Contributions à l'étude des corrélations quantiques.

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DOCTORAL THESIS

presented by

Fatimazahra SIYOURI

Discipline: Theoretical Physics.

Speciality: Physics and Technologies of Information.

Contributions to the study of quantum correlations

In front of the jury : President:

Yassine HASSOUNI - PES- Faculty of Sciences, Rabat Examiners:

Morad El BAZ - PH- Faculty of Sciences, Rabat

Olivier Giraud - Professor of Physics, Paris 11 university, France. Noureddine Fettouhi - PES- Faculty of Sciences, Rabat

Az-EddineMARRAKCHI - PES- Faculty of Sciences, Fes El Houssaïne EL RHALEB - PES- Faculty of Sciences, Rabat Hassan Tahri - PES- Faculty of Sciences, Oujda

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1

ACKN OWLEDGEMEN T

The work we presented in this thesis was achieved in the Laboratory of Theoretical Physics (LPT) at the Faculty of Sciences of Rabat.

At rst, i thank God for his blessing and for giving me strength and patience to do this work.

This thesis was led by Mr Morad EL BAZ and Mr Yassine HASSOUNI, Professors at the Faculty of Sciences Rabat. They have guided my rst steps in research with valuable advice. I would like to express my sincere gratitude to them for the continuous support of my Ph.D study and related research, for their extreme kindness, motivation, and immense knowledge. I want to think them also for their countless hours of reecting, reading and most of all patience throughout the entire process, this thesis would not have been complete without their guidance and constant feedback.

I want to thank Mr EL Houssaïne EL RHALEB, Professor at the Faculty of Sciences Rabat, to accept to be with the jury members.

I want to thank Mr Noureddine FETTOUHI, Professor at the Faculty of Sciences Rabat, to accept to be with the jury members. I express my gratitude and my thanks also for his encouragement and his kindness.

I also want to thank Mr Hassan TAHRI, Professor at the Faculty of Sciences Oujda, to accept to be with the jury members.

A very special thank you to Mr Olivier GIRAUD, Professor of Physics at the University Paris-11 France, to accept to come to morocco for being present with the jury members. Also for his extreme kindness and his relevant remarks.

Many thanks also to Mr Adil BELHAJ, Professor at the Multidisciplinary faculty Beni mellal, for his advices and his moral support.

I would also like to say a heartfelt thank you to my family for always believing in me and encouraging me to follow my dreams.

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1 Auteur : Fatimazahra SIYOURI

Titre : Contributions à l'étude des corrélations quantiques Directeur de thèse : Pr. Morad El BAZ et Pr. Yassine Hassouni

Résumé

Les corrélations quantiques ont attiré beaucoup d'attention en raison de leur potentiel d'utilité dans plusieurs applications de la théorie de l'information quantique. Les principaux intérêts de recherche de corrélations quantiques portent sur l'étude de leur comportement, de leur dynamique dans les systèmes quantiques ouverts ainsi que l'inuence de leur environnement sur ces corrélations et leur quantication.

Dans cette thèse, nous nous concentrons sur l'étude des corrélations quantiques dans les systèmes quantiques fermés et ouverts. Notre objectif est de montrer leur pertinence pour une compréhension approfondie du traitement de l'information quantique et des processus dynamiques sous-jacents. Dans ce contexte, nous avons fait une étude compar-ative de leur comportement pour les états comprimés GHZ. Nous avons traité les eets environnementaux sur leur dynamique dans le cas des états de Bell diagonaux. Nous avons emprunté la fonction de Wigner de l'optique quantique pour explorer son rôle dans la détection et la quantication de ces corrélations. À cet égard, en utilisant la négativité de la fonction de Wigner nous avons étudié les corrélations quantiques présentes dans les états de Werner en états bipartites à variables continues (états comprimés et états cohérents).

L'étude réalisée dans cette thèse représente un grand pas en avant dans la compréhen-sion des corrélations quantiques à savoir d'une part l'intrication quantique et la discorde quantique et d'autre part, les propriétés associées et leur comportement dans le cas des états mixtes à variables continues.

Mots clés: Corrélations quantiques; Discorde quantique; Intrication ; Les états co-hérents; Les états comprimés; Les états de Werner; Fonction de Wigner; Les états de Bell, GHZ.

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Contributions to the study of quantum correlations

Abstract

Quantum correlations are very important for the fact that they are considered as potentially useful in several applications of quantum information theory. The main interests studies of quantum correlation are the investigation of their behavior, their dynamics in open quantum systems and the inuence of their environments on it, and also their quantication.

In this thesis, we focus our attention on the study of general quantum correlations in both closed and open quantum systems. Our objective is showing their relevance for a thorough understanding of the quantum information processing and of the underlying dynamical processes. In this context, we investigated comparatively their behavior for Greenberger-Horne-Zeilinger (GHZ)-type squeezed states. Moreover, we processed the environmental eects on their dynamics in the case of Bell diagonal states. On the other hand, in the same direction, we borrowed the Wigner function from quantum optics to address its pertinence in detecting and quantifying these correlations. In this respect, by using the negativity of Wigner function we studied the quantum correlations that are present in Werner states made of continuous variables bipartite states (squeezed states and coherent states).

We deem that the study performed in this thesis does represent a major step forward in understanding the quantum correlations, quantum entanglement and quantum discord, the associated properties and their behavior in continuous variable mixed states.

Keywords: Quantum correlations; Quantum discord; Quantum entanglement; Co-herent states; Werner states; Wigner function; Squeezed states, Bell-diagonal states, GHZ.

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3 Auteur : Fatimazahra SIYOURI

Titre : Contributions à l'étude des corrélations quantiques Directeur de thèse : Pr. Morad El BAZ et Pr. Yassine Hassouni

Résumé étendu

Les corrélations quantiques ont attiré beaucoup d'attention en raison de leur po-tentiel d'utilité dans plusieurs applications de la théorie de l'information quantique. L'intrication quantique et la discorde quantique sont considérées comme les deux aspects les plus importants de ces corrélations. L'évaluation quantitative et qualitative de ces corrélations est la tâche essentielle dans les études conceptuelles de la mécanique quantique, elle a aussi une importance décisive dans la théorie de l'information quantique opératoire. Les principaux intérêts de recherche de corrélations quantiques portent sur l'étude de leur comportement, de leur dynamique dans les systèmes quantiques ou-verts ainsi que l'inuence de leur environnement sur ces corrélations et leur quantication. Dans cette thèse, nous nous concentrons sur la compréhension des aspects les plus importants des corrélations quantiques, l'intrication quantique et de la discorde quan-tique, en comparant leur comportement et en étudiant leur dynamique. De plus, nous examinons la pertinence de la fonction de Wigner dans la détection et la quantication de ces corrélations. Plus précisément, les travaux présentés dans cette thèse se focalisent sur trois sujets: Tout d'abord, nous étudions le comportement des corrélations quantiques (intrication quantique, discorde quantique normale et super discorde quantique) pour les états tripartites comprimés GHZ (chapitre 2). Dans cette étude, nous vérions la monotonie de la super discorde quantique en fonction des mesures faibles. Nous conrmons que la discorde quantique peut exister même en l'absence de l'intrication. En eet, le traitement quantique de l'information à l'aide de la discorde sera plus ecace que son traitement en utilisant l'intrication. Ensuite, nous étudions le comportement dynamique des corrélations quantiques pour les états de Bell diagonaux interagissant avec des réservoirs de déphasage indépendants (chapitre 3). Dans ce travail, nous comparons le comportement de la discorde quantique dans les régimes Markovien et non-Markovien

