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Dynamics of Quantum Correlations

with Photons

Experiments on bound entanglement and contextuality for application in quantum information

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Nature publishing group (papers) ISBN 978-91-7447-421-3

Printed in Sweden by Universitetsservice US-AB, Stockholm, Stockholm 2011 Distributor: Department of Physics, Stockholm University

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...the “paradox” is only a conflict between reality and your feeling

of what reality “ought to be.”

– Richard Feynman, in The Feynman Lectures on Physics, vol III, pp. 18-9 (Addison-Wesley, 1964).

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Abstract

The rapidly developing interdisciplinary field of quantum information, which merges quantum and information science, studies non-classical aspects of quantum systems. These studies are motivated by the promise that the non-classicality can be used to solve tasks more efficiently than classical methods would allow. In many quantum informational studies, non-classical behaviour is attributed to the notion of entanglement.

In this thesis we use photons to experimentally investigate fundamental questions such as: What happens to the entanglement in a system when it is affected by noise? In our study of noisy entanglement we pursue the chal-lenging task of creating bound entanglement. Bound entangled states are cre-ated through an irreversible process that requires entanglement. Once in the bound regime, entanglement cannot be distilled out through local operations assisted by classical communication. We show that it is possible to experi-mentally produce four-photon bound entangled states and that a violation of a Bell inequality can be achieved. Moreover, we demonstrate an entanglement-unlocking protocol by relaxing the condition of local operations.

We also explore the non-classical nature of quantum mechanics in several single-photon experiments. In these experiments, we show the violation of various inequalities that were derived under the assumption of non-contextuality. Using qutrits we construct and demonstrate the simplest possible test that offers a discrepancy between classical and quantum theory. Furthermore, we perform an experiment in the spirit of the Kochen-Specker theorem to illustrate the state-independence of this theorem. Here, we investigate whether or not measurement outcomes exhibit fully contextual correlations. That is, no part of the correlations can be attributed to the non-contextual theory. Our results show that only a small part of the experimental generated correlations are amenable to a non-contextual interpretation.

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List of Abbreviations

Symbol Description

LOCC local operations assisted by classical communication

BE bound entanglement

PT partial transpose

PPT positive partial transpose

HV hidden variable models, also referred to as classical models

EPR Einstein-Podolsky-Rosen

CHSH Clauser-Horne-Shimony-Holt

KS Kochen-Specker

H horizontal polarization

V vertical polarization

SMF single mode fibre

HWP half-wave plate

QWP quarter-wave plate

BS beam splitter, refers to a general or 50/50 BS

SPBS special polarized BS, 100/0 for H and 33/66 for V polarization

PBS polarized beam splitter

FWHM full width half maximum

PS phase shift

SPDC spontaneous parametric down conversion

APD avalanche photo diode

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Contents

Abstract . . . i

List of Abbreviations . . . iii

Sammanfattning på svenska . . . vii

List of Accompanying Papers . . . ix

Preface . . . xi

My Contributions to the Accompanying Papers . . . xiv

Acknowledgements . . . xv

Part I: Background Material and Results 1 Quantum Information Basics . . . 3

1.1 Bits, Qubits and Entanglement . . . 3

1.1.1 The Qubit . . . 4

1.1.2 Multi-Qubit . . . 5

1.1.3 Mixed States . . . 6

1.1.4 No-cloning and LOCC . . . 8

1.1.5 Entanglement in Pure States . . . 9

1.1.6 Entanglement in Mixed States . . . 11

1.1.7 Distillation and Bound Entanglement . . . 12

1.2 State and Entanglement Verification . . . 13

1.2.1 State Fidelity . . . 14

1.2.2 Witness Method . . . 14

1.2.3 PPT-Criterion . . . 15

1.3 Hidden Variable Models . . . 17

1.3.1 Bell Inequality . . . 17

1.3.2 Kochen-Specker . . . 20

1.3.3 Fully Contextual Correlations . . . 22

1.3.4 Klyachkoet al.and Wright . . . 26

2 The Art of Quantum Optics and Data Analysis . . . 31

2.1 Implementation of Qubits . . . 31

2.1.1 Photon Polarization . . . 31

2.1.2 Path Encoding . . . 32

2.2 Distribution of Photons . . . 32

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2.3.1 Wave Plates . . . 34

2.3.2 Beam Splitters . . . 35

2.4 Linear Optical Two-Qubit Gates . . . 37

2.4.1 Polarization-Path Gate and Polarization Analysis . . . 38

2.4.2 Two-Photon Sign-Shift Gate . . . 39

2.5 Down-Conversion . . . 44

2.5.1 Pump Laser . . . 44

2.5.2 Two-Photon SPDC . . . 45

2.5.3 Multi-Photon Product States . . . 49

2.5.4 Creation of Mixed States . . . 50

2.6 Detection and Data Analysis . . . 51

2.6.1 Detectors . . . 52

2.6.2 Multi-Channel Coincidence Unit . . . 53

2.6.3 Detection Efficiencies . . . 54

2.6.4 Expectation Values . . . 55

2.6.5 Quantum State Tomography . . . 56

3 Experimental Bound Entanglement . . . 61

3.1 Bit-Flip and Phase-Flip Error Channel . . . 62

3.1.1 Probability Tetrahedron . . . 64 3.1.2 Witness . . . 65 3.1.3 Bell Inequality . . . 66 3.2 The Experiment . . . 70 3.2.1 State Characterization . . . 70 3.2.2 Unlocking Entanglement . . . 74

4 Experiments on the Foundation of Quantum Mechanics . . . 77

4.1 Kochen-Specker Inequality and Fully Contextual Correlations . . . 79

4.1.1 Results of the Experiment on the Kochen-Specker Inequality . . . 83

4.1.2 Results on Fully Contextual Correlations . . . 85

4.2 Klyachkoet al.and Wright . . . 86

4.2.1 Results on Klyachkoet al.and Wright . . . 91

5 Conclusions and Outlook . . . 93

Bibliography . . . 97

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vii

Sammanfattning på svenska

Det sägs att John Wheeler i sitt sökande efter den kvantmekaniska principen döpte om det till ”Merlin principen”. Legenden berättar om trollkarlen Merlin som kunde ändra form om och om igen när han var förföljd. Detta är likt kvantmekanikens många tolkningar och den mystik som förknippas med den. Det snabbt växande tvärvetenskapliga forskningsfältet kvantinformation är en sammanslagning av kvantmekanik och informationsvetenskap. Detta forsk-ningsfält försöker förstå sig på de icke-klassiska aspekterna av kvantmekani-ken. En av drivkrafterna är förhoppningen att kunna använda dessa system för att lösa informations teoretiska problem mer effektivt än vad som är möj-ligt enmöj-ligt den klassiska fysiken. Denna icke-klassiska del av kvantmekaniken benämns oftast sammanflätning, eller entanglement på engelska.

I denna avhandling använder vi fotoner för att utföra experiment som un-dersöker grundläggande frågor som: Vad händer med sammanflätningen i ett system som är under bruspåverkan? I denna studie av brusets påverkan på sammanflätningen strävar vi efter att skapa bunden sammanflätning. Detta är ett exempel på en irreversibel process där sammanflätning behövs för att skapa kvanttillstånden. Men när bruset har drivit tillståndet till den bundna regimen kan man inte destillera ut någon sammanflätning med hjälp av lokala opera-tioner och klassisk kommunikation mellan parterna. Detta trots att systemet fortfarande är sammanflätat. Vi visar i studien att det är möjligt att experimen-tellt framställa bundna sammanflätade kvanttillstånd med ett system bestående av fyra fotoner. Dessutom visar vi att dessa tillstånd även kan bryta en Bell olikhet. Genom att bryta villkoret för lokala operationer visar vi även att det är möjligt att låsa upp den bundna sammanflätningen.

