P (+ + |ab) = | 〈ψ | | θa, +〉 |φb, +〉 |2 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 per-

formed 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 pho-

tons emitted by calcium atoms. Later, a test of the CHSH inequality was imple-

mented 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 typeproposed by Clauser and Horne [65] and original Bell inequality [1].


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 parti-

cles. When the distance between two local measurements is small it conflicts

with the assumption of no communication between one observer’s measure-

ments 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 mea-

surement 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]

|Ecoinc1,1 + Ecoinc

1,2 + Ecoinc2,1 − Ecoinc

2,2 | ≤ 4

η− 2, (2.19)

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 assump-

tion 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 quan-

tum 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.


Figure 2.2. Schematic diagram of the ploytope in the joint probability spaceor alternatively in the correlation function space. The inside region of thepolytope represents the accessible region of local-realistic (LR) theories andoutside region contains the region of quantum mechanics (QM). Each facetof the polytope corresponds to a tight Bell inequality. A non-tight Bellinequality including bias at the boundary is represented by a line deviatingfrom 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 mea-

surement 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 geo-

metric 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 sub-

systems of a composite body. Let us consider a state that is in a Hilbert space of

two qubits Ha ⊗Hb,

|Ψ〉ab =1√2(| 0〉a | 1〉b + | 1〉a | 0〉b). (2.20)

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

|Ψ〉ab = |ψ〉a |ψ′〉b , (2.21)

where |ψ〉a and |ψ′〉b are the states defined in the Hilbert space Ha and Hb respec-

tively. 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

density operators ρi1 and ρi

2:

ρab =∑

i

piρia ⊗ ρi

b,∑

i

pi = 1 (2.22)

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 sepa-

rated 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 sys-

tems is already well understood. The entanglement criterion in a simple model is

clearly defined. A quantum state with density matrix ρ 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 quan-

tum 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 quan-

tum 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 entanglementmodel.

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

ρd can not be represented by a direct sum of any lower dimensional states, i.e.

the density matrix ρd cannot be written as a block-diagonal matrix, ρd 6=⊕

i Ciρi

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

|ψ4〉 = | 00〉 + | 11〉 + | 22〉 + | 33〉 and a mixed state ρ = |ψ2〉〈ψ2 | ⊕ |ψ′2〉〈ψ′2 |where |ψ2〉 = | 00〉 + | 11〉 and |ψ′2〉 = | 22〉 + | 33〉 (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 state ρN can not be represented by a product state of any lower-partite

2.3. Quantum entanglement 27

states, i.e. ρN 6= ∑i Ciρ

iN1⊗ ρi

N2where 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 entangle-

ment 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 entan-

glement can play different roles in quantum information processing.

2.3.3 Entanglement Witness

An entanglement witness (EW) is an observable which reacts differently to entan-

gled 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 entan-

gled and separable states. For example, let us consider a Hermitian operator Wwhose average expectation value by separable states is bounded by a maximal value

Wmax:

|〈W〉| ≤ Wmax. (2.23)

If an expectation value of a quantum state ψ exceeds this bound |〈W〉ψ| > Wmax,

we can conclude that the quantum state ψ is entangled. Note that if there is a

separable state satisfying |〈W〉| = 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 oper-

ator 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 Bellinequality (BI). EW detect entangled states including some states which donot violate the local realistic (LR) theories, while BI detect non-local stateswhich 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 sepa-

rable 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

linear combination σAB = pρAB1 + (1− p)ρAB

2 of two separable states, ρAB1 and ρAB

2

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 vari-

able states as they provide insights into the boundaries between quantum and clas-

sical 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]

W (α; s) =2

π(1− s)Tr[ρΠ(α; s)], (2.24)

where

Π(α; s) =∞∑

n=0

(s + 1

s− 1

)n

|α, n〉〈α, n | , (2.25)

and |α, n〉 is the number state displaced by the complex variable α in phase space.

It is produced by applying the Glauber displacement operator D(α) to the number

state |n〉. 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

W (α; s) =2

π|s|∫

d2β W (β) exp

(−2|α− β|2

|s|)

. (2.26)

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 func-

tions [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, β′) + Q(α′, β)−Q(α′, β′)]− π[Q(α) + Q(β)] ≤ 0, (2.27)

2.4. Phase space representations 31

-20

2Α

-2

0

2

Β

0

1Π2

��4

W

-20

2Α

-1.5-1-0.5 0 0.5 1 1.5 2Α

-1.5

-1

-0.5

0

0.5

1

1.5

2

Β

BW-type Hr=1L

-

+ +

+

-20

2Α

-2

0

2

Β

0

1Π2

��4

W

-20

2Α

-1.5-1-0.5 0 0.5 1 1.5 2Α

-1.5

-1

-0.5

0

0.5

1

1.5

2

Β

BW-type Hr=0.2L

-

+ +

+

HaL r=0.2

HbL r=1.0

Figure 2.5. Plot of the two mode Wigner function of the TMSS with differentsqueezing 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 isgiven in the contour plots for each squeezing rate.