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et nous testons la possibilité d'apparition d'un changement soudain dans la dynamique de la discorde quantique dans le cas de trois qubits. Enn, nous étudions la possibilité d'utiliser la fonction de Wigner pour détecter et quantier les corrélations quantiques en général (chapitre 4). Notre objectif dans cette étude est non seulement de conrmer les résultats précédents (fonction de Wigner détecte les corrélations quantiques), mais aussi montrer que la fonction de Wigner n'est pas sensible à toutes les corrélations quantiques.

Le premier chapitre est consacré à présenter les outils fondamentaux sur les cor-rélations quantiques. Ce chapitre contient tous les éléments nécessaires au lecteur pour pouvoir comprendre la suite. Il contient ainsi un rappel sur les outils nécessaires de la mécanique quantique, et les diérentes manières de décrire l'état quantique d'un système physique. Le chapitre contient également une introduction à la théorie de l'information classique et quantique. C'est dans ce dernier contexte que s'inscrivent toutes les études qui suivent dans les chapitres suivants.

Le deuxième chapitre traite des dénitions mathématiques des corrélations quantiques. Le cas particulier et le plus connu de ces corrélations, est l'intrication quantique qui est un phénomène essentiel à plusieurs processus de l'information quantique. Ce phénomène est dénit au début du chapitre et ses propriétés sont présentées par la suite. Ensuite le chapitre, traite d'une manière plus générale des corrélations quantiques, qu'est la discorde quantique et la super discorde quantique. En eet, il a été établit dans la littérature que l'intrication quantique ne présente pas les seules corrélations quantiques existantes et qu'il en existe d'autres. La discorde quantique est une mesure qui permet de capter toutes ces corrélations quantiques. En relaxant les conditions de mesures quantiques, on obtient la super discorde quantique qui peut ainsi être vue comme une mesure encore plus générale des corrélations quantiques. Ce chapitre contient aussi une contribution à l'étude de ces trois types de mesures dans le cas d'un système physique tripartite d'états comprimés.

Le troisième chapitre traite le comportement dynamique des corrélations quantiques. L'évolution des corrélations quantiques présentes dans un système physique donné est étudiée quand ce dernier est couplé à un environnement non-Markovien. Ce chapitre se base aussi sur une contribution originale, où nous avons étudié la dynamique non-Markovienne des corrélations quantiques quand le système est décrit par les états de Bell diagonaux. Ce travail a permis d'élucider des questions pertinentes quant au comportement des corrélations quantiques. Ainsi la disparition et la réapparition soudaine des corrélations quantiques a été mise en évidence dans le cas multipartite.

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5 Le quatrième chapitre s'articule autour de deux contributions majeures, et attaque le problème de détection et quantication des corrélations quantiques. La méthode proposée dans ces travaux originaux consiste à utiliser la fonction de Wigner pour ce rôle là. En traitant les diérents cas dans nos travaux de recherche, nous avons démontré que cette fonction peut jouer ce rôle. Cependant, nous avons montré que la fonction de Wigner n'est pas sensible à toutes les corrélations quantiques mais seulement à l'intrication, cela veut dire que la fonction de Wigner ne peut détecter ecacement que l'intrication quantique. Dans ce cas, la négativité de la fonction de Wigner indique la présence d'intrication quan-tique dans le système étudié. En outre, cette fonction peut être utilisé pour quantier la quantité d'intrication présente dans un système donné. En eet nous avons démontré que le volume de la partie négative de la fonction de Wigner indique la quantité d'intrication présente dans le système.

Mots clés: Corrélations quantiques; Discorde quantique; Intrication ; Les états co-hérents; Les états comprimés; Les états de Werner; Fonction de Wigner; Les états de Bell, GHZ.

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Chapter 0 GENERAL INTRODUCTION

Quantum mechanics is the branch of physics that was developed in the early decades of the 20th century to study and describe phenomena highlighted experimentally on elementary particles and the mystery of the structure of the atom. It turns out to be very successful in results and various applications [1, 2, 3]. It is also regarded as the basis for understanding quantum information and solving dierent problems that classical physics failed to explain. Quantum information is the basic entity of study in quantum information theory which regroup the quantum mechanics and the Shannon's information theory (classical information theory). The quantum information theory is a new discipline that was born with the idea that, the classic physical supports of information will be replaced by quantum supports [4].

Generally, quantum mechanics yields an appropriate mathematical structure for describing a quantum systems. Unlike classical physics, its language implies density operators and wave-functions. However, in 1932, Eugen Wigner formulated quantum mechanics in terms of the Wigner function [5] in order to account for quantum corrections to classical statistical mechanics (the mechanical phase space is connected to classical mechanics) [6, 7]. In trying to approach it in a way one is inclined to fall into the semi-classical eld. This allows to compute expectation values of quantum mechanical observables in a classical manner (as an integral) rather than by the operator formalism. The Wigner function is a measurable quantity and it is a very well behaved function but nevertheless, it can also take negative values for a quantum state and therefore is a quasi-probability function. Its negativity is taken as a strong indication of a non-classical state [8]. So far, the Wigner function nds many applications in both classical and quantum optics [9, 10, 11, 12].

The strangeness of quantum mechanics reposes on the concept of quantum correlation that is originated from the principle of superposition. Quantum entanglement, as a special type of quantum correlation, is extensively recognized as the key resource of quantum communication and quantum computation [13]. Its role in various tasks in quantum information theory is incontestable. Historically, the term 'entanglement' was

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9 rst introduced in 1935 by Erwin Schrodinger [14] after he discovered certain of its bizarre properties and consequences. There is no true classical analog of quantum entanglement. It represents the strong non-classical correlations that two or more particles can possess, these particles may or may not be spatially separated. Generally, if we consider two entangled systems (correlated), propagating in two opposite directions, measuring the rst system in a certain basis always gives a result entirely determined by the outcome of the second in the same base. So, there is a perfect correlation of measurements performed on two entangled systems even if they are not causally connected. This result shocked the father of relativity Einstein, who had a local realistic vision of physics. This vision leads to the conclusion that if the act of measurement inuences the two systems, so, there must exist an inuence that propagates from one system to the other at a speed faster than that of light. In fact, the quantum formalism implies that the inuence of the measuring act on the two components of an entangled system is instantaneous (non-causal). In 1935, Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen (E.P.R) imagined a thought experiment in a resounding article which aims to demonstrate the incompleteness of quantum mechanics. This experiment leads to a paradox if one accepts entangled states really exist; this paradox is known as "EPR paradox" [15]. The argumentation was then reformulated by David Bohm in 1952 [16] for the straightforward quantum mechanical system of two spin 1

2 particles.