Vi utforskar även kvantmekanikens icke-klassiska inslag med hjälp av en-skilda fotoner genom att bryta mot ett flertal icke-kontextuella olikheter. Här visar vi hur man kan konstruera det enklaste testet där det finns en diskre-pans mellan klassisk och kvant-fysik. Förutom denna studie utför vi ett expe-riment som är i samma anda som Kochen-Specker teoremet, syftet är att belysa tillståndsoberoendet i teoremet. Dessutom undersöker vi korrelationer mellan mätutfallen som är helt icke-kontextuella. Detta innebär att teoretiskt kan ing-en del av de korrelationer som uppkommer tillskrivas ding-en icke-kontextuella delen av teorin.

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ix

List of Accompanying Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Experimental four-qubit bound entanglement, E. Amselem,

and M. Bourennane, Nature physics 5, 748 (2009).

II Reply to ’Experimental bound entanglement?’, E. Amselem,

and M. Bourennane, Nature physics 6, 827 (2010).

III Experimental multipartite entanglement through noisy

Quantum Channel, E. Amselem, M. Sadiq and M. Bourennane, submitted (2011).

IV State-Independent Quantum Contextuality with Single

Photons, E. Amselem, M. Rådmark, M. Bourennane, and A. Cabello, Phys. Rev. Lett. 103, 160405 (2009).

V Two Fundamental Experimental Tests of Non-Classicality

with Qutrits, J. Ahrens, E. Amselem, M. Bourennane, and A. Cabello, submitted (2011).

VI Experimental fully contextual correlations, E. Amselem, L. E.

Danielsen, A. J. López-Tarrida, J. R. Portillo, M. Bourennane, and A. Cabello, arXiv:1111.3743v1, submitted (2011).

Reprints were made with permission from the publishers.

Related Papers Not Included

1. Proposed experiments of qutrit state-independent contextuality and two-qutrit contextuality-based nonlocality, A. Cabello, E. Am-selem, K. Blanchfield, M. Bourennane, and I. Bengtsson, submitted (2011).

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xi

Preface

It is said that John Wheeler in his search for the quantum principle renamed it the ”Merlin principle”. According to legend, the magician Merlin could change form again and again when pursued. Similarly, quantum mechanics can be interpreted in many ways and is surrounded by mystery.

The study of quantum mechanics began at the end of the 19th and the begin-ning of the 20th century with the idea of quantizing the energy levels of black-body radiation. The birth of quantum mechanics required the introduction of new concepts that do not easily lend themselves to an intuitive understand-ing. One of these curious concepts is the superposition principle, which states that a non-separable system of states can be constructed, so-called entangled states. In the early days of quantum mechanics the existence of non-separable states was strongly criticized by the three eminent physicists Einstein, Podol-sky, and Rosen (EPR). In a publication from 1935 they opposed the idea of the existence of such states [1]; they believed that quantum mechanics cannot completely describe physical reality. This was the starting point of the so-called hidden variables debate. During the 1960s two important results were presented that shed new light on the problem of hidden variables. First John Bell managed to construct a test [2] that could rule out models favouring the ideas shared by EPR. Then Simon Kochen and Ernst Specker proved theoreti-cally that the predictions of quantum mechanics cannot be reconciled with the basic assumptions of the hidden variable theories [3]. They showed that for any quantum system with a dimension higher than two there is always a set of tests that will give rise to quantum mechanical predictions that are different from those derived through classical logic. During the following years further progress was achieved by Clauser, Horne, Shimony, and Holt, who gener-alised Bell’s results to obtain the CHSH inequality [4]. Alan Aspect’s exper-iment in 1981 was an even more important milestone. He performed the first experimental test of Bell’s inequality [5, 6]. His results showed a violation of the inequality, thereby confirming the quantum mechanical predictions. Since then, his experimental test has been refined, generalized, and experimentally verified for many different scenarios. However, until today no complete Bell test has been experimentally realized. Loopholes in the experiments still allow the results to be explained by hidden variable models.

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These developments gave rise to the new field of quantum computation and communication. Efforts were made to harness the non-classicality of quantum mechanics in order to solve practical problems. Here the concept of entangle-ment began to play an important role, especially for quantum teleportation and superdense coding [7], which cannot be realized without it. In these experi-mental advances the fragile nature of quantum states became more and more an issue, since the distribution of entanglement to several distant parties was crucial for communication tasks. To solve this experimental problem, a distil-lation protocol that takes several copies of a noisy entangled state as input and produces higher quality entanglement as output was proposed by Bennet and coworkers [8, 9]. Thanks to their result high quality entanglement can now, in principle, always be achieved for small and low dimensional systems. It was believed that for larger systems a generalization existed and would eventually be found. But the Horodecki family discovered a set of states in 1998 [10] that are entangled but not distillable. These interesting states became known as bound entangled states (BE). They illustrate an irreversible process that limits the usage of the entanglement in the system. Owing to their mixed structure, they are considered to be closer to separable classical states, that is, they are regarded as being less quantum than distillable entangled states. However, it was found that BE states can maximally violate Bell type inequalities [11,12], thereby making them as much non-classical as any other entangled state. Ini-tially, BE states were not suitable for experiments, but in 2001 John Smolin discovered a state [13] that is more suitable and even keeps it bound entangled in the presence of moderately high noise.

Regarding the Kochen-Specker theorem there are several experiments [14, 15] attempting to catch the spirit of the proof, but they were incomplete. These experiments utilize inequalities that are derived under the assumption of non-contextual hidden variables and a violation occurs only for specific quantum states. But one of the most striking features of the Kochen-Specker proof is that it does not refer to any quantum state. Only recently it was found that it is possible to convert this counterfactual logic to an experimentally testable inequality that is state-independent [16]. The theoretical toolbox for deriving this kind of inequalities has greatly improved and has produced property spe-cific tests where there is a discrepancy between classical and quantum [17].

In this thesis, we study the experimental creation and characterization of bound entanglement, and we also investigate several tests on non-contextual hidden variables with single photons. To understand the ideas around the ex-periments we begin by describing the basic concepts and necessary theoretical tools in chapter 1. This chapter is not meant as a complete account but more as an introduction of the concepts. It serves as a guide for later chapters where the experiments are presented. First, we discuss the concept of the qubit and

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troduce entanglement both in its pure and mixed form, in section 1.1. We then look briefly into the idea of distillation that brings us to bound entanglement, see section 1.1.7. In section 1.2 we discuss different methods to characterize an experimentally generated state. In the following we discuss the idea of hid-den variable models and contextuality, section 1.3, where we introduce and discuss several inequalities that will be experimentally tested.

This theoretical section is followed by an experimental part, chapter 2, which describes the experimental toolbox. Here we introduce two encoding schemes for qubits together with ways to distribute and implement one-qubit gates, see sections 2.1 - 2.3. Our experiments also requires two-qubit gates, these are discussed in section 2.4, where we investigate how to implement both two-qubit single-photon gates coupling between degrees of freedom and two-photon gates coupling between qubits encoded in separated photons. For the bound entanglement experiments a four-photon source that produces en-tanglement between pairs of photons is needed. Section 2.5 is dedicated to the subject of spontaneous parametric down conversion (SPDC) and the creation of mixed states. Chapter 2 ends with introducing the toolbox for detection and data analysis, specifically, we describe the method of maximum likelihood quantum state tomography (QST) for the reconstruction of a density matrix from a set of measurement data.

All tools are combined in chapters 3 and 4 where each experiment with its results are presented. We end with a conclusion and outlook in chapter 5.

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My Contributions to the Accompanying Papers

Below I comment on my contributions to the papers accompanying this thesis. The lab was under construction when I began my Ph.D. studies in the fall of 2006. Hence, I have been involved in all aspects of building up a modern research lab, including purchasing and characterising optical components and other necessary instruments. Furthermore, I have developed various computer programs for data communication, control systems for the experiments, and data analysis.

Paper I: In this experiment I made a major contributions in designing the experiment. I performed all the experimental work and data analysis. The paper was written by all co-authors.

Paper II: A concern of the bound entanglement experiment was raised by J. Lavoie, et al. [18]. The data analysis was performed by me and the reply was written by all co-authors.