32 Basic Concepts

where Q(α, β) = (1/π2)〈α, β|ρ|α, β〉 is the two-mode Q-function for a quantum state

ρ and Q(α) is the marginal distribution. Here α, α′, β, and β′ 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

4|W (α, β) + W (α, β′) + W (α′, β)−W (α′, β′)| ≤ 2, (2.28)

where α, α′, β, and β′ 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

at α = −α′ = 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 mea-

surement between two separated systems [1, 8, 37]. It was shown that these con-

straints, 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 me-

chanics 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 di-

mensionality 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 sim-

ilar 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 rep-

resented 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 proba-

bilities. 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 representa-

tion 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)B∗b (λ). Based on the local hidden-variable description, the correlation func-

tion is the average over many trials of the experiment as

Cab =

∫dλ ρ(λ)Aa(λ)B∗

b (λ), (3.1)

1Instead of the complex eigenvalues, one may consider real eigenvalues, but then one has toemploy a rather complicated form of the correlation weights to obtain Bell inequalities equivalentto the ones derived in the present paper.


where ρ(λ) is the statistical distribution of the hidden variables λ with the properties

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

Cab =d−1∑

k,l=0

ωk−lP (Aa = k, Bb = l), (3.2)

where ωk−l 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:

∑

k

µk,l = 0, ∀l and∑

l

µk,l = 0. ∀k

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


as ωα where α ≡ k − l ∈ {0, 1, ..., d− 1} and it obeys∑

α ωα = 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

C(n)ab =

∫dλ ρ(λ) (Aa(λ)B∗

b (λ))n

=d−1∑

k,l=0

ωn(k−l)P (Aa = k, Bb = l)

=d−1∑α=0

ωnαP (Aa.= Bb + α), (3.3)

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

C(d+n)ab = C

(n)ab 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

B =∑

a,b

d−1∑n=0

fab(n)C(n)ab , (3.4)

where coefficients fab(n) are functions of the correlation order n and the measure-

ment configurations a, b. They determine the constraint of local-realistic theories

with a certain upper bound and its violation by quantum mechanics will be inves-

tigated in Sec. 3.3. The zero-th order correlation has no meaning as it simply shift

the value of B by a constant and is chosen to vanish, i.e.,∑

a,b fab(0) = 0. The Bell


function in Eq. (4.1) is rewritten in terms of the joint probabilities given in Eq. (3.3),

as

B =∑

a,b

d−1∑α=0

εab(α)P (Aa.= Bb + α), (3.5)

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

We note that the coefficients εab(α) are obtained by the Fourier transformation

of fab(n) based on the kernel of a given correlation weight as

εab(α) =d−1∑n=0

fab(n)ωnα, (3.6)

fab(n) =1

d

d−1∑α=0

εab(α)ω−nα. (3.7)

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 gener-

alised structure, including previously proposed Bell inequalities in bipartite systems.

In the case of d = 2, CHSH-type inequalities can be obtained with coefficients as

f(1) = (1, 1,−1, 1) and εab(α) = fab(1)(−1)α where α ∈ {0, 1}. For arbitrary d-

dimensional systems, the two types of Bell inequalities, CGLMP and SLK, are rep-

resented 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


coefficients are given as

ε11(α) = 1− 2α

d− 1, ε12(α) = −1 +

2 ˙(α− 1)

d− 1,

ε21(α) = −1 +2α

d− 1, ε22(α) = 1− 2α

d− 1, (3.8)

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

f11(n 6= 0) =2

d− 1

(1

1− ω−n

),

f12(n 6= 0) =2

d− 1

(1

1− ωn

),

f21(n 6= 0) =2

d− 1

(1

ω−n − 1

),

f22(n 6= 0) =2

d− 1

(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 char-

acteristics of the Bell inequality.

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

coefficients are given by

f11(n 6= 0) = (ωnδ + ω(n−d)δ)/4,

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

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

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

fab(n = 0) = 0 ∀a, b, (3.10)


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

ε11(α) = S(δ + α),

ε12(α) = S(δ + η1 + α),

ε21(α) = S(δ + η2 + α),

ε22(α) = S(δ + η1 + η2 + α), (3.11)

where

S(x 6= 0) =1

4(cot

π

dx sin 2πx− cos 2πx− 1),

S(x = 0) =1

2(d− 1). (3.12)

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 de-

terministic 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

BmaxLR = max

αab

[∑

a,b

εab(αab)|α11 + α22.= α12 + α21]. (3.14)

The quantum expectation value for arbitrary quantum state ρ is written by

BQM(ρ) = Tr(Bρ)

=∑

a,b

d−1∑n=0

fab(n)Tr(C(n)ab ρ), (3.15)

where B is the Bell operator defined by replacing the correlation function in Eq. (4.1)

with correlation operator,

C(n)ab =

∑

k,l

ωn(k−l)Pa ⊗ Pb (3.16)

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

tation value of any quantum state exceeds the local realistic bound BmaxLR , i.e., the

Bell inequality is violated by quantum mechanics, the composite system is entan-

gled and shows nonlocal quantum correlations. The maximal quantum expectation

is called quantum maximum BmaxQM 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

can obtain the quantum maximum, BmaxQM = 2

√2, which is in agreement with the


Cirel’son bound [101].

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

ρ = p|ψm〉〈ψm|+ (1− p)1

d2(3.17)

where p is the probability that the state is unaffected by noise. Then, the minimal

probability for the violation is pmin = BmaxLR /BQM(|ψm〉). We now investigate the

violation of two types of Bell inequalities, CGLMP and SLK, and compare them as

follows.

CGLMP inequality - The local-realistic upper bound, BmaxLR = 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-

dimensional system, the quantum maximum is BmaxQM ' 2.9149 for the non-maximally

entangled state,

1√n

(|00〉+ γ|11〉+ |22〉), (3.18)

where γ ' 0.7923 and n = 2 + γ2. It is higher than the expectation by maximally

entangled state, B(|ψm〉) ' 2.8729. The expectation of the CGLMP is shown in

Fig. 3.1 against the entanglement degree γ, once the local measurements are cho-

sen 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 vary-

ing δ and η1,2. All the variants of the SLK have the same quantum maximum d− 1

for a maximally entangled state |ψm〉, BmaxQM = BQM(|ψm〉), as we prove as follows.