The essential argument of EPR is based on the following three requirements (see, e.g. Ref. [17]):

• Completeness: The theory is said to be complete, if each element of physical reality has a counterpart in the physical theory.

• Realism: The physical quantity has physical reality, if the value of this quantity can be predicted with certainty (probability equal to unity) without disturbing the system.

• Locality: Suppose that A and B are separated subsystems. Measurements on the subsystem A do not aect measurements on subsystem B that is far away. We conclude that there is no instantaneous action at a distance.

To explain the EPR problem, Albert Einstein proposed a local deterministic theory including "hidden variables"; this was not accepted by Niels Bohr who denied the existence of quantum states as long as they have not been measured. Bohr's response was qualitative about this paradox, and technology didn't allow to decide in a unquestionable way between the two views at the time. Therefore, the reality of the phenomenon of entanglement remained a question of viewpoint without direct experimental support until Alain Aspect's experiments. In 1964, the physicist John Bell proposed a mathematical description of the EPR thought experiment [18, 19] to prove the conditions upon which the type of incompleteness of quantum theory suspected by EPR was possible. He de-veloped a series of mathematical inequalities, called Bell inequalities, using the quantum

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spin instead of the position and momentum which were used in the EPR paradox. These inequalities have been dened as relations that the measurements on correlated states (entangled) must satisfy in the hypothesis of a local deterministic theory with hidden variable. If the inequality is not respected, this will be an indication of result of the nonexistence of hidden variables so the world is not local. Bell showed that the inequality can be violated if the postulates of quantum physics are true. In 1972, Freedman et al realized the rst experiment to verify Bell's inequalities [20]. They developed in this experiment a theory of non-local hidden variables using polarized photons. Then in 1982, for the same purpose Alain Aspect and his colleagues have devised a second experiment [21] by using polarized photons emitted by a source of Calcium. This experiment involves performing measurements on two entangled particles suciently remote from one another to avoid any transmission of information at the speed of light that can distort the mea-surement result. So Alain Aspect's team demonstrated in this experiment denitively the violation of the Bell inequality [21], thus he conrmed the non-locality of quantum physics. However, according to various studies [22, 23, 24], quantum entanglement is not the only type of quantum correlations and consequently the measures of quantum entanglement cannot be considered as a complete measure of quantum correlations for any quantum system. In this respect, it was concluded that there may be some other measures which can better quantify the quantum correlations for quantum systems. For any correlated bipartite quantum system Henderson et al indicated how one can get the classical part which can be used as a measure of classical correlations [22]. In 2001, Harold Ollivier and Wojciech H. Zurek [25, 24] and, independently L. Henderson and Vlatko Vedral [22] introduced a new surprising phenomenon called quantum discord. Olliver and Zurek referred to it also as a measure of quantumness of correlations[25]. This surprising phenomenon is dened as the dierence between two expressions which each represent the mutual information in the classical limit, one obtained by using conditional entropy and the other by performing local measurements on any one of the subsystems. The quantum discord is proposed to describe the quantum correlations, thus is not limited to entanglement. It must be non-negative, indeed the states with vanishing quantum discord can be identied with pointer states. For pure states, the quantum discord is equivalent to quantum entanglement.

During the last two decades, quantum correlations have been of renewed interest as the realm of quantum information science emerged and matured. Quantum entanglement and quantum discord are considered as the two paramount aspects of these quantum correlations.The quantitative and qualitative evaluation of such correlations is essential task in conceptual studies of quantum mechanics, and it also has decisive importance in operative quantum information theory. Recently, the quantum discord behavior and the quantum discord dynamics in several physical systems have attracted great attention both theoretically ([26]-[31]) and experimentally ([32]-[35]). Generally, a real quantum system is open and therefore will inevitably be aected by its environment [36]. The open quantum system is much brittle and easy to lose its coherence, due to the interaction with

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11 its environment. This loss of coherence is the principal problem for the implementation of quantum information processing. Therefore, it is worthwhile to study the dynamics of quantum correlations in open systems and try to search adequate approaches to protect these correlations from the inuence of their environments. Until now, there have been a lot of studies on the dynamics of quantum entanglement in realistic quantum systems ([37]-[46]). However, the eects of the environments (with and without memory) on the dynamics of quantum discord was relatively less investigated. The quantum discord dynamics was investigated initially under the Markovian approximation [47, 48], then it has been generalized to consider the non-Markovian environments with memory ([49]-[56]). On the other hand, recently, the study of quantum correlations by using the Wigner function has attracted considerable attention [9, 57, 58, 59, 60]. In this direction, [57] and [58] showed that entanglement can be detected by using the negativity of the Wigner function. In fact, the negativity of the Wigner function has played an important role with the rise of quantum information science. This important role is mainly due to the fact that it is regarded as a resource in quantum computing and simulation [61, 62]. Also, because it has been suggested as a suitable measure of quantumness [9]. In view of this, the following questions arise, whether Wigner function is sensitive to all kinds of quantum correlations or only entanglement? The volume occupied by the negative part of the Wigner function can be used as a measure of quantum correlations?

In this thesis we focus on understanding the most important aspects of quantum correlations, quantum entanglement and quantum discord, by comparing their behavior and studying their dynamics. Moreover, we address the pertinence of the Wigner function in detecting and signaling the presence of these correlations. More precisely, the works presented in this thesis revolve around three subjects: First, we study the quantum correlations behavior- quantum entanglement, normal quantum discord and super quantum discord, for tripartite GHZ-type squeezed states [63] (see chapter 2). In this study we verify that weak measurements reveal more quantumness in quantum systems [64], and we have armed the notion of quantum discord, which comes from the increasing of super quantum discord with decreasing measurement strength. We conrm that quantum discord may exist even without entanglement. In fact, quantum information processing using the discord will perform better than quantum information processing with entanglement. This work has permitted to characterize some properties of quantum correlations of tripartite GHZ-squeezed states, a study which can be useful in those quantum information protocols exploiting these states as a resource. Then, we study the dynamical behavior of quantum correlations for Bell diagonal states (BDS) interacting with independent dephasing reservoirs [65] (see chapter 3). In this work, we compare the quantum discord behavior in both Markovian and non-Markovian regime and we test the possibility of occurrence of sudden change in the dynamics of quantum discord in the case of three qubits of BDS. This study provides comparisons with Ref. [53], where the authors found that for some BDS which interact with their environments the calculation of quantum discord could experience a sudden transition in its dynamics.