Paper III: The basic concept and building blocks did come from the experiment of article I. When building the experiment I did a major contribution to the designing of the new set-up. The experiment was built by me and Muhamad Sadiq. I performed the entire data analysis and wrote the paper with supervision of the co-authors.

Paper IV: The cascaded schema and the measurement with repreparation of the state where developed and designed by me and Magnus Rådmark, with equal contributions. This holds true also for all the laboratory work, as well as the entire data analysis. The paper was written by all co-authors.

Paper V: Designing the time encoding scheme and set-up of the experiment was performed by me and Johan Ahrens with equal contributions. All the laboratory work, as well as the entire data analysis was preformed by both of us. The paper was written by all co-authors.

Paper VI: I designed the experiment and performed all the laboratory work with equal contributions together with Magnus Rådmark. The entire data analysis was performed by me and the paper was written by all co-authors.

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Acknowledgements

The work in this thesis could not have been completed without the help and support from many fantastic people. First of all, I would like to express my deep and sincere gratitude to my supervisor Mohamed Bourennane for believ-ing in me and introducbeliev-ing me to the intrigubeliev-ing world of applied quantumness. Special thanks go to him for the opportunity to let me more or less freely muck about in a brand new and very shiny laboratory. I would also like to give a very special thanks to Adán Cabello who with great enthusiasm discussed and an-swered all my questions about contextuality. Thanks go to Ingemar Bengtson, Piotr Badziag, Gunnar Björk and Hoshang Heydari for discussions and col-laborations.

To all present and former members of the group THANKS for an inspir-ing and joyful atmosphere. In particular, a great deal of gratitude goes to my lab mates Magnus Rådmark, Hatim Azzouz, Johan Ahrens, Christian Kothe, Muhamad Sadiq, Alley Hameedi, Atia Amari and Hannes Hübel for making the dark lab hours seem to pass faster by having mind-boggling discussions. Also, big thanks to Isabelle Herbauts and Sören Holst who read through the thesis and providing many important comments. For the more social aspects of the Ph.D life thanks Kate Blanchfield, Klas Marcks von Würtemberg, Olof Lundberg, Istvan Zoltan Jenei and Thor Wikfeldt for the many adventures we endured. There are many more at Albanova that deserves my gratitude for helping me and providing a pleasant atmosphere, to all of you thanks.

Finally, I would like to thank my family for the love and support that they always provide. Most of all my deepest gratitude and love goes to my lovely Maria who is and will always be by my side!

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Part I:

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3

1. Quantum Information Basics

Classical information is usually measured in units called bits, where bit is the abbreviation for binary digit. Claude E. Shannon, often referred to as the founder of information theory, first used this term in a landmark article [19], where he introduced the basic elements of communication. Today, informa-tion theory has truly revoluinforma-tionized our way of living because it enabled the construction of immensely powerful electronic devices for the manipulation of strings of bits. Stimulated by these developments, a generalization of the term to the quantum regime emerged, the quantum bit, or qubit in short. Through this generalization new and even more powerful methods of manip-ulating data beyond the presently used classical ones have emerged. Several theoretical proposals for a quantum computer have already been put forth as well as many other more specialized protocols dealing with problems ranging from the factorization of large numbers to the secure communication between distant parties. Aside from pure application-oriented proposals, new tools for investigating more fundamental questions about the quantum world have been proposed.

Below we will introduce the basic concepts of quantum information the-ory, from the qubit and entanglement to the more specialized concept of non-contextual inequalities, as well sa tools for detecting bound entanglement. Throughout this chapter many of these concepts will be illustrated with ex-amples. All the presented tools will be used experimentally in later chapters.

1.1

Bits, Qubits and Entanglement

A bit can be defined through any two level system, whereby the two levels are often denoted by the abstract binary numbers 0 or 1. Nevertheless, all infor-mation has to be encoded in a physical system, for example as an electrical potential difference, 0 volts and 3.3 volts. This is a common encoding in dig-ital electronic circuits. Other common encoding options are pulsed light used in optical fibre networks and free space set-ups such as for example remote TV-controls.

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1.1.1 The Qubit

Contrary to a bit that can only encode two different states, a qubit can encode an infinite number of states. Nevertheless, the qubit is a two-level system like the bit. The two levels of the qubit are here represented by the two orthogonal quantum states |0i and |1i. These two states constitute a basis for the qubit, which is referred to as the computational basis. By the superposition principle any qubit state |ψi can be represented by

|ψi = α|0i + β |1i, or in matrix form, |ψi = α

β !

, (1.1)

where α and β are complex numbers satisfying the normalization condition |α|2+ |β |2= 1. A qubit can be encoded in a physical system in many ways,

for example by the spin of an electron, by two atomic energy levels, or by the polarization of photons. We will discuss the encoding in more detail later in the experimental chapter 2.

The normalization condition allows us to rewrite |ψi, disregarding a global phase factor, into a more illustrative form,

|ψi = cos(θ

2)|0i + sin( θ 2) · e

|1i, (1.2)

where θ and φ range from 0 to 2π. In this form, |ψi describes points on the surface of a sphere, the Bloch sphere, see fig. (1.1). The axes x, y, and z represent the eigenstates of three observables known as the Pauli matrices,

σx= 0 1 1 0 ! , σy= 0 −i i 0 ! , and σz= 1 0 0 −1 ! . (1.3)

Each of the matrices has two eigenvalues, +1 and −1. The eigenstates of the three observables are listed in table (1.1). We will denote the basis constructed out of the set of eigenstates of each operator as the σx, σy, or σzbasis,

respec-tively. The computational basis consists therefore of the eigenstates of the σz

operator. Note that orthogonal states lie opposite to each other on the Bloch sphere and that the state orthogonal to |ψi in (1.2) can be written as |ψ⊥i = sin(θ

2)|0i − cos( θ 2) · e

−iφ|1i. The corresponding observable for the states |ψi

and |ψ⊥i is σ (θ , φ ) = sin(θ ) cos(φ )σx+ sin(θ ) sin(φ )σy+ cos(θ )σz.

From the relation given by (1.3) it is easy to see that σxcorresponds to a bit

flip operation in the computational basis: applying a σx-operation on a qubit

α |0i + β |1i will result in the state α |1i + β |0i. Similarly, the operation σz

shifts the phase of the state by π, thereby multiplying the state |1i by minus 1. The operations described above are only valid in the computational basis. For example, in the σxbasis these actions are reversed.

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1.1 Bits, Qubits and Entanglement 5

Table 1.1: Pauli matrices and their eigenvectors and eigenvalues.

Observable Eigenvalues Eigenstates

σx ±1 |x, ±1i =√12(|0i ± |1i)

σy ±1 |y, ±1i =√1

2(|0i ± i|1i)

σz ±1 |z, +1i = |0i, |z, −1i = |1i

1.1.2 Multi-Qubit

There are important single qubit protocols for quantum communication and computation, but expanding to multi-qubit systems brings about new oppor-tunities for more complex and richer tasks. When working with multi-qubit states, all the two-qubit Hilbert spaces are combined to form a bigger space. This new Hilbert spaceH⊗n=H1⊗ · · · ⊗Hncontains all possible n

multi-qubit pure states. The dimensionality of the space grows exponentially with the amount of qubits used, for n qubits the dimensionality is 2n. For two qubits the computational basis consists of the four vectors,

|0i1⊗ |0i2= |00i =

      1 0 0 0      

, |0i1⊗ |1i2= |01i =

      0 1 0 0       ,

|1i1⊗ |0i2= |10i =

      0 0 1 0      

, |1i1⊗ |1i2= |11i =

      0 0 0 1       . (1.4)

Here, the subscripts 1 and 2 indicate the two different Hilbert spaces in the tensor productH1⊗H2. Generalizing this notation to include more qubits is

straightforward. Then the position in the Dirac bracket formalism refers to the different local single qubit Hilbert spaces where each qubit lives.