0 0.5 1 1.5 2Γ

2

2.2

2.4

2.6

2.8

3

BQ

M

HaL

0 0.5 1 1.5 2Γ

1.2

1.4

1.6

1.8

2

2.2

BQ

M

HbL

Figure 3.1. The expectation value of (a) the CGLMP and (b) the optimalSLK for d = 3 as varying the value γ for the quantum state, (1/

√n)(|00〉+

γ|11〉+|22〉) where n = 2+γ2. The SLK takes the maximum 2 when the stateis maximally entangled (γ = 1), whereas the CGLMP takes the maximum2.9149 for a partially entangled state (γ ' 0.7923). The dashed lines indicatethe local-realistic upper bounds.

The Bell operator of the SLK variants can be written as

BS =1

2

d−1∑n=1

α · β, (3.19)

where α = (A†n1 , A†n

2 )T and β = Uβ with β = (Bn1 , Bn

2 )T and U is a 2 × 2 unitary

matrix with elements,

U11 = (ωnδ + ω(n−d)δ)/2,

U12 = (ωn(δ+η1) + ω(n−d)(δ+η1))/2,

U21 = (ωn(δ+η2) + ω(n−d)(δ+η2))/2,

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

where η1,2 ∈ {1/2,−1/2}.


The expectation of the Bell operator is given by

1

2

∣∣∣∣∣d−1∑n=1

〈ψ |α · β |ψ〉∣∣∣∣∣ ≤

1

2

d−1∑n=1

∣∣∣〈ψ |α · β |ψ〉∣∣∣

≤ 1

2

d−1∑n=1

(∣∣∣〈ψ |α1 ⊗ β1 |ψ〉∣∣∣ +

∣∣∣〈ψ |α2 ⊗ β2 |ψ〉∣∣∣)

≤ 1√2

d−1∑n=1

√∣∣∣〈ψ |α1 ⊗ β1 |ψ〉∣∣∣2

+∣∣∣〈ψ |α2 ⊗ β2 |ψ〉

∣∣∣2

=1√2

d−1∑n=1

√√√√2∑

i=1

∣∣∣〈ψ |αi ⊗ βi |ψ〉∣∣∣2

, (3.21)

where we consecutively used the triangle inequality and the arithmetic-geometric

means inequality, 2|a||b| ≤ |a|2 + |b|2. Note that

2∑i=1

∣∣∣〈ψ |αi ⊗ βi |ψ〉∣∣∣2

≤2∑

i=1

〈ψ |(α†i ⊗ β†i

)(αi ⊗ βi

)|ψ〉

=2∑

i=1

〈ψ |1⊗ β†i βi |ψ〉 , (3.22)

where we used that αi is unitary. Here the above inequality is obtained by reasoning

that Q ≡ 1 − |ψ〉〈ψ | is a positive operator as 〈φ | Q |φ〉 = 1 − | 〈φ |ψ| |〉2 ≥ 0

for any |φ〉, and | 〈ψ | C |ψ〉 |2 = 〈ψ | C† |ψ〉〈ψ | C |ψ〉 = 〈ψ | C†(1 − Q)C |ψ〉 =

〈ψ | C†C |ψ〉 − 〈ψ | Q |ψ〉 ≤ 〈ψ | C†C |ψ〉, where C ≡ αi ⊗ βi. Since∑

i β†i βi =

∑jk

∑i U

∗ijUikβ

†jβk =

∑jk δjkβ

†jβk =

∑i β

†i βi = 21, it is clear that

2∑i=1

∣∣∣〈ψ |αi ⊗ βi |ψ〉∣∣∣2

≤ 〈ψ |1⊗2∑

i=1

β†i βi |ψ〉 = 2. (3.23)

Hence the upper bound for all variants of the SLK is

∣∣∣〈ψ | BS |ψ〉∣∣∣ ≤ d− 1. (3.24)


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

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

periodicity, BmaxLR (δ) = Bmax

LR (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

minδ

[BmaxLR (δ)] =

1

4(3 cot

π

4d− cot

3π

4d)− 1, (3.25)

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

ε) = BmaxLR (1/4 − ε) for 0 < ε ≤ 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 maxi-

mally entangled states as being consistent with our intuition whereas the CGLMP

is maximally violated by non-maximally entangled states. We remark that the coef-


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 poly-

tope 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 predeter-

mined measurement outcome called local-realistic configuration. All interior points

of the polytope are given by the convex combination of generators and they repre-

sent 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 =

k,Bb = l) where k, l = 0, 1, ..., d − 1 and a, b = 1, 2, can be arranged in a 4d2-

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, B1〉 ⊕ |A1, B2〉 ⊕ |A2, B1〉 ⊕ |A2, B2〉 (3.26)

where |n〉 stands for |n mod d〉 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 inde-

pendent. 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

the local-realistic upper bound, BmaxLR . Thus the first condition T.1 is fulfilled. Sec-

ond, we examine whether there are h linearly independent generators which give the

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

αab, the generators (3.26) become

|A,A− α11〉 ⊕ |A,A− α12〉 ⊕ |A− α12 + α22, A− α11〉

⊕|A− α11 + α21, A− α12〉, (3.27)

where A ∈ {0, 1, ..., d − 1} and the number of linearly independent generators is