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Finally, we investigate the possibility to use the Wigner function to detect and quantify quantum correlations in general [66, 67] (see chapter 4). Our objective in this study is not only to conrm the previous results [9, 57, 59, 58] (Wigner function does detect quantum correlation) but also answer the question of what kind of quantum correlations it does detect. This work allows us to describe the best feature of Wigner function for bipartite continuous states (coherent and squeezed states), which gives more eciency in quantum information theory.

This thesis is organized according to the following plan:

• In chapter I, we review the necessary tools and concepts of quantum mechanics and information theory. In particular, the postulates of quantum mechanics, the description of a quantum state, the essential phenomena of quantum physics: entanglement and superposition, and the presentation of classical information theory and quantum information theory.

• In chapter II, we shed some light on the study of quantum correlations and the associated measures. So, we present the entanglement and separability criteria, also we introduce several measures associated with this type of correlations in bipartite and tripartite systems. After that, we introduce the quantum discord - normal discord, super discord and geometric discord- as a measure of the total correlations. Finally, we present our paper [63], where we study the behavior of quantum correlations for tripartite GHZ-type squeezed states.

• In chapter III, we review all necessary concepts of the quantum theory of open systems, and we discuss the non-Markovianity and the corresponding measure for the degree of memory eects in terms of information ow between the system and its environment. Then, we present some of the results we obtained during the preparation of this thesis [65], where we investigate the dynamics of quantum correlations for a system of three qubits.

• In chapter IV, we present our paper [66], in which we consider the negativity of Wigner function as a measure of quantum correlations.

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Chapter 1 FUNDAMENTALS OF QUANTUM

INFORMATION THEORY

Quantum mechanics is without doubt, of all current physical theories, the one whose mathematical formulation seems more mysterious [68]. First, quantum information theory has been developed to give a better explanation for quantum mechanics. For that fact, this theory has attracted a lot of interest and it is considered as a rapidly developing eld which touches profoundly the basics of quantum physics. On the other hand, the interpretation of quantum basics in terms of quantum information oers the computer scientists the possibility of appropriating quantum theory with their tools and the physicists the possibility of apprehending the quantum mechanics as an information theory.

In this chapter we focus on studying the basics of quantum information theory, based on the necessary physical tools and the concepts that will help us better understand all that will come thereafter in the next chapters.

So, by using the mathematical formalisms we introduce the important principles of quantum mechanics. In particular, the description of state space using the wave-function, the evolution of quantum system, and the quantum measurements. But it seems that this wave-function cannot describe all quantum states, except pure ones. For that fact, we introduce then other tools to describe any arbitrary quantum state completely; the density matrix which is particularly appropriate when a system is not perfectly controlled and the Wigner function (continuous variables) for which the physical interpretation is often easier than the density matrix. After that, we give briey the formal denition of the most important phenomena of quantum mechanics, which are the superposition and the entanglement that will be more detailed in chapter two. Afterward, we dene the concepts of the classical information and Shannon's entropy which is the central point of information theory introduced by Claude Shannon [69, 70] in the early 1940's, this entropy consists to evaluate the maximum amount of information contained in a event. Then, we dene the quantum information theory, including the von Neumann entropy

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[71] and the associated quantum quantities using the foundations of classical information and mathematical tools of quantum mechanics.

1.1 Principles of quantum mechanics

1.1.1 State space description

In general, any isolated quantum mechanical system is associated with a complex vector space with inner product (a Hilbert space H), known as the state space of the system. The state of the system at time t is completely specied by a vector |ψ(t)i (Ket) of the complex Hilbert space, called state vector (or wave-functionψ(x, t)). It will be convenient to choose the |ψ(t)i unitary, such that:

kψ(t)k2 = hψ(t)|ψ(t)i = 1

In order for |ψ(t)i to be well dened, certain conditions are required: - |ψ(t)i must be a single-valued vector of the spatial coordinates.

- In order to satisfy the Schrödinger equation, the second derivative of |ψ(t)i must exist, which means that the rst derivative must be continuous.

- |ψ(t)i must have a nite amplitude over a nite interval. One of the simplest quantum mechanical system is the qubit.

1.1.2 Evolution of a quantum system

The temporal evolution of the state |ψ(t)i of an isolated physical system is described by the Schrödinger equation:

i¯h∂

∂t|ψ(t)i = H |ψ(t)i (1.1) where H is the hermitian Hamiltonian operator of the system.

Evolution in terms of a unitary operator

Consider |ψ(t1)i and |ψ(t2)i the states of the system at times t1 and t2 respectively.

The evolution of an isolated quantum system is described by a unitary operator U dened by the Schrödinger equation:

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1.1. Principles of quantum mechanics 15

|ψ(t2)i = U (t2− t1) |ψ1i (1.2)

where U depends only on the time interval between the start and end times. The expression of |ψ(t2)i is obtained by solving the corresponding Schrödinger equation, and

it gives:

|ψ(t2)i = e−iHU (t2−t1)/¯h|ψ1i , (1.3)

On the other hand, can evolve in dierent manners that cannot be describe by unitary operators. Consider for example an open quantum system interacting with the environ-ment or an interaction with a measuring device. This last kind of evolution is described next.

1.1.3 Quantum measurements

A general quantum measurement is described by an ensemble {Mm} of measurement

operators acting on the state vector of the measured system. If the state of the system before measurement is |ψi, then outcome m occurs with probability p(m) given by

p(m) = hψ| M_m†Mm|ψi (1.4)

and the state of the system just after measurement is |ψ0i = _q Mm|ψi

hψ| Mm†Mm|ψi

(1.5) The sum of the probabilities of all possible outcomes is equal to one (Pmp(m) = 1),

which implies that PmM †

mMm = 1

Projective Measurements

A special and widely used class of quantum measurements consists in the so-called projective measurement. Any projective measurement operation of a physical quantity is described by a hermitian operator A = A† _{acting in the state space, called observable.}

The spectral decomposition of this observable is given by A =X m λmPm = X m λm|ψmi hψm| (1.6)

where Pm is a projection operator onto the eigenspace associated with the eigenvalue λm,

and |ψmi is the orthonormal eigenvector of the operator A. The projectors satisfy the

closure relation PmPm = I.

Let |ψi denote the state of a system just before the measurement, so the probability of obtaining the outcome m is

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p(m) = hψ| Pm|ψi (1.7)

When the outcome m is obtained, the state of the system just after the measurement becomes ´ ψE= Pm|ψi pp(m) = Pm|ψi phψ| Pm|ψi (1.8) Note that in the case of projective measurements (von Neumann measures) described by a set of projectors Pm, the measurement operator satisfy the completeness relation

P

mPmPm† = I, as well as being an orthogonal operator, i.e hermitian operator satisfying

the following properties: Pm = Pm† and PiPj = δijPi.