At this point it should be stressed that each qubit in a multi-qubit system is not required to be physically separable from the other qubits, like for exam-ple two atoms or two photons. We can construct several qubits by using one

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Figure 1.1:The Bloch Sphere describes the set of states a qubit can take. An arbitrary state|ψiwith its parametrizationθ andφis illustrated.

system with many degrees of freedom where each represents a qubit. Alterna-tively, we can regard a n = 2mlevel system as being an m-qubit system, even

though no qubits can be physically identified directly. Our view of qubits is therefore more shaped through the physical restrictions on the type of opera-tions that can be applied and the possibilities of measuring the results of these operations.

1.1.3 Mixed States

Quantum states are fragile and are easily disturbed by the environment. This mechanism is called decoherence and couples the pure state to its surround-ings, thereby adding information that is not available for the participants using only the quantum system. This coupling transforms a pure state into a statisti-cal ensemble of pure states that can not be described by the pure-state formal-ism. In a laboratory environment it is practically impossible to create a perfect pure state, especially when transporting the states through noisy channels. The resulting mixtures can be described mathematically by density operators. We define the density operator ρ = |QihQ| for a pure state |Qi. An ensemble of a set of pure states can now be described by a weighted sum of density

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opera-1.1 Bits, Qubits and Entanglement 7

tors. Generally, a density operator can be described as,

ρ = ∑iωiρi= ∑iωi|QiihQi|, where ∑iωi= 1, (1.5)

where the weights ωi are interpreted as the probability distributions of the

pure states, ρi= |QiihQi|. A density operator can be represented in the matrix

formalism, where the elements of the matrix are given by hm|ρ|ni for a given basis in the N-dimensional space with the indices n, m ∈ {1, 2, ..., N} for the basis. The matrix representation is basis-dependent, but usually the compu-tational basis is used, which simplifies the indexing for multi-qubit systems. In the computational basis the indices are given by the decimal representation of the string of binary numbers that represents each separate qubit. A mixed quantum state ρ possesses, among others, the following properties:

• ρ is Hermitian and positive semi-definite. That is, the eigenvalues λi

of ρ are all real and greater than or equal to 0. • Normalization condition, Tr(ρ) = 1.

• For pure sates ρ2= ρ.

• Tr(ρ2) ≤ 1, where the equality holds only for pure states.

• The expectation value h A i of an operator A is given by h A i = Tr(A · ρ ).

Here, Tr() is the trace, which is the summation over the diagonal elements of a density operator in matrix-form.

Since we set out to investigate mixed states, it is crucial to understand den-sity operators. To gain some insight, let us investigate what happens to a single qubit ρ, when it is affected by depolarizing noise. Suppose we begin with a pure state |x, 1i = (|0i + |1i)/√2. In the matrix formalism its density operator is given by, ρ = |x, 1ihx, 1| =1 2 1 1 1 1 ! . (1.6)

The off-diagonal elements indicate that the state is in a coherent superposition and can admit interference effects between its components if it is rotated by some operator. If the state undergoes a depolarizing stage where it loses its interference properties then, depending on how strong this depolarization is, the state will be transformed into,

ρnoisy= p 11

2 + (1 − p)ρ, where 0 ≤ p ≤ 1. (1.7)

If the noise parameter is p = 0, the state is still described by the pure state |x, 1i. A measurement in the computational basis will give equal probability of finding |0i and |1i, whereas in the σxbasis only |x, 1i will be found.

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The noise can render the state ρ into a complete mixture11/2, where the original state is completely washed out. This happens when the parameter p is 1. Again, a measurement in the computational basis will have equal probabil-ity of finding |0i and |1i, but in the σx basis there the two outcomes |x, 1i and

|x, −1i will be also equally probable. This is due to the lack of interference between |0i = (|x, 1i + |x, −1i)/√2and |1i = (|x, 1i − |x, −1i)/√2, which are not in a coherent superposition. For p > 0 there will be a contribution of the incoherent part11/2, which will washout some effects. This is always the case for an imperfect experimentally generated quantum state.

Considering the Bloch sphere, fig. (1.1), a pure qubit state is represented on the surface of the sphere, through vectors with length 1. The set of mixed qubit states are be represented in the Bloch ball, which is the interior of the sphere. A mixed qubit state is represented by a shorter vector in the sphere pointing from the origin, which represents the complete mixed state,11/2.

1.1.4 No-cloning and LOCC

When working with multi-qubit systems one can easily be lured to believe that almost any operation on the states is possible. This is of course not true. A sim-ple examsim-ple, which has great impact on quantum cryptography and quantum computation, is the no-cloning theorem. The theorem states that it is not pos-sible to construct a copying machine that takes an arbitrary unknown state and makes a perfect copy of it. The proof [7] considers a perfect copying machine that takes an arbitrary input state |φ i, which is to be copied, together with a target state |si, a blank paper, and converts the target |si to |φ i. The copying process is performed in the machine by a unitary operation U . This copying operation U can thus copy any two states,

U|φ i ⊗ |si = |φ i ⊗ |φ i,

U|ψi ⊗ |si = |ψi ⊗ |ψi. (1.8)

However, since the inner product is preserved for unitary operations, these two equations give hψ|φ i = (hψ|φ i)2, which has only two solutions. Either |φ i and |ψi are orthogonal or they are equal. Thus if the copying machine can faith-fully copy one state, then it can only faithfaith-fully copy one other state, namely the orthogonal one. This restriction is one of the cornerstones of quantum cryp-tography and allows for the detection of an eavesdropper in a communication line [7].

The restriction described above is due to quantum mechanics itself and can-not be changed. Other restrictions can be applied more artificially to suit a certain scenario. An important set of operations in quantum communication and quantum computation are local operations assisted by classical

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communi-1.1 Bits, Qubits and Entanglement 9

cation (LOCC). The above restriction emerges naturally when quantum states are distributed to separated parties. Each party can thus only manipulate the qubits which are locally available. Any type of manipulation is allowed, from making measurements on the qubits to only storing them or measuring them together with previously received qubits. Regardless of the operations per-formed on the qubits locally, it is supposed that the local results can be com-municated classically to the other parties. In doing so, the parties can try to convert their quantum state to something that might be more useful for a par-ticular task. For two parties A and B sharing a state ρ ∈HA⊗HB, a general

LOCC operation [20] can be described by,

ρ → 1 M∑iAi⊗ Bi· ρ · A † i ⊗ B † i,

where M = Tr(∑iAi⊗Bi·ρ ·A†i⊗B†i) is the normalization, and the operators Ai

and Biare the local operations of the parties A and B. Each operator can occur

with a certain probability that can either be induced by the parties involved or by the environment that is outside the parties’ control. With LOCC it is possible to describe transmission channels that induce errors when distributing qubits. Alternatively, LOCC can be used to the opposite effect, to clean up a set of states that are noisy in order to retrieve more pure quantum states that can be used for a multiparty quantum protocol.

1.1.5 Entanglement in Pure States

It is possible to create quantum states that consist of several qubits and are inseparable. In such a quantum system it is not possible to consider each out-come of a measured qubit to be independent from the measurement outout-comes of the remaining qubits, regardless of the physical distance between the qubits. Correlations arising from these kinds of systems can be stronger than those achievable in classical physics. This strange phenomenon, called entangle-ment, has important consequences and is used in many quantum informational tasks such as quantum teleportation. The definition of entanglement for pure states is,

Definition: A pure state |Qi over the partitions Pi, where i∈ {1..n}, is called

entangled if it cannot be represented as a product of pure states|φiiPi. That is,|Qi 6= |φ1iP1⊗ · · · ⊗ |φniPn. A state that can be represented by a product of pure sates over this partition is called a separable state. For pure and mixed two-qubit systems entanglement is well characterized [21]. But as soon as more than two qubits are involved the task of characteriza-tion becomes significantly harder, because of the rapid increase in complexity

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of the states. In the two-qubit case there are four well-known entangled states, the so-called Bell states. They are defined as,

|ψ±i =√1

2(|01i ± |10i), |φ

±i =1

2(|00i ± |11i). (1.9)

These four states constitute an orthonormal basis spanning the two-qubit Hilbert space. It is possible using only local operations to convert each of the Bell states to any of the three others, for instance,

|ψ−i =11 ⊗11|ψ−i, |ψ+i = σz⊗11|ψ−i

|φ−i =11 ⊗ σx|ψ−i, |φ+i = σz⊗ σx|ψ−i.