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,A〉 ⊕ |A,A〉 ⊕ |A− 1, A〉 ⊕ |A− 1, A〉

|A,A〉 ⊕ |A,A〉 ⊕ |A, A〉 ⊕ |A,A〉

|A,A〉 ⊕ |A,A− 1〉 ⊕ |A− 1, A〉 ⊕ |A− 1, A− 1〉

|A,A + 1〉 ⊕ |A,A〉 ⊕ |A,A + 1〉 ⊕ |A,A〉 (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 in-

equalities 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 the-

ories,” 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 concep-

tual foundations of quantum theory and also for investigating quantum entangle-

ment. 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 out-

comes 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 exam-

ple, 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 quan-

tum 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 inequal-

ities 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 inequal-

ities for arbitrary d-dimensional bipartite systems [17]. Suppose that two parties,

Alice and Bob, independently choose one of two observables A1 or A2 for Alice, and

B1 or B2 for Bob. Possible measurement outcomes are denoted by ka for Aa and la

for Bb with a, b = 1, 2, where ka, lb ∈ V ≡ {0, 1, ..., d − 1}. A general Bell function

is then written as [17]

B =2∑

a,b=1

d−1∑

ka,lb=0

εab(ka, lb)Pab(ka, lb), (4.1)

where Pab(ka, lb) is the joint probability for outcomes ka and lb, and εab(ka, lb) are

their weighting coefficients (here assumed to be real). For local-realistic (LR) sys-

tems 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

BmaxLR = max

C

{ 2∑

a,b=1

εab(ka, lb)

}, (4.2)

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

the bound BmaxLR . The flexibility in choosing the coefficients εab(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 εab(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


local-realistic from non-local quantum regions in correlation or joint probabil-

ity 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

| j〉 with j = 0, · · · , d− 1 in which this state reads |ψmaxd 〉 =

∑d−1j=0 | jj〉 /

√d.

As a general method, one could choose the coefficients εab(ka, lb) freely and ex-

amine 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

ζR(k) =

+1 if outcome k ∈ R,

−1 otherwise,

(4.3)

where R is an arbitrarily chosen subset of all possible outcomes, i.e. R ⊂ V . The

coefficients are then given by

ε11 = ζR1(k1)ζS1(l1), ε12 = ζR1(k1)ζS2(l2),

ε21 = ζR2(k2)ζS1(l1), ε22 = −ζR2(k2)ζS2(l2), (4.4)

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

Eq. (4.2) we find the local-realistic upper bound BmaxLR = 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


are described by 4d2 dimensional linearly independent vectors

G = |k1, l1〉 ⊕ |k1, l2〉 ⊕ |k2, l1〉 ⊕ |k2, l2〉 (4.5)

where |n〉 stands for |n mod d〉 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 acces-

sible to local-realistic theories. We now only consider extremal points associated

with configurations giving the maximal local-realistic value BmaxLR 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

extremal points yielding BmaxLR define a facet of the polytope distinguishing local-

realistic from non-local quantum mechanical correlations [33]. We assume the num-

ber of elements in the sets R1, R2, S1, S2 to be n1, n2, m1, m2, respectively, where

0 ≤ n1, n2,m1,m2 ≤ d− 1. We then count the number of configuration giving BmaxLR

and find

M = d2(d2 − 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 dif-

ferent tight Bell inequalities obtained via the above method. The corresponding

choices of the coefficients εab 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


k

(T1) (T2) (T3)

l

(a)

(b)

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æææææææææææææææææææææææææææææææ

à

à

à

àà

à

ààà

àààà

àààà

àààà

àààà

àààààààà

àààààààà

àààààààà

ì

ì

ì

ì

ììììììììììììììììììììììììììììììììììììììììììììì

2 25 50

2

2.5

3

d

BQ

M

T1æ

T2à

T3ì

Figure 4.1. (a) The coefficient distributions of Bell inequalities (T1), (T2),and (T3) for d = 8 are shown with weighting +1 (white for ε11,ε12,ε21 andgrey for ε22) and -1 (grey for ε11,ε12,ε21 and white for ε22) 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) decreasesbelow the local-realistic upper bound 2 (dashed line).


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

coefficients εab 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 εab 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-

tangled state |ψmaxd 〉 with increasing dimension d. The measurements Aa and Bb

are performed in the bases

| a, k〉 = (1/√

d)d−1∑j=0

ω(k+αa)j | j〉

| b, l〉 = (1/√

d)d−1∑j=0

ω(l+βb)j | j〉 (4.7)

obtained by quantum Fourier transformation and phase shift operations on | j〉. Here

ω = exp(2πi/d), and αa and βb are phase factors differentiating the observables of

each party Aa and Bb, respectively. The expectation value of the Bell function is


then given by

BQM =2∑

a,b=1

d−1∑

k,l=0

εab(k, l)

2d3 sin [πd(k + l + αa + βb)]

. (4.8)

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

B =∑

a,b

∑

k,l

εab(k, l) | a, k〉〈a, k | ⊗ | b, l〉〈b, l | . (4.9)

From Eq. (4.4),

B2 = 41d ⊗ 1d + [P1, P2]⊗ [Q2, Q1] (4.10)

where Pa =∑

k ζRa(k) | a, k〉〈a, k |, Qb =∑

l ζSb(l) | b, l〉〈b, l | and 1d is the d-

dimensional identity operator. Since ‖[P1, P2]‖ ≤ ‖P1P2‖+‖P2P1‖ ≤ 2‖P1‖‖P2‖ = 2

and likewise for ‖[Q2, Q1]‖ where ‖·‖ indicates the supremum norm, we finally obtain

‖B2‖ = ‖41d ⊗ 1d + [P1, P2]⊗ [Q2, Q1]‖

= 4 + ‖[P1, P2]‖‖[Q2, Q1]‖ ≤ 8, (4.11)

or ‖B‖ ≤ 2√

2.