Positive Operator Valued Measure

Positive Operator Valued Measure (POVM) is the most general formulation of a quan-tum measurement in the quanquan-tum physics theory. It is described by a measurement operator Em, such that

Em = Mm†Mm (1.9)

The operators Em are known as the POVM elements associated with the measurement

and the complete set {Em} is known as a POVM. The operators Mm are not necessary

hermitian, but they satisfy to the following equation: X

m

MmMm† = I (1.10)

In this case, the probability of getting the result m, knowing that the state |ψi is measured, is given by

p(m) = hψ|Em|ψi =ψ|MmMm†|ψ

(1.11) the set of operators Em are sucient to determine the probabilities of the dierent

measurement results.

Generally, the state obtained after the measurement is not easily expressed in terms of operators Em. In fact let us dene Mm = Mm† =

√ Em, whence |ψ0i = √ Em|ψi pp(m) (1.12)

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1.2. Description of a quantum state 17

1.1.4 Composite systems

The state space of a composite physical system is the tensor product of the state spaces of the composing physical systems.

Generally, the composite system consists of a plurality of physical subsystems that may interact one with the other. Let HA and HB be two Hilbert spaces forming a composite

system. The composite space HAB is given by the tensor product of the two spaces HA

and HB:

HAB = HA⊗ HB (1.13)

Furthermore, if |ψAi and |ψBi are two states of the spaces HA and HB, respectively,

the state of the composite system is given by

|ψABi = |ψAi ⊗ |ψBi (1.14)

Note that if the spaces HAand HBhave nite dimensions dimAand dimB, respectively,

the dimension of the composite space HAB is

dimAB = dA.dB (1.15)

Because a wave-function (or state vector) cannot describe all possible states, except those that are pure, we need other tools that allow a perfect and complete description of any arbitrary quantum state. In the case of a mixed state, the description is better made using the density matrix operator, described next.

1.2 Description of a quantum state

1.2.1 Density matrix

As discussed the rst section, in quantum mechanics, the state of a system is described by a state vector |ψi ∈ H. The knowledge of this state allows us to dene the probabilities of all possible outcomes of a given measure. Such a state is called pure state. However, in many cases, we do not have a complete knowledge about the quantum-mechanical state of the system. For this aim, the physicist and mathematician John von Neumann introduced the density matrix, also called density operator [71], it is often represented by ρ.

A)Density matrix for a pure state

For a pure state, the density matrix is dened as:

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and it obeys several properties: 1- Completeness: T r(ρ) = 11

2- The mean value of an operator is given by hAi = T r(Aρ) = T r(ρA) = hψ|A|ψi, 3- ρ is a hermitian operator: ρ† _{= ρ}_,

4- Positivity: ρ is a positive operator: hφ|ρ|φi ≥ 0, ∀i |φi ∈ H. 5- Projection operator: ρ2 _{= ρ}_{, implies that T r(ρ}2_{) = 1}_.

B) Density matrix for a mixed state

In practical situations, it is almost impossible to associate without ambiguity a unique state vector to a system (lack of information about the preparation of the physical system). However, it is possible to associate it with a density operator if we are given a probability distribution pi (i = 1, ..., n), giving the probability of nding the physical

system in one of the n pure states |ψii.

For such a mixed state, the density operator is dened as a convex sum of projectors |ψii hψi|, whose states |ψiiare not necessarily pairwise orthogonal. In this case, the density

matrix is dened as:

ρ = n X i=1 pi|ψi hψ| (1.18) with Pn i=1pi = 1, and pi ≥ 0.

A mixed state appears as a statistical mixture of pure states. Moreover, note that a pure state is only a particular case of a mixed state for which there is only one probability pi, that is non-null and is equal to one.

All the properties of the density matrix for a pure state are still applicable in the case of a mixed state, except one:

-The fth property (T r(ρ2_{) = 1)} _{is not valid in the case of a mixed state. Actually it}

1_{In fact, consider an orthonormal basis {|φ}

ii}for H, whose dimension is d and |ψi = P d

i=1ci|φiithe decomposition of |ψi in this basis, with Pd

i=1|ci| 2 = 1. We have T r(ρ) = d X i=1 hφ|ρ|φi = d X i=1 |ci| 2 = 1. (1.17)

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1.2. Description of a quantum state 19 is a dening property of a pure state. In other words, the system is in a mixed state if and only if the trace of ρ2 _{is less than 1 (0 ≤ T r(ρ}2 _{≤ 1)}_{, if it's not the case this system is}

in a pure state (T r(ρ2_{) = 1)}. This property is interesting because it allows to distinguish

between a pure state and a mixed state, also it allows to measure the degree of mixture for a mixed state through the purity T r(ρ2₎ _[72].

Characterization of a density operator

For any quantum system, an operator ρ is an associated operator with a set of pure states {pi; |ψii}, whose pi are the associated probabilities with the pure states |ψii

(Pn

i=1pi = 1 and pi ≥ 0 ∀i), if and only if,

- ρ is a positive operator;

- ρ has a trace equal to 1 (T r(ρ) = 1).

Indeed, consider ρ a density operator which describes the system as follows: ρ = Pn

i pi|ψii hψi|. According to the properties 3 and 4, ρ is a positive operator and it has

a trace equal to 1. Now, inversely, suppose that ρ satises the two previous conditions. This hermitian operator has a spectral decomposition, as

ρ =

d

X

i

λi|ii hi| (1.19)

where the vectors |ii are d eigenvectors relative to the eigenvalues λi. The eigenvectors

|ii are normed and those of them which are relative to dierent eigenvalues are pairwise orthogonal, because ρ is a hermitian operator. However, note that it is always possible to apply a unitary transformation to ρ in order that all its eigenvectors become pairwise orthogonal. Since ρ is a positive operator, we have

λi = hi|ρ|ii ≥ 0, (1.20)

its eigenvalues λi are positive or null. Thus, the condition T r(ρ) = 1 implies that d

X

i=1

λi = 1 (1.21)

So one can deduce that the state of a system |ii with the probability λi, is described

by a density operator ρ. Then, the postulates of quantum physics can be reformulated in terms of this density operator.

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The postulates of quantum mechanics can be reformulated in terms of the density operator. So by using the density operator we can study the evolution of a system and also its measurements.

At any time t, the state of a physical system is completely determined by a density operator ρ(t) of the Hilbert space. The temporal evolution of the density operator ρ(t) for an isolated physical system is described by the Schrödinger equation:

i¯h∂ρ(t)

∂t = [H, ρ(t)] (1.22)

where H is the hermitian Hamiltonian operator of the system.

On the other hand, the measurements of quantum systems are characterized by an ensemble of operators noted {Mk}. Consider |ψiithe state of the quantum system before

the measurement, the probability of obtaining the measurement result k corresponding to the operator Mk is:

p(k | i) =Dψi|M † kMk|ψi

E

= T r(M_k†Mk|ψii hψi|) (1.23)

Immediately, after the measurement giving the result k, the state of the system is ψ_ik = _q Mk|ψii hψi| M † kMk|ψii (1.24) Note that the measurement operators must satisfy the completeness relation P

iM †

kMk = I, as the sum of all the probabilities must equal one.