(1.10) As can be seen, only the flip operation σx and the π phase shift operation σz

are needed. Note that11 ⊗ σy gives the same result as σz⊗ σx up to a global

phase.

One might wonder what is so special and strange with these states. Let us take a look at |ψ−i. If we choose to measure the state in the σz basis we

would see that the measurement results from the two qubits are always op-posite, that is, if one is +1 then the other is −1 and vice versa. This is only a normal correlation and one can argue that the particles are prepared in this way. But if we choose to measure the state in the σx basis the same type of

correlation will be found in the measurement outcome. In fact, every time we measure these two qubits in the same basis we will obtain perfect anticorre-lations. Somehow the two qubits seem to communicate to align themselves according to how they will be measured, even though the involved qubits can be separated miles from each other with no means of communicating. These correlations are stronger than similar non-communicating parts of distributed and seemingly simple systems in everyday life.

It is worth noting that if one of the Bell states is distributed to two parties, Alice and Bob, and no classical communication is established between them, then all their measured data will indicate that they have each been given a completely depolarized qubit state. This lack of communication that results in ignorance of the parties involved can be accounted for by taking the par-tial trace over of the ignored parties, ρB= TrA(|ψ(i)ihψ(i)|). The partial trace

reduces the two-qubit state to a one-qubit state by summing over elements of the density operator to create a new operator, ρBj,k= ∑ihi, j|ρ|k, ii, where

i, j, k ∈ {0, 1}, referring to the qubit basis for each party. Performing this op-eration when Bob ignores Alice will leaves Bob’s qubit in the state ρB=11/2

which is the completely mixed state. A consequence of this is that if one of the parties is not willing to collaborate, the reduced state of one party will not include any information of the other. The parties need to collaborate to obtain any usable correlation.

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1.1 Bits, Qubits and Entanglement 11

A collaboration is also needed when creating entanglement. Two parties, Alice and Bob, sharing a product state |ψ1iA⊗ |ψ2iB can never create

en-tanglement by LOCC. Each local operation UA and UB will affect only their

local qubit Hilbert space, UA⊗ UB|ψ1iA⊗ |ψ2iB= |ψ10iA⊗ |ψ20iB, and the

re-sult will still be a product state between Alice and Bob. Therefore, the par-ties need to meet and perform a joint measurement to create entanglement, or one of the parties needs to send parts of an pre-entangled state through a communication channel. One way of creating entanglement is through a con-ditioning gate, which is similar to a control gate in electronics but operating in the quantum regime. Suppose we begin with the product state |x, 1i|z, 1i = (|0i|0i + |1i|0i)/√2, then using a quantum control NOT-gate,

11 + σz 2 ⊗11 + 11 − σz 2 ⊗ σx=       1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0       , (1.11)

the state |x, 1i|z, 1i is converted into |φ+i. In the control NOT-gate one qubit is used as control and the other as target. If the control is |0i nothing will happen to the target, but if it is |1i the target will undergo a flip realized by the σxoperator in (1.11). Note also that operating with the control not-gate on

an entangled state |φ+i will transform it back into a product state. In this way one can design a two-qubit measurement device that maps the Bell states to product states which are experimentally easier to measure, see section 2.4 for an experimental realization of the control NOT-gate.

1.1.6 Entanglement in Mixed States

Entanglement in mixed states has a slightly different definition than entangle-ment in pure states.

Definition: A mixed state ρ is called entangled if it can not be written as a sum of product states, that is,

ρ 6= ∑iωiρ1i⊗ · · · ⊗ ρni, (1.12)

where1 to n refers to the local Hilbert space that the i state is living in. States that satisfy the right hand side of (1.12) are called separable. Also, we say that there is a separable cut in a state ρ ∈HA⊗HB if we can

write ρ = ∑iωiρiA⊗ ρiB, with ρiA and ρiB being inHA andHB, respectively.

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in general when more parties are involved more cuts can exist. Specifically, for a separable state such as the right-hand side of (1.12) we have n − 1 cuts, 1|2|..|n.

To show whether a state is entangled or not is in general a difficult task. In the previous section 1.1.5 we have seen that the Bell states (1.9) constitute a basis and thus any pure separable state can be constructed by linear com-binations of (1.9). Thus entanglement can be lost by coherently combining entangled pure states. A similar situation can occur when mixing pure entan-gled states. The equal weighted mixture of the Bell states |φ+i and |φ−i is an

example illustrating this fact. Even though it is constructed by two entangled states, the equal mixture is separable and no entanglement is present,

1 2|φ +ihφ+| +1 2|φ −ihφ| =1 2|0ih0| ⊗ |0ih0| + 1 2|1ih1| ⊗ |1ih1|.

We observe that a separable cut is present between the qubits. The same hap-pens for an equal mixture of |ψ±i. An equal mixture of all four Bell states results in the completely depolarized two-qubit state11/4.

1.1.7 Distillation and Bound Entanglement

Many quantum protocols rely on pure maximally entangled states such as the Bell states. One important example is quantum teleportation. In fact, quantum teleportation is often the underlying effect for the protocols to work. Generat-ing and distributGenerat-ing perfect maximally entangled states between long distant parties is difficult. The surrounding environment induces decoherence and renders the pure quantum state to a mixed state. Thus reliable teleportation cannot directly be achieved by distributing the resources through these noisy channels. To solve this dilemma one can use a distillation protocol. Distilla-tion is the ability to extract from many noisy states fewer states that are closer to one of the Bell states (1.9). Bennett and collaborators [8, 9] showed that it is possible for two parties to distil n < N purer entangled states from N noisy entangled states. With a sufficient amount of copies the two parties can come arbitrary close to one of the Bell states in (1.9).

A general distillation protocol can be described as follows. Suppose that

two parties Alice and Bob have a large number N of pairs ρ ∈HA⊗HB

which are noisy but entangled. The joint N pairs are then described by ρ⊗N. By performing LOCC operations they can try to reduce ρ⊗N to a set of n two-qubit states with purer entanglement between them. As described in section 1.1.4, operations of this form, omitting the normalization, can generally be described by [10, 20], b ρ⊗n=

i Ai⊗ Bi· ρ⊗N· A†i ⊗ B † i.

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1.2 State and Entanglement Verification 13

Hereρb

⊗ndenotes the distilled states and A† i, B

i are Alice’s and Bob’s

opera-tions in their separateHA/B⊗N Hilbert space. The operations A†i and B†i project the state ρ⊗Nto a sub-space of (HA⊗HB)⊗Nwhich is non-separable. In short,

the idea is to use a big Hilbert space and project down to a smaller one where entanglement is more concentrated between the parties. It is assumed in dis-tillation protocols that only LOCC are used between all parties since they are separated and cannot transport their qubits to another laboratory.

For a set of two-qubit states that are inseparable it has been shown [22] that regardless of how small its amount of entanglement is it is always possible to distil out a Bell state. One might falsely conjecture that any inseparable state can be distilled. Surprisingly, this is not true in multi-qubit and higher-dimensional scenarios. There are states that are entangled and do not admit any distillation protocol [23]. These are the so-called bound entangled (BE) states, which are defined as,

Definition: If a state is entangled but is not distillable by LOCC it is called a bound entangled state.