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

(T1) by writing the coefficients as ε11 = ε12 = ε21 = (−1)k+l and ε22 = −(−1)k+l.

For even d we use∑d−1

k,l=0(−1)k+l/2d3 sin [πd(k + l + αa + βb)] = cos π(αa + βb) and


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

as Eab = Πa⊗Πb. with the local measurement Πa =∑d−1

k=0(−1)k | a, k〉〈a, k |. Finally,

we obtain the optimal Bell inequality

B = E11 + E12 + E21 − E22 ≤ 2, (4.13)

where Eab = 〈Eab〉 =∑

k,l(−1)k+lPab(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.


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 in-

stance, be realised by non-degenerate optical parametric amplifiers [103] in photonic

systems. It is written as |TMSS〉 = sechr∑∞

n=0 tanhn r |n, n〉 where r > 0 is the

squeezing parameter and |n〉 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 prob-

lems arise: First, we obtain the local measurement basis by applying the quan-

tum Fourier transformation to |n〉. This is equivalent to the phase states | θ〉 =

(1/√

2π)∑∞

n=0 exp (inθ) |n〉 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 prop-

erties [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 meth-

ods 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

| θ, k〉 =1√

q + 1

q∑n=0

exp (inθk) |n〉 (4.14)

where θk = θ + 2πk/(q + 1) and k = 0, 1, ..., q. Note that q is a cutoff param-

eter (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 E(θ, φ) = Π(θ) ⊗ Π(φ) using the phase parity operator

Π(θ) =∑q

k=0(−1)k | θ, k〉〈θ, k |.We consider a truncated TMSS

|ψq〉 =sechr√

1− tanh2q+2 r

q∑n=0

tanhn r |n, n〉 (4.15)

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

BQM = 〈ψq | E(θ, φ) + E(θ, φ′) + E(θ′, φ)− E(θ′, φ′) |ψq〉

= 4√

2tanh

q+12 r

1 + tanhq+1 r, (4.16)

when θ = 0, θ′ = π/(q +1), φ = −π/(2q +2), and φ′ = 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

r ≥ 1

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

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


0 1

1

2

3

tanh HrL

BQ

M

HaL

1 49 991

2

3

4

5

qr

HbL∆=0.0001

∆=0.001

∆=0.01

Figure 4.2. (a) Expectation values of the Bell operator for truncated TMSSwith 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 inequalityis 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 in-

equality proposed in [38, 19, 39]. There, local measurements are performed in the

basis obtained by applying a Glauber displacement operator D(α) on the num-

ber states |n〉. The measurement basis is written as |α, n〉 = D(α) |n〉 with

α an arbitrary complex number. The displaced number operator is defined as

nα ≡ D(α)nD†(α). Since nα |α, n〉 = n |α, n〉, the correlation operator is given

by E(α, β) = Π(α)⊗ Π(β), where Π(α) =∑∞

n=0(−1)n |α, n〉〈α, 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 viola-

tion 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 in-

equality 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 state ρ 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 pos-

itive. 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 ex-

hibit 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 Wodkiewicz (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]

W (α; s) =2

π(1− s)Tr[ρΠ(α; s)], (5.1)

where Π(α; s) =∑∞

n=0((s+1)/(s− 1))n |α, n〉〈α, n |, and |α, n〉 is the number state

displaced by the complex variable α in phase space. It is produced by applying

the Glauber displacement operator D(α) to the number state |n〉. 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

W (α; s) =2

π|s|∫

d2β W (β) exp

(−2|α− β|2

|s|)

. (5.2)

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 formu-

late 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 in-

cludes 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.


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-

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

A1, A2, B1, B2 can be written as

Aa = O(αa; s), Ba = O(βb; s), for a, b = 1, 2

using a Hermitian operator

O(α; s) =

(1− s)Π(α; s) + s1 if −1 < s ≤ 0,

2Π(α; s)− 1 if s ≤ −1,

(5.3)

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 O(α; s)

are given by its eigenvalues,

λn =

(1− s)( s+1s−1

)n + s if −1 < s ≤ 0,

2( s+1s−1

)n − 1 if s ≤ −1,

(5.4)

and their eigenvectors are the displaced number states. The maximum and minimum

measurement outcomes of O(α; s) for any non-positive s are λmax = 1 and λmin = −1,

respectively. For s = 0 we have O(α; 0) = Π(α; 0) =∑∞

n=0(−1)n |α, n〉〈α, n |, the

displaced parity operator, while for s = −1 we find that O(α;−1) = 2 |α〉〈α | − 1

5.2. Generalised Bell inequalities of quasi-probability functions 69

projects onto the coherent states.