The probability of getting the result k after using the laws of probability, is given by: p(k) = n X i p(k | i)pi = n X i T r(M_k†Mk|ψii hψi|)pi = T r(M † kMkρ) (1.25)

After the measurement, the outcome k is corresponding to the state |ψii with the

According to Bayes formula, p(i | m) = p(m, i)/p(m) = p(m | i)p(i)/p(m), the above equation (1.26) become:

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1.2. Description of a quantum state 21 ρk = X i pi Mk|ψii hψi| M † k T r(M_k†Mkρ) = MkρM † k T r(MkM † kρ) . (1.27)

C) Reduced density operator

Consider two subsystems A and B forming a composite system AB. The Hilbert space associated with AB is HAB = HA⊗ HB. Let the density operator ρAB describe

the complete system in the space HAB. The reduced density operator of subsystem A is

dened by:

ρA= T rB(ρAB), (1.28)

In the case where the system is described by a pure state |ψABi, the description of

the states of subspaces HA and HB is given by the partial trace of |ψABi hψAB| over the

subsystems of A and B, such that:

ρA= T rB(|ψABi hψAB|) (1.29)

where T rB represents the partial trace operation over subsystem B, dened for any

op-erator O acting on the space HAB and any orthonormal basis φBj

of HB by T rA(O) = dB X j=1 φB j |O|φ B j = OA (1.30)

with OA an operator acting on HA. It is easily veried that the operator ρA is a density

operator because it is positive and its trace is equal to 1: T rA(ρA) = T rB(ρB) = 1 =

T r(ρ) = 1.

The reduced density operator describes completely all the properties or outcomes of measurements of the subsystem A, given that subsystem B is left unobserved. To illustrate the properties mentioned previously, we consider the following example,

Considering a Bell state, described by |ψ+_{i =} _√1

2(|00iAB + |11iAB). The density

operator of this state is dened by

ρ = |00iABh00| + |00iABh11| + |11iABh00| + |11iABh11|

2 (1.31)

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ρA= T rB(ρ) =

|0i_Ah0| + |1i_Ah1|

2 (1.32)

This subsystem is mixed, since T r((ρA)2) = 1/2 < 1.

1.2.2 Wigner function

The Wigner function was introduced by Wigner [5] as a useful tool to express quantum mechanics in a phase space formalism. Although it was derived rstly for technical reasons, this approach has recently attracted much interest, because it is very suited to analyze the transition from classical to quantum dynamics [6, 7]. It provides a specic description for some measures performed on quantum states, but this comes at the expense of being potentially negative. Consequently, one cannot interpret the Wigner function as a regular probability distribution for a random variable, but rather it is generally interpreted as a quasi-probability distribution. As for the density matrix, this function completely characterizes a quantum state, although it is not often used (at least less than the density matrix). Thus, among the various transforms of the wave-function or in general of the density matrix, which furnish a phase space view of the quantum state, the Wigner function is the one that has set up more applications, mostly in statistical mechanics.

Now, let's consider a system that exists in a continuous one-dimensional Hilbert space. Any state |ψi in the space can be represented as a complex wave-function: |ψi = R |qi ψ(q)dq, where hq|ψi = ψ(q). Similarly, a general mixed state is expressed as a density matrix ρ = R ρ(q1, q2) |q1i hq2|. The Wigner function W (q, p) provides an

equivalent representation of any quantum state in the quadrature phase space (q, p). In the general case, which includes mixed states, the Wigner function is dened as [73]:

W (q, p) = 1 2π¯h Z ∞ −∞ dνe−2ipν¯h hq +ν 2|ρ|q − ν 2i (1.33)

with the change of variables y = −ν/2, and by taking ¯h = 1 one can nd the following equivalent denition W (q, p) = 1 π Z ∞ −∞ dye2ipyhq + y|ρ|q − yi (1.34) For the special case of a pure quantum state, the Wigner function can be expressed via the wave-function as follows:

W (q, p) = 1 π

Z ∞

−∞

ψ∗(q + y)ψ(q − y)e2ipydy (1.35) The wave-function of the n-th harmonic oscillator can be given as

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1.2. Description of a quantum state 23 hn|qθi = einθ 1 p (√π2n_n!)Hn(q)e −q2_/2 (1.36) where Hn is the n-th Hermite polynomial and θ is the phase angle. For each element of

the density matrix [74] one can write W|kihl|(q, p) = (−1)n π r l! k!( √ 2)k−l(q − iy)k−le−(q2+p2)Lk−l_l (q 2_{+ p}2 2 ) (1.37) with Lk−l

l being Laguerre polynomial of order n. This equation is true for k ≥ l, otherwise,

we exchange l by k and y becomes −y. The Wigner function is easily calculated by W (x, y) =X

k,l

ρklW|kihl|(q, p) (1.38)

The generalization to a multimode state ρ = ρ1 ⊗ ...ρn is obvious and yields:

W (q1, p1, ..., qn, pn) = W1(q1, p1)...Wn(qn, pn) (1.39)

in this case, the (π) must be replaced by (π)n_{, with n is the number of modes.}

For pure and mixed states respectively, W (q1, p1, ..., qn, pn)is written as:

W (q1, p1, ..., qn, pn) = 1 (π)n Z ∞ −∞ ... Z ∞ −∞ dy1...dyne2i(p1y1...pnyn)ψ∗(q1+y1..., qn+yn)×ψ(q1+y1..., qn+yn) (1.40) and W (q1, p1, ..., qn, pn) = 1 (π)n Z ∞ −∞ ... Z ∞ −∞ dy1...dyne2i(p1y1...pnyn)hq1+ y1...qn+ yn|ρ|q1− y1...qn− yni (1.41) Interestingly, negativity of the Wigner function provides us the signatures of non-classicality, but we cannot obtain any idea about the amount of non-classicality through it. There exist several quantitative measures of non-classicality. In this thesis we may utilize the most pertinent quantitative measure of non-classicality which is known as the non-classical volume. The non-classical volume is dened as a doubled volume of the integrated negative part of the Wigner function that can be expressed as [9]

δ = Z Z

|W (q, p)| − W (q, p) dp dq= Z Z

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A non-zero value of δ designates a quantum state. As we will see in chapter four, we may use it to get δ for dierent choices of parameters and investigate how the non-classicality (or the amount of quantumness) varies with the change of a particular parameter.

A) Properties

The Wigner function has many properties, we present below the most important ones:

- Marginal distribution: Starting from Z

dpW (q, p) = hq|ρ|qi , and Z

dqW (q, p) = hp|ρ|pi (1.43) This properties must stay true whether one rotates the axes in phase space, as in

qθ = U†(θ)qU (θ) = q cos(θ) + p sin(θ),

pθ = U†(θ)pU (θ) = −q sin(θ) + p cos(θ).

or,

q = qθcos(θ) − pθsin(θ),

p = qθsin(θ) + pθcos(θ).