These are indeed curious states since they require entanglement when created but then the entanglement is not available for distillation. This situation has been compared to thermodynamics [10], where there is free energy that can perform work and the bound energy which is unavailable to perform work. In the case of entanglement, the equivalent to work is for example reliable data transmission through quantum teleportation. Thus two different types of entanglement can be considered to exist in a noisy quantum system, a free and a bound type. Furthermore, BE is an example of an irreversible process since many BE states can be generated from a pure state affected by LOCC, but then this process can not be reversed once the state is brought into a bound entangled regime. In chapter 3 we will discuss a set of BE states that we then experimentally investigate.

1.2

State and Entanglement Verification

To experimentally generate a desired state is difficult and in the end we are left with something that is hopefully close to the desired theoretical state. If the state is too complicated and a reconstruction of the full density matrix is not possible, we are obliged to only look at certain characteristics of the state. The verification of important characteristics such as entanglement and that these properties are the desired ones requires tools that can detect and quantify or discard different properties. Here we will introduce different methods that can verify the quality of an experimental state and characterise its entanglement

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properties in different ways. The discussion of a very powerful method which allows for the reconstruction of a complete density matrix is postponed to the experimental part, section 2.6.5, because it requires some understanding of the measurement process and the output format of measured data.

1.2.1 State Fidelity

If the density matrix is available then a measure quantifying the distance be-tween two states is the fidelity. The fidelity bebe-tween two state ρ and δ is defined as,

F(ρ, δ ) = Tr(p√

ρ · δ ·√ρ ). (1.13)

If one of the states is a pure state the fidelity reduces to

F(ρ, |φ ihφ |) = phφ|ρ|φi. Thus the fidelity is related to the overlap

between the states ρ and |φ ihφ |, which is simply the probability to project the state ρ onto the state |φ i. The fidelity ranges from 1 for perfect resemblance between the states to 0 when no resemblance exists between the states. No resemblance is here equivalent to orthogonality and perfect resemblance means that the prepared state is equal to the desired one.

1.2.2 Witness Method

To verify if an experimental state has the proper entanglement properties one can use a witness operator. This is a powerful technique which tests with rather few measurements the entanglement properties of an experimental state. A witness operator ω is defined as an observable with negative expectation value Tr(ωρ) < 0 for a set of states ρ that has the desired entanglement properties. A positive expectation value Tr(ωρ) ≥ 0 indicates that the state might not have the right entanglement properties but it is not conclusive. It has been shown that for each inseparable state ρ in a bi-party system inH1⊗H2there exists

an operator ω such that Tr(ωρ) < 0 and Tr(ωρ) ≥ 0 for all separable states [24]. This would not be particularly useful if it had not also been found that a witness ω could be optimised [25,26] and decomposed by local measurements [26, 27].

Maximum Overlap Witness

Finding a witness can be a tedious task, but fortunately there are some general results which simplify the search. Maybe the most common witness optimiza-tion for an expected state is the maximum overlap witness [28],

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1.2 State and Entanglement Verification 15

Here α = max|φ i∈ϒ(|hψ|φ i|2) is the maximum overlap of |ψihψ| calculated for

all states in the set ϒ that are to be disregarded. Finding α gives an operator that can indicate whether a state is close to |ψi and has the same entanglement properties.

In the two-qubit case the optimization is over the set ϒ of all separable states. For the two qubit Bell states four witnesses can be constructed, these are given by,

ωψ±=1

2·11 − |ψ ±ihψ±|

ωφ±=12·11 − |φ

±ihφ±|. (1.15)

The only task left is to decompose these witnesses into local measurable op-erators. In this case this is easy since the Bell states can be rewritten in a form containing only squares of Pauli matrices (1.3) and the identity matrix.

Stabilizer Witness

Another method is to use stabilizers [26] to find a witness. Instead of using the state |ψi one uses the operators that stabilizes the state. A stabilizer Si is

an operator which has |ψi as an eigenstate and 1 as eigenvalue, Si|ψi = |ψi.

The idea is that many N qubit entangled states are uniquely defined by N sta-bilizers which are composed of local sigma matrices. By only knowing some of the stabilizers a witness can be constructed. Furthermore, for mixed states this method can simplify the search by finding stabilizers that stabilize the mixed state and not only some of the pure states in the mixture. An impor-tant condition for constructing a witness with stabilizers is that the stabilizers cannot commute over the set of states which is used in the optimization. The reason for this is that two stabilizers commute if and only if there is a pure product state among their common eigenstates [26]. A witness can be found by replacing the state in (1.14) by stabilizers,

ω = α ·11 − ∑iSi, (1.16)

where α = max|φ i∈ϒ(hφ | ∑iSi|φ i) is the maximum expectation value

calcu-lated for all states in the set ϒ that are to be disregarded. It is not necessary to optimize over mixed states since α will also give a bound for all mixed states that can be constructed from the set ϒ. This witness approach will be used later on to find entanglement witnesses for bound entangled states.

1.2.3 PPT-Criterion

Asher Peres [21] derived a powerful and useful separability criterion called the positive partial transpose (PPT) criterion. It states that a state ρ ∈HA⊗HB

is entangled if its partial transpose ρtB =11 ⊗ T

B· ρ has a negative

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|ai, bji ∈HA⊗HB. Each position of the elements ρi, j,k,l= hai, bj|ρ|ak, bli of

ρ is transposed such that ρtB elements are ρtB

i, j,k,l= ρi,l,k, j. Partial transposition

is basis-dependent but the spectrum is not.

To see how this criterion can be used let us assume we have a bipartite separable state ρ = ∑iωiρiA⊗ ρ

B

i . Taking the partial transpose means that we

transpose only one of the subsystems, say B, ρtB= ∑

iωiρiA⊗ (ρiB)t. (1.17)

Since ρ and ρiB are quantum states they have real and positive eigenvalues. The transposition does not effect the eigenvalues, thus (ρB

i )t is a legitimate

state and also ρtB, which means that it must have positive eigenvalues. This shows that one always has a positive partial transpose over any separable cut. As a simple counter-example let us apply the PPT-criterion on the state ρφ+ = |φ+ihφ+|, where |φ+i ∈HA⊗HB is one of the Bell states in (1.9). In

the matrix formulation expressed in the computational base we obtain,

ρφ+=       1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1       =⇒ ρtB φ+=       1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1       . (1.18)

The eigenvalues of ρφ+ are 0 and 1. After partial transposition of qubit B the

density operator ρtB

φ+has the eigenvalues −0.5 and 0.5. As expected, we obtain

a negative eigenvalue indicating that the state is entangled.

For systems of dimension 2 ⊗ 2 and 2 ⊗ 3 it is a sufficient and necessary condition, thus giving a complete characterization of separability. In higher dimensions the criterion is no longer sufficient. The insufficiency is related to the fact that the PPT-criterion is not only a powerful method for detect-ing entanglement, but it is closely related to entanglement distillation [10]. As mentioned above, separable states do not violate the PPT-criterion and are also non-distillable since no entanglement can be created by LOCC over a separa-ble cut. More generally it was shown that the violation of the PPT-criterion is a necessary condition for distillation [10]. But if a state violates the PPT cri-terion, then entanglement is present and there might be a distillation protocol that can be used. Thus entangled states that do not violate PPT are not distill-able and are bound entangled by definition. It is still an open question if there are bound entangled states which violate the PPT-criterion. A complete char-acterization of the experimental density matrix is required to experimentally evaluate the partial transpose.

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1.3 Hidden Variable Models 17

1.3

Hidden Variable Models

Regarding measurements, quantum mechanics differs greatly from classical physics. In contrast to the perfect measurement outcomes predicted by classi-cal physics, only statisticlassi-cal predictions can be deduced by quantum mechan-ics. Nevertheless it is possible to construct correlations between measurement outcomes which are stronger than allowed by classical physics. These differ-ences allowed Einstein, Podolsky, and Rosen (EPR) to propose a paradoxical example [1], which suggests that quantum mechanics only gives an incom-plete description of nature. This example started the debate on whether quan-tum mechanics can be completed with hidden variables (HV). Schrödinger pointed out the fundamental role of quantum entanglement in EPR’s example and concluded that entanglement is “the characteristic trait of quantum me-chanics” [29]. However, Bohr argued that similar paradoxical examples occur every time we compare different experimental arrangements, without the need of entanglement nor composite systems [30].