A Bell operator can be constructed using the measurement operators Aa, Bb by

way of a construction similar to the CHSH combination

B = C1,1 + C1,2 + C2,1 − C2,2, (5.5)

where Ca,b = Aa ⊗ Bb is the correlation operator. Since the expectation values

of the local observables are bounded by |〈Aa〉| < 1 and |〈Bb〉| ≤ 1 for any non-

positive s, the expectation value of the Bell operator defined in Eq. (6.5) is bounded

by |〈B〉| ≡ |B| ≤ 2 in local-realistic theories. Note that the expectation value of

Π(α; s) for a given density operator ρ is proportional to the s-parameterised quasi-

probability function [35, 36, 89]

W (α; s) =2

π(1− s)Tr[ρΠ(α; s)]

=2

π(1− s)

∞∑n=0

(s + 1

s− 1

)n

〈α, n | ρ |α, n〉 , (5.6)

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(α; s) are not bounded. We thus obtain the following generalised


Bell inequality

|B|{−1<s≤0} =

∣∣∣∣π2(1− s)4

4[W (α1, β1; s) + W (α1, β2; s)

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

× [W (α1; s) + W (β1; s)] + 2s2

∣∣∣∣ ≤ 2,

(5.7)

|B|{s≤−1} = |π2(1− s)2[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[ρΠ(α; s)⊗Π(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 de-

riving 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 mea-surement is carried out after mixing the incoming field with a coherent state(denoted by | ξ〉 for Alice and | δ〉 for Bob) in a beam splitter (BS) of hightransmissivity T . The photon number detectors (PNDs) have efficiency ηd.

A pair of entangled states generated from a source of correlated photons is dis-

tributed 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 | ξ〉 and | δ〉 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

the displacement operators D(α) and D(β) respectively, where α = ξ√

(1− T )/T

and β = δ√

(1− T )/T [91]. Measurements ((s+1)/(s−1))n with n =∑

n n |n〉〈n |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 ob-

tained by the parity measurements (−1)n (s = 0) and the Q-function by on-off (i.e.

photon presence or absence) measurements (s = −1).


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

Pη(m) =∞∑

n=m

P (n)

(n

m

)(1− η)n−mηm (5.8)

P (n) =∞∑

m=n

Pη(m)

(m

n

)(η − 1)m−nη−m, (5.9)

where(

nm

)= n!/(m!(n−m)!).

For the measurement of ((s+1)/(s−1))n, the measured distribution of outcomes

is given by

∞∑m=0

(s + 1

s− 1

)m

Pη(m) =∞∑

m=0

(s + 1

s− 1

)m ∞∑n=m

P (n)

(n

m

)(1− η)n−mηm

=∞∑

n=0

(1− η)n

∞∑m=0

(s + 1

s− 1

)m

(1− η)−mηm

(n

m

)P (n)

=∞∑

n=0

(1− η)n

∞∑m=0

((s + 1)η

(s− 1)(1− η)

)m (n

m

)P (n)

=∞∑

n=0

(1− η)n

(1 +

(s + 1)η

(s− 1)(1− η)

)n

P (n)

=∞∑

n=0

(1− η +

s + 1

s− 1η

)n

P (n)

=∞∑

n=0

(s′ + 1

s′ − 1

)n

P (n), (5.10)

where we used (i) the equivalence∑∞

n=m P (n)(

nm

)(1−η)n−mηm =

∑∞n=0 P (n)

(nm

)(1−

η)n−mηm, (ii) the relation∑∞

m=0 αm(

nm

)= (1 + α)n, 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

Wη(0; s) =2

π(1− s)

∞∑m=0

(s + 1

s− 1

)m

Pη(m)

=2

π(1− s)

∞∑n=0

(1− η + η

s + 1

s− 1

)n

P (n)

=W

(0;−1−s−η

η

)

η≡ W (0; s′)

η. (5.11)

The s-parameterised quasi-probability function measured by a detector with effi-

ciency η can therefore be identified with the quasi-probability function with pa-

rameter s′ = −(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

〈O(α; s)〉η =

π(1−s)2

2ηW (α; s′) + s if −1 < s ≤ 0,

π(1−s)η

W (α; s′)− 1 if s ≤ −1,

(5.12)

where 〈·〉η 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


of s and η is given by

〈B{−1<s≤0}〉η =π2(1− s)4

4η2

[W (α1, β1;−1− s− η

η) + W (α1, β2;−1− s− η

η)

+W (α2, β1;−1− s− η

η)−W (α2, β2;−1− s− η

η)

]

+πs(1− s)2

η

[W (α1;−1− s− η

η) + W (β1;−1− s− η

η)

]+ 2s2,

(5.13)

〈B{s≤−1}〉η =π2(1− s)2

η2

[W (α1, β1;−1− s− η

η) + W (α1, β2;−1− s− η

η)

+W (α2, β1;−1− s− η

η)−W (α2, β2;−1− s− η

η)

]

−2π(1− s)

η

[W (α1;−1− s− η

η) + W (β1;−1− s− η

η)

]+ 2.

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]

|Ψ〉 =1√2(| 0, 1〉+ | 1, 0〉), (5.14)

where |n,m〉 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

-1.5

-1.0

-0.5

0.0

s

0.8

0.9

1.0

Η

2.0

2.4

2.8

ÈBÈmax

Figure 5.2. Maximum Bell expectation value |B| = |〈B〉| for the single pho-ton entangled state. Only the range of parameters s and detector efficienciesη with |B| > 2 is shown.

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

WΨ(α, β; s) =4

π2(1− s)2

(−1 + s

1− s+

2

(1− s)2|α + β|2

)

× exp

[−2(|α|2 + |β|2)

1− s

], (5.15)

and its marginal single-mode distribution is

WΨ(α; s) =1

π(2− 2η + 4η2|α|2) exp[−2η|α|2]. (5.16)

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.