So, we can express P (qθ) as:

P (qθ) =

Z

W (qθcos(θ) − pθsin(θ), pθcos(θ) + qθsin(θ))dpθ (1.44)

Then, one gets

P (qθ) =q|U (θ)ρU†(θ)|q

(1.45) The denition (1.44) which yields the probability distribution for qθ in terms of

the function W (q, p), is Known as a Radon transform. So, if one knows P (qθ) for all

angles θ, one can determine W (q, p), through the so-called Radon inverse transform [75]. Thus from (1.44) and (1.45) it seems that the marginal distributions coincide with the quadrature probability distributions.

- Probability completeness: The quasi-probability distribution sums up to one, then the Wigner function satises the normalization condition

Z Z

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- Hemitian operators: If the operator is hermitian (like for density matrix ρ = ρ†_),

the Wigner function has only real values.

- Linearity: The Wigner function is linear in terms of the density operators:

W (q, p)p1ρ1+p2ρ2+... = p1Wρ1 + p2Wρ2 + ... (1.47)

- Fidelity: The delity between any state ρ and any pure state |ψi is expressed in terms of Wigner function as follows:

F2 = (phψ|ρ|ψi2 = π Z ∞ −∞ Z ∞ −∞ dq dp W (q, p) W|ψihψ|(q, p) (1.48)

For states with delity equal to 0, the integral cannot be null except if there exists areas such that one of the two Wigner functions is negative.

-Transpose operation: Since ρT _{= ρ}∗_{, the transposition of the density matrix is}

translated in terms of the Wigner function in phase space as follows

W_ρT(q, p) = W_ρ(q, −p) (1.49)

-Purity: The purity of a state is expressed in terms of the Wigner function P = T r(ρ2) = 2π

Z ∞

−∞

dqdpW2(q, p) (1.50) In fact, the matrix elements can be obtained in any base, which shows that the Wigner function completely determines the quantum state and that its data is completely equiv-alent to that of the density matrix:

ha|ρ|bi = 2π Z ∞

−∞

dqdpW (q, p)W|aihb|(q, p) (1.51)

Wigner function for operators: The generalization of the Wigner function of a single mode for any operator is written as

WA(q, p) = 1 π Z ∞ −∞ dye2ipyhq + y|A|q − yi (1.52) The trace of an operator product from their Wigner functions is calculated as follows

T r {AB} = π Z ∞ −∞ Z ∞ −∞ dx dp WA(x, p) WB(x, p) (1.53)

This formula is undoubtedly important. For instance, one can calculate the average value of an operator,

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hAi = T r {ρA} = π Z ∞ −∞ Z ∞ −∞ dx dp WA(x, p) W (x, p). (1.54)

B) Other formula for Wigner function

There are other formulations for the Wigner function, and the expression (1.34) is not the only formula. Here, we will dene the interesting ones.

Since the value of the Wigner function at the origin of the phase space only depends on the diagonal elements of the density matrix, we have

W (0, 0) = 1 π

X

n

(−1)nρnn (1.55)

and by the help of the parity operator P = eiπN_{, we obtain}

W (0, 0) = 1

πT r[ρP ] (1.56)

Therefore, as applying the displacement operator D(α) is equivalent to a translation of the Wigner function in phase space, one can express the Wigner function as follows [76]. W (α) = 2 πT r(D(−α)ρD(α)P ) (1.57) where D(α) = e(αâ†_−α∗_ˆ_a) , â = (ˆq+ ˆ_√p) 2 ,â † ₌ i( ˆp−ˆ_√q)

2 are respectively the annihilation and

creation operators, and α is the amplitude of a coherent state, where, α = (q + ip)/√2. It is also expressed with the conjugated values W (α, α∗₎_:

W (α, α∗) = 2T r[D(α, α∗)ρD−1(α, α∗)P ] = 2W (q, p) (1.58) In this case the Wigner function is still dened as a function of two real variables per mode. Therefore, there is a relation between quadrature and complex amplitude.

C) Examples of Wigner functions

The Wigner function which corresponds to special states of the electromagnetic eld may be obtained by using either (1.34) or (1.58). We choose in this PhD thesis the normalization corresponding to (1.34).

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1.2. Description of a quantum state 27 For the vacuum state, one has the Gaussian

W0(q, p) =

1 πe

−q2_−p2

(1.59) where q and p are position and momentum respectively. In this case the Wigner function is a round hill centered at the origin of phase space.

• Coherent state:

The Wigner function for a coherent state is written as Wc(q, p) =

1 πe

−(q−q0)2−(p−p0)2 (1.60)

this function is a Gaussian centered at the phase space point (q0, p0), where the

parameters q0 and p0 are mean values of position and momentum. In this case,

the Wigner function is the same as for the vacuum state dened above, but it is displaced in the phase space.

• Squeezed state:

For a single mode squeezed state, with squeezing parameter r, the Wigner function is given by [77]

Wξ(q, p) =

1 πe

(e−rq−q0)2_e(erp−p0)2 _, (1.61)

As for coherent state, this function is a Gaussian centered at the phase space point (q0, p0), where the parameters q0 and p0 are mean values of position and momentum

in the state ρ. • Fock state:

For a given Fock state |ni, the general expression of the Wigner function is expressed as W|ni(q, p) = (−1)n π e −(q2_+p2₎ Ln( q2_{+ p}2 2 ) (1.62)

For the Fock state |1i, one gets W1(q, p) =

1 πe

−q2_−p2

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This function vanishes for q2_{+ p}2 _{= 1/2}_{, and is negative at the origin of the phase}

space. This negative value recalls us of the highly quantum nature of a Fock state. For higher photon numbers, the Wigner function exhibits more oscillations, the number of zeros coinciding with n.

• Schrodinger's cat states:

For a Schrodinger's cat state that is a coherent superposition of two coherent states localized in two distant points of the conguration space, ±q0. The wave-function

of such a state reads in the position representation ψ(q) = √N

2[φ+(q) + φ−(q)] (1.64) where φ±= (1_π)1/4exp(−1₂(q ± q0)2+ ip0(q ± q0)).

Inserting (1.64) into the Wigner function (1.52) one obtains

Wψ(p, q) = W+(q, p) + W−(q, p) + Wint(q, p) (1.65)

where W±(p, q) represent two peaks of the distribution centered at the classical

phase space points (q0, p0), it is expressed as:

W±(p, q) = N2 2πexp(−(q ± q0) 2_{− (p − p} 0)2) (1.66) while, Wint(p, q) = N2 π cos(2pq0)exp(−q 2_{− (p − p} 0)2) (1.67)

is the interference structure that is showed between both peaks. Normalizing (1.64) gives

N = (1 + cos(2p0q0)exp(−q20))

−1/2 _(1.68)

In Figures 1.1 and 1.2 we plot the Wigner function of some of these examples

• Superposition of coherent states:

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(a) q0= 0.5and p0= 1 (b) q0= 0and p0= 0

(c) r = 0.5, q0= 0.5and p0= 1

Figure 1.1: Wigner function W (q, p) for (a) single mode coherent state, (b) vacuum state and (c) single mode squeezed state.

|ψi = N (|αi + |βi) (1.69) where N = (2 + exp(−1 2|α| 2 −1 2|β| 2 + α∗β) + exp(−1₂|α|2−1 2|β| 2 + αβ∗))−1/2 is the normalization constant. But it is very interesting to consider a superposition of n coherent states whose Wigner function is written as

W = N0π−3/2

n

X

i,j=1

Wij (1.70)

where the terms Wij are the Wigner function of each coherent state. For i 6= j, Wij

are the interference terms.