The underlying idea in the HV models is to consider reasonable assump-tions of the world, for example that each particle has preestablished quanti-ties, and then investigate if quantum mechanics could be substituted by this theory and satisfy all assumptions. John Bell constructed a hidden variable model [31] in 1966 that could predict all outcomes of any measurement on a two-level system like a qubit. This was followed by two proofs of the con-trary for systems with higher dimensionality than a qubit, the Kochen-Specker theorem [3, 32] and Bell’s inequality [2].

Here we will discuss and present several inequalities that will be used later on in the experiments. First we look at a Bell-type inequality derived by Clauser-Horne-Shimony-Holt (CHSH) where each party is at separated lo-cations. This inequality will then be expanded from two to four parties for the experiments on bound entanglement. Then three inequalities are presented which are more in the spirit of Kochen-Specker. These can be tested by se-quential measurements on a single system. Each of these inequalities tries to capture different aspects of the non-classicality of quantum mechanics and will subsequently be subjected to experimental verification.

1.3.1 Bell Inequality

Entanglement admits correlations in a system that are stronger than correla-tions in classical physics. This gives rise to quantum systems that look like they can affect each other without any link between them. This is captured in Einstein’s correspondence with Born where he famously derided entangle-ment as “spooky action at a distance”. To resolve this matter hidden variable models with desired constrains where proposed. But John Bell managed to

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derive inequalities that give a bound for the correlations allowed by the hid-den variable assumptions [2] but when applied to an entangled quantum state a violation occurs. This violation indicates that no HV model that satisfies all the assumptions can reproduce all predictions asserted by quantum mechan-ics. Besides indicating that the system at hand cannot be described by a HV model this also gives us a tool to verify that entanglement is present in an experimental state.

Derivation of Clauser-Horne-Shiminy-Holt Inequality

The derivation of the Clauser-Horne-Shiminy-Holt (CHSH) inequality for two parties proceeds as follow [4, 33–35]: Let us put aside quantum mechanics for the derivation. Suppose two parties Alice and Bob are situated in different laboratories. In both laboratories there is some type of equipment with two lamps on it, one indicating the value +1 and the other −1. There also is a knob that can be set in two different ways. For Alice this adjustable parameter is denoted by a and for Bob similarly by b. Let us refer to this equipment as a measurement box; we do not concern ourselves with its function or inner workings. Both parties are monitoring the blinking lamps and are keeping records of the events, ±1, depending on the settings, a and b. The probability that the outcome is i ∈ {+1, −1} when the setting a is set on Alice’s side is P(i|a). This probability can be calculated from the records that are kept. In the same way Bob can calculate P( j|b) from the setting b and from j ∈ {+1, −1}. After completion of the experiment they can together through their records calculate the joint probability P(i, j|a, b) of obtaining the outcome i with the setting a on Alice’s side and j with the setting b on Bob’s side. We impose two crucial assumptions about locality and realism in our experiment:

• Realism: The probabilities do not need to depend solely on the rameter a and b but also on some set of parameters λ . These pa-rameters characterise any other dependency that the probability of an event can depend on but are unknown or disregarded by the parties; λ is usually referred to as the hidden variable. Let P(λ ) be the proba-bility that the parameter λ will occur. The result of Alice’s statistical measurement P(i|a) is then given by ∑λP(i|a, λ ) · P(λ ) and similarly

for Bob with his parameters. Any correlations between the systems come from the parameter λ and are described by joint probabilities of the form P(i, j|a, b) = ∑λP(i, j|a, b, λ ) · P(λ ).

• Locality: The probability measured on Alice’s (Bob’s) box is in-dependent of any distant system such as Bob (Alice), it only de-pends on its local environment. This implies that the joint probability P(i, j|a, b, λ ) can be factorized as P(i|a, λ ) · P( j|b, λ ). This is usually

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1.3 Hidden Variable Models 19

guaranteed with a spacelike separation between the parties when they choose the measurement settings and agree upon the definition of a coincidence.

The first assumption allows us to assume that Alice’s (Bob’s) probabilities are governed by her (his) choice of settings and some parameter λ . Note that we have not assumed that Alice (Bob) can deterministically know which out-come will happen even if she (he) knows λ , but it has been shown [36, 37] that one can always extend a hidden variable of a non-deterministic model to a deterministic one where the probabilities of the form P(i|a, λ ) are either 1 or 0. Together with the second assumption the result is that the two systems are completely decoupled from each other aside from the classical link that the parameter λ offers. Correlations between Alice and Bob are thus described by joint probabilities of the form P(i, j|a, b) = ∑λP(i|a, λ )P( j|b, λ )P(λ ). We

will refer to correlations that are built up by joint probabilities of this type as classical correlations, and in this context they describe local-realistic models. The expected average outcome between Alice and Bob with the settings a and b is then given by,

E(a, b) = ∑i, ji· j · P(i, j|a, b)

= ∑i, j∑λi· j · P(i|a, λ )P( j|b, λ )P(λ )

= ∑λE(a, λ )E(b, λ )P(λ ).

(1.19)

Here, E(a, λ ) is for example from the local expectation value given by E(a) = ∑i∑λi· P(i|a, λ )P(λ ) = ∑λE(a, λ )P(λ ), where E(a, λ ) = ∑ii· P(i|a, λ ).

Ob-serve that the functions E(a, λ ), E(b, λ ) and E(a, b) range from a perfect op-posite result −1 to a perfect equal result 1. Now let us look at the following measurement sequence,

| E(a, b) + E(a, b0) | + | E(a0, b) − E(a0, b0) | ≤ ∑λ(| E(a, λ ) || E(b, λ ) + E(b

0, λ ) | + | E(a0, λ ) || E(b, λ ) − E(b0, λ ) |) P(λ )

≤ ∑λ(| E(b, λ ) + E(b

0, λ ) | + | E(b, λ ) − E(b0, λ ) |) P(λ ) ≤ 2 ,

where we have used | E(a, λ ) |≤ 1 and that the sum of real values is less then the sum of the absolute values. For the last step we used the lemma that | x + y | + | x − y |≤ 2 for x, y ∈ [−1, 1]. Thus the inequality becomes

| E(a, b) + E(a, b0) | + | E(a0, b) − E(a0, b0) |6 2 , (1.20) which is the CHSH inequality. How is this related to quantum mechanics? The expected average outcome E(a, b) is defined in quantum mechanics as E(a, b) = Tr(A(a) ⊗ B(b) · ρ), where A(a), B(b) are quantum operators and ρ

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the state produced by a source that is distributed to the parties. Experimentally, the quantity E(a, b) is usually calculated through the formula E(a, b) = ∑i, ji·

j· P(i, j|a, b).

To illustrate the violation of (1.20) by quantum mechanics we use the oper-ators, A(a = x) = σx, A(a0= z) = σz B(b = +) =σx+σz 2 , B(b 0= −) =σx−σz 2 . (1.21)

If the source produces the pure state ρ = |ψ−ihψ−|, the terms on the left-hand side of (1.20) are all equal to −1/√2except for the last term which is equal to 1/√2. The sum of these numbers gives | −2√2 | 2. This violation can be shown to be the maximum allowed by quantum mechanics [38]. Thus for any quantum mechanical systems and irrespective of the way of measurement, one can never obtain a value greater than 2√2in a bipartite scenario as described above.

In the above derivation there is no reference to what type of state and mea-surements the inequality is supposed to be optimized in order to obtain a vio-lation. For instance violation is also obtained for |φ+i, but not for all mixtures or superpositions between the two states |ψ−i and |φ+i. In contrast, we

ob-tain no violation for |ψ+i and |φ−i when using the operators in (1.21); the

left-hand side of the inequality (1.20) is then 0. But by switching b and b0 in (1.21), B(b = −) and B(b0= +), a maximal violation is again obtained.