The maximum expectation values |B|max = |〈B〉|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


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

|TMSS〉 = sech r

∞∑n=0

tanhn r |n, n〉 , (5.17)

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

-1.5

-1.0

-0.5

0.0

s

0.7

0.8

0.9

1.0

Η

2.0

2.4ÈBÈmax

r = 0.6

-1.5

-1.0

-0.5

0.0

s

0.7

0.8

0.9

1.0

Η

2.0

2.4ÈBÈmax

r = 0.8

-1.5

-1.0

-0.5

0.0

s

0.7

0.8

0.9

1.0

Η

2.0

2.4ÈBÈmax

Figure 5.3. Demonstration of quantum non-locality for two-mode squeezedvacuum states (TMSSs). Maximum Bell values are shown for differentsqueezing r in the range of s and η where the Bell inequality is violated.


s = 0

0.5 1.0 1.5 2.0 2.5

2.0

2.2

2.4

r

ÈBÈ m

ax

s = -0.5

0.5 1.0 1.5 2.0 2.5

2.0

2.2

2.4

r

ÈBÈ m

ax

s = -0.7

0.5 1.0 1.5 2.0 2.5

2.0

2.2

2.4

r

ÈBÈ m

ax

s = -1.0

0.5 1.0 1.5 2.0 2.5

2.0

2.2

2.4

r

ÈBÈ m

ax

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

states is given by

WTMSS(α, β; s) =4

π2R(s)exp

(− 2

R(s){S(s)(|α|2 + |β|2)

− sinh 2r(αβ + α∗β∗)})

, (5.18)


WTMSS(α; s) =2

πS(s)exp

(−2|α|2

S(s)

), (5.19)

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.


The parameter s determines the characteristics of the detected non-local correla-

tions. 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 par-

ity 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]

W (α, τ) =1

t(τ)2

∫d2βW th(β)W

(α− r(τ)β

t(τ), τ = 0

). (5.20)

Here the parameters r(τ) =√

1− e−γτ and t(τ) =√

e−γτ are given in terms of the

energy decay rate γ, and

W th(β) =2

π(1 + 2n)exp

(− 2|β|2

1 + 2n

)(5.21)

is the Wigner function for the thermal state of average thermal photon number n.

5.6. Discussion and Conclusions 81

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

the parameter

s(τ) ∼ −r(τ)2

t(τ)2(1 + 2n) = (1− eγτ )(1 + 2n). (5.22)

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, sin-

gle 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 realisa-

tion 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 Schrodinger cat states [127, 128].


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 quan-

tum 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 quan-

tum 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 wit-

nessing entanglement since their violation guarantees the existence of entanglement.

Banaszek and Wodkiewicz (BW) [39, 38, 19] suggested a Bell-type inequality (re-

ferred 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 applica-

tions. 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 for-

malism, 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 experi-

mentally measured Wigner function. For this, we include effects of detector efficiency

into possible measurement outcomes. Possible expectation values of the entangle-

ment 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:

O(α) =

1ηΠ(α) + (1− 1

η)1 if 1

2< η ≤ 1,

2Π(α)− 1 if η ≤ 12,

(6.1)

where Π(α) =∑∞

n=0(−1)n |α, n〉〈α, n | is the displaced parity operator and 1 is the

identity operator. |α, n〉 = D(α) |n〉 is the displaced number state produced by

applying the Glauber displacement operator D(α) to the number state |n〉.

Let us then consider the expectation value of observable (6.1) when the measure-

ment 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

Bernoulli transformation [112]: Pη(m) =∑∞

n=m P (n)(

nm

)(1 − η)n−mηm. Thus the

expectation value of the parity operator is obtained as

〈Π(α)〉η =∞∑

m=0

(−1)mPη(α, m) =∞∑

n=0

(1− 2η)nP (α, n), (6.2)

where 〈·〉η 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

W η(α) ≡ 2

π〈Π(α)〉η, (6.3)

which is given as a rescaled Wigner function by Gaussian smoothing. Note that a


smoothed Wigner function can be identified with a s-parameterised quasi-probability

function as W η(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 observ-

able (6.1) as

〈O(α)〉η =

π2η

W η(α) + 1− 1η

if 12

< η ≤ 1,

πW η(α)− 1 if η ≤ 12,

(6.4)

which is bounded as |〈O(α)〉η| ≤ 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,

denoted by A1, A2 for Alice and B1, B2 for Bob. All observables are variations of the

operator (6.1) as Aa = O(αa) and Ba = O(βb) with a, b = 1, 2. We then formulate

a Hermitian operator as a combination of each local observable Aa, Bb in the form

W = C1,1 + C1,2 + C2,1 − C2,2, (6.5)

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 B 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 W or B. 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

states. For a separable state ρsep =∑

i piρAi ⊗ ρB

i where pi ≥ 0 and∑

i pi = 1, the

expectation value of the correlation operator measured with efficiency η is given by

〈Ca,b〉sepη =∑

i

pi

∞∑n,m

(1− 2η)n+m 〈α, n | ρAi |α, n〉〈β,m | ρB

i | β, m〉

=∑

i

pi〈Aa〉iη〈Bb〉iη. (6.6)

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

|〈Aa〉iη|, |〈Bb〉iη| ≤ 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:

|〈W〉sepη | =

∣∣∣∣∑

i

pi(〈A1〉iη〈B1〉iη + 〈A1〉iη〈B2〉iη + 〈A2〉iη〈B1〉iη − 〈A2〉iη〈B2〉iη)∣∣∣∣

≤ 2∑

i

pi = 2 ≡ Wsepmax. (6.7)

Therefore, if |〈W〉ψη | > Wsepmax = 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(α) is measured as −1.