1.3.2 Kochen-Specker

The Kochen-Specker (KS) theorem illustrates with great precision Bohr’s in-tuition that each time we compare different experimental arrangements para-doxical conclusions can be drawn. The theorem states that, for every physical system with dimension higher than two there is always a finite set of tests such that it is impossible to assign them predefined non-contextual results in agreement with the predictions of quantum mechanics. Remarkably, the proof of the KS theorem requires neither a composite system nor any special quan-tum state, it holds for any physical system with more than two internal levels, independent of its state.

Here we like to assign values to a set of observables that are in a non-contextual setting. To understand what non-contextuality means let us first consider compatible measurements.

• Compatible measurement: If a physical system is prepared in such a way that the result of test [experiment] A is predictable and repeat-able, and if a compatible test B is then performed (instead of test A)

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1.3 Hidden Variable Models 21

a subsequent execution of test A shall yield the same result as if test Bhad not been performed [39].

• Non-contextuality: A non-contextual model is a model where the measurement of A does not depend on which context A is measured in. If B and C are compatible with A but not necessarily with each other then the two contexts, A measured together with B or A mea-sured together with C, are not changing the outcome of A.

In quantum mechanics, compatibility means that the operators A and B com-mute, that is, there is at least one basis that diagonalises both operators. For a two-level system, two non-equal tests can never be found to be compatible since if two operators commute they must be the same up to a global phase, but for three and higher levels it is possible. Note also that in classical physics we assume that any set of experiments can always be made to be compati-ble by carefully performed measurements. The proof that we will describe is based on counterfactual logic in a four level system derived by Mermin [40]. We will later restate this KS proof in form of an inequality which can be tested experimentally.

Proof of the KS Theorem

Consider the nine dichotomic observables in (1.22). Each observable can have the value +1 or −1.

A= σz⊗11, B=11 ⊗ σz, C= σz⊗ σz,

a=11 ⊗ σx, b= σx⊗11, c= σx⊗ σx,

α = σz⊗ σx, β = σx⊗ σz, γ = σy⊗ σy.

(1.22)

Let us assume that we can ascribe the values υ(A), υ(B), ...,υ(γ) to each ob-servable. It is possible to construct several constraints that need to be satisfied by observing that each row and column of (1.22) constitutes of compatible observables. Due to the fact that if a functional relation F(A, B, ..) = 0 holds for a set of compatible observables then the results of measuring the set of observables must also satisfy F(υ(A), υ(B), ..) = 0, we obtain the constraints:

υ (A)υ (B)υ (C) = 1, υ (a)υ (b)υ (c) = 1, υ (α )υ (β )υ (γ ) = 1, υ (A)υ (a)υ (α ) = 1, υ (B)υ (b)υ (β ) = 1, υ (C)υ (c)υ (γ ) = −1 .

(1.23) All constrains need to be satisfied simultaneously for a given set of values υ (A), υ (B), ..., υ (γ ) ascribed to the observables. Thus the product of all left-hand sides in (1.23) must give a result of 1, since each value appears twice. But now we get a contradiction with the product of the right-hand side which is equal to −1. Thus it is not possible to fulfil all constraints in (1.23) by assuming that the operators have fixed values and are non-contextual.

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State-independent KS Inequality

In contrast to a Bell inequality that is violated by certain quantum states, the KS theorem uses counterfactual logic without referring to any particular state. It was found by Cabello [16] that one can construct an inequality that captures the impossibility to assign values υ(A), υ(B), ...,υ(γ) to the observables in (1.22). The inequality reads,

h χ i = h ABC i + h abc i + h αβ γ i + h Aaα i + h Bbβ i − hCcγ i ≤ 4 , (1.24) where the bound can be found by performing an exhaustive search over all 29 possible ways of ascribing the values ±1 to υ(A), υ(B), ..., υ(γ) . Quantum mechanically, the operator product of each row or column gives the identity

operator up to a sign, ABC = abc = αβ γ = Aaα = Bbβ = −Ccγ =11. This

means that for each measured state the obtained expectation value of each term in (1.24) will always be 1 except for the last term which is −1, thus h χ i = 6 regardless of the state.

1.3.3 Fully Contextual Correlations

As already mentioned, quantum correlations can be stronger than those al-lowed by classical physics. This leaves the question of what part of a violation of an inequality can be attributed to the “classical” model. Here we identify a simple non-contextual inequality, where the quantum violation cannot be im-proved by any hypothetical post-quantum resource. This will bound the part which can be attributed to a “classical” model to zero. The simplicity of the inequality offers an experimental approach to give a very low bound on the content. This will be discussed in detail in section 4.1, where the experiment is presented.

To reveal that there are some contextual correlations in an experiment, it is common to violate an inequality, which is an expression like,

I(P) =

Ta1...anx1...xnP(a1. . . an|x1. . . xn) ≤ ΩNC≤ ΩQ≤ ΩC, (1.25) where Ta1...anx1...xn are real valued numbers and ΩNC is the maximum value of the left-hand side for non-contextual correlations. Similarly, ΩQ and ΩC

denote the maximum value of the left-hand side for quantum and contextual correlations, respectively. Also, P(a1. . . an|x1. . . xn) are the joint probabilities

of obtaining outcomes a1. . . an, when compatible measurements x1. . . xn are

performed. The non-contextual theories are those for which we can write P(a1. . . an|x1. . . xn) = ∑λP(λ ) ∏

n

i=1P(ai|xi, λ ). Note that this is only a

gen-eralization of the local hidden variable models obtained during the deriva-tion of the CHSH-inequality. The difference here is that we must

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explic-1.3 Hidden Variable Models 23

itly assume that the measurements x1. . . xn are compatible. For the

CHSH-derivation, compatibility is guaranteed by a spacelike separation between the parties. When a separation can be accomplished then (1.25) is a Bell inequal-ity instead.

The joint probabilities P(a1. . . an|x1. . . xn) for contextual models, satisfying

P(a1|x1) = ∑a2. . . ∑anP(a1a2. . . an|x1x2. . . xn) for all x2. . . xnand similarly for any other P(ai|xi), can be expressed in terms of a non-contextual and a

con-textual part as:

P(a1. . . an|x1. . . xn) = wNC· PNC(a1. . . an|x1. . . xn)

+(1 − wNC) · PC(a1. . . an|x1. . . xn),

(1.26)

where 0 ≤ wNC≤ 1 is the fraction of non-contextual correlations and (1 −

wNC) is the fraction of contextual correlations. To quantify the amount at-tributed to the non-contextual part in the joint probabilities P(a1. . . an|x1. . . xn)

we can use wNC. But the decomposition (1.26) might not be unique, therefore

we focus on the decompositions that maximizes wNC.

Definition: We call the maximum of wNCover all possible decompositions of

the form (1.26) thenon-contextual content and denote it by WNC.

Note that the decomposition (1.26) and the definition is parallel to the ones introduced in [37, 41]. In fact, for correlations generated through spacelike separated experiments, the non-contextual content is exactly the local content defined in [41].

Any experimental violation of an inequality of the form (1.25) provides an upper bound on WNC, specifically we have the relation,

WNC≤

ΩC− Ωexp

ΩC− ΩNC

. (1.27)

This follows from the fact that we can divide left-hand side of (1.25) for any experiment into a part containing the non-contextual correlations and into an-other part containing the contextual correlations as in (1.26),

Ωexp≡ WNC·I(PNC) + (1 −WNC) · I(PC) ≤ WNC·ΩNC+ (1 −WNC) · ΩC. (1.28)

If we believe in quantum mechanics then the maximal experimental violation occurs when Ωexp= ΩQ, thus the smallest value of the numerator of (1.27)

is obtained when we saturate the quantum bound. To reveal fully contextual correlations, WC= 0, the best option is to test a non-contextual inequality such

that it is violated by quantum mechanics and its maximum quantum value equals its maximum contextual value, ΩNC< ΩQ= ΩC.

References

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