Thus the expectation value of (6.5) is bounded by local realism as |〈B〉η| ≤ BLRmax =

2(1−2/η)2. Likewise for η ≤ 1/2, we can obtain BLRmax = 18. Note that BLR

max ≥ Wsepmax

for all η, and BLRmax = Wsep

max 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]. 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:

|〈W〉η> 12| =

∣∣∣∣π2

4η2[W η

1,1 + W η1,2 + W η

2,1 −W η2,2]

+π(η − 1)

η2[W η

a=1 + W ηb=1] + 2(1− 1

η)2

∣∣∣∣ ≤ 2,

(6.8)

|〈W〉η≤ 12| = |π2[W η

1,1 + W η1,2 + W η

2,1 −W η2,2]− 2π[W η

a=1 + W ηb=1] + 2| ≤ 2,

where W ηa,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

function W η(αa, βb) with the notation a, b for simplicity), and W ηa(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 entangle-

ment 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 |Ψ〉 = (| 0, 1〉+

| 1, 0〉)/√2 where | 0, 1〉 (| 1, 0〉) is the state with zero (one) photons in the mode of

6.4. Testing single photon entangled states 89

HaL

0.5 0.6 0.7 0.8 0.9 1.0

2.0

2.4

2.8

Η

ÈXW\È

HbL

0.5 0.6 0.7 0.8 0.9 1.0

2.0

2.4

2.8

Η

HcL

0.5 1.0 1.5 2.0 2.51.9

2.0

2.1

2.2

2.3

2.4

2.5

r

ÈXW\È

Η=1

Η=0.99

Η=0.7

Η=0.5

Figure 6.1. Maximum expectation value of the entanglement witness oper-ator in Eq. (6.5) for an input of (a) a single photon entangled state and (b) atwo-mode squeezed state with r = 0.4 (black) and r = 0.8 (grey). Entangle-ment exists if the expectation value exceeds the dashed line Wsep

max = 2. Notethat the shaded region which exceeds BLR

max = 2(1− 2/η)2 (for 1/2 < η ≤ 1)is the criterion of non-locality. (c) Witnessing entanglement with varyingsqueezing 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).


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

W ηa,b =

4

π2(1− 2η + 2η2|αa + βb|2) exp[−2η(|αa|2 + |βb|2)] (6.9)


W ηa =

1

π(2− 2η + 4η2|αa|2) exp[−2η|αa|2]. (6.10)

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-

cal parametric amplifiers [103], and be written as |TMSS〉 = sech r∑∞

n=0 tanhn r |n, n〉where r > 0 is the squeezing parameter. The measured Wigner function with effi-

ciency η for a two-mode squeezed state is given by

W ηa,b =

4

π2η2R(η)exp

(− 2

R(η){S(η)(|αa|2 + |βb|2)

− sinh 2r(αaβb + α∗aβ∗b )}

), (6.11)

and its marginal single-mode Winger function is

W ηa =

2

πηS(η)exp

(−2|αa|2

S(η)

), (6.12)

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 ex-

pectation 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 param-

eter 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


Η=0.55

0.55 0.60 0.65 0.70 0.75 0.80

2.0

2.2

2.4

2.6

Ε

ÈXW\È

HaL Η=0.55

0.55 0.60 0.65 0.70 0.75 0.80

2.0

2.2

2.4

2.6

Ε

HbL

Figure 6.2.Witnessing entanglement with a real efficiency η = 0.55 as vary-ing the estimated efficiency ε for an input of (a) a single photon entangledstate 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

|〈W〉η> 12| =

∣∣∣∣π2

4ε2[W η

1,1 + W η1,2 + W η

2,1 −W η2,2] (6.13)

+π(ε− 1)

ε2[W η

a=1 + W ηb=1] + 2(1− 1

ε)2

∣∣∣∣ ≤ 2.

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

expectation values of separable states are not bounded by Wsepmax = 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 correc-

tion. 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 pre-

cisely 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 mea-

surement scheme of Wigner function in such systems [133].

In summary, we proposed an entanglement witness which can detect entangle-

ment 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 struc-

ture can be represented either in correlation function space or joint probabil-

ity 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 func-

tions 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 opti-

mal Bell inequality are obtained asymptotically by the EPR state, and showed

that a parity measurement in the Pegg-Barnett phase basis is an optimal mea-

surement. 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 di-

mensional 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

Schrodinger cat 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 inef-

ficient 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 lo-

cal 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 entan-

glement. 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 ef-

ficiency 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 quan-

tum 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 infor-

mation 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 quan-

tum 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 free-

dom 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. Schrodinger. Discussion of probability relations between separated systems.

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

[7] E. Schrodinger. 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 inequal-

ities. 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 en-

tanglement. 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. Acın, 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 Crepeau, 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 dis-

tributions. 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. 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 gener-

alized 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 ho-

modyne 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. Gomez, C. H. Monken, C. Saavedra, and S. Padua.

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 communica-

tion 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. Lutkenhaus. 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. 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. Guhne, D. Bruß, and M. Lewenstein. Relations between entan-

glement 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. Wodkiewicz. Direct probing of quantum phase space by

photon counting. Phys. Rev. Lett. 76, 4344 (1996).

108 Bibliography

[92] K. Banaszek, A. Dragan, K. Wodkiewicz, and C. Radzewicz. Direct measure-

ment 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 sys-

tems 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. 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. Brukner, M. Zukowski, and A. Zeilinger. Quantum communication com-

plexity 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. 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 gen-

eration 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. Bjork. 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 detec-

tor 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).

Documents

Bell tests in physical systems - Department of Physics at … · 2010. 7. 8. · Bell tests in physical systems Seung-Woo Lee, St. Hugh’s College, Oxford Trinity Term, 2009 Abstract