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(a) n = 1 (b) n = 3

(c) n = 6

Figure 1.2: Wigner function W (q, p) for Fock state

coherent state from the expression of the same function for the superposition. Thus, the Wigner function for the coherent state |αi is:

W|αi(q, p) =

√

πexp(−p2+√2i(α − α∗)p − 2 |α|2− q2₊√_{2(α + α}∗

)q) (1.71) it can be also obtained from the equation (1.60) by using α = q0_√+ip0

2 .

In Fig. 1.3-a, Fig. 1.3-b, Fig. 1.3-c and Fig. 1.3-d we plot the Wigner function of superposition of single mode coherent states

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(a) α = 3 and β = −3 (b) α = 3, β = −3 and γ = 5

(c) α = 3, β = −3, γ = 4,γ = −4 (d) α = 2.5, β = −2.5, γ = 5, γ = −5 and σ = 4

Figure 1.3: Wigner function for a) two coherent states b) three coherent states c)four coherent states d) ve coherent states.

1.2.3 Fidelity

The delity is a measure of the similarity between two quantum states. Let's consider two pure states |ψ1i and |ψ2i, the delity can be written as:

F = |hψ1|ψ2i| (1.72)

if we have two states |ψi and ρ, such that one of them is pure , the delity between these states is dened as

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F = T r(p|ψi hψ| ρ |ψi hψ|) = phψ |ρ| ψiT r(p|ψi hψ|) = phψ |ρ| ψi _(1.73) with √•the square root is positive.

Note that the delity is equal to one when the states are identical and zero when they are orthogonal.

In the case of mixed states, the general formulation of delity between two given states ρ1 and ρ2 [78] can be written as:

F = T r[ q√

ρ2ρ1

√

ρ2] (1.74)

The most important information resource in quantum information science, that distinguishes it from classical information theory, is entanglement phenomenon [15, 14]. This latter is based on the superposition principle, the simplest example of this principle is the qubit. We will discuss briey these phenomena in the next section.

1.3 Essential phenomena of quantum mechanics

The quantum superposition principle [3], which expresses the idea that a system can exist simultaneously in two or more mutually exclusive states is at the heart of the mystery of quantum mechanics. It is very necessary for quantum entanglement. However, the superposed states are not necessary entangled.

1.3.1 Superposed states

Quantum mechanics is a linear theory, this means that any linear combination of possible states is also a possible state, known as superposed states. The existence of these states is without a doubt one of the most intriguing aspects of quantum information, it appears clearly in the quantum interference. The idea of the superposition principle is that a system is in all possible states simultaneously with various probability (at the same time ), until it is measured.

On the other hand, the quantum state contains all the information about a system, for instance, it can describe an atom that is at the same time in two energy levels, an electron which is located at the same time in two dierent positions. So, the superposition concept states that a quantum state |ψi can be expanded as a linear combination of the normalized vectors |αniof a particular operator that constitutes

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1.3. Essential phenomena of quantum mechanics 33 a basis of the space occupied by |ψi. For the discrete case, the superposition of a quantum state |ψi can be expressed as:

|ψi = c1|α1i + c2|α2i ...cn|αni (1.75)

where cn are the complex coecients of the linear combination and are given by,

cn= hαn|ψi (1.76)

1.3.2 The Qubit

The qubit (quantum bit)[79] is a quantum state which represents the smallest unit of storage of the quantum information. By denition a quantum bit is a linear superposition of two basic states, noted |0i and |1i.

The general expression of a qubit is given by,

|ψi = α |0i + β |1i (1.77) where α and β are two complex numbers, that represent the probability amplitudes to obtain the states |0i and |1i, respectively, during a measurement of the state |ψi. These two complex numbers satisfy the following normalization condition

|α|2

+ |β|2 = 1 (1.78)

The states |0i and |1i constitute an orthogonal basis in the Hilbert space. A qubit diers signicantly from a classical bit by the fact that a classical bit can take only the values 0 or 1 corresponding to the states |0i and |1i. A qubit doesn't have this restriction, consequently it is much richer than a classical bit.

The Bloch sphere

In quantum mechanics, the bloch sphere [80] gives the geometric representation (Fig. 1.4) of the state space of a 2-level quantum system. Moreover, the points on the surface of the sphere correspond to pure states of the system, while points in the interior correspond to mixed states [81]. The bloch sphere is indeed geometrically a sphere and the correspondence between elements of the bloch sphere and pure states can be explicitly given.

From the formula (1.78), a qubit can be rewritten as follows:

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with −π

2 ≤ θ < π

2, 0 ≤ φ < 2π. In fact, the variation of these two parameters

allows a quantum state to take all the values of the Bloch sphere.

A two-qubit system has four computational basis states noted |00i; |01i; |10i and |11i. In the classical computing, if we have two bits, the total number of a possible states that these two bits can take is four: 00, 01, 10 or 11. Due to the superposition principle, a pair of qubits may also exist in a superposition of these four states, such as the following state vector,

|ψi = α00|00i + α01|01i + α10|10i + α11|11i (1.80)

The state of the qubit after a measurement is |xi, such that x = 00, 01, 10 or 11, the probability of getting |xi is |αx|2.

Figure 1.4: Geometric representation of a qubit in a Bloch sphere

1.3.3 Entangled state

Quantum entanglement is a fundamental phenomenon of the quantum mechanics. Moreover it is of particular importance in quantum computation and quantum infor-mation [13]. This phenomenon is characterized by the fact that the quantum state of two entangled states must be described globally, although they may be spatially separated. When quantum objects are placed in an entangled state, there exist correlations between the observed physical properties of these objects which would not be present if these properties were local. Indeed, even if they are separated by large distances, two entangled systems A and B are not independent and should be considered as a single system {A + B}.

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1.3. Essential phenomena of quantum mechanics 35 To dene this more rigorously let A ∈ HA and B ∈ HB be two subsystems of a

bipartite system |ψABidescribed by the Hilbert space HA⊗ HB. Let the subsystems

A and B be prepared in pure states |ψAi and |ψBi, respectively. The state of the

composite system |ψABi is written as,

|ψABi = |ψAi ⊗ |ψBi (1.81)

Suppose that one might carry out only local measurements on the system. Then, after a measurement of any local observable ˜A ⊗ IB on the subsystem A, where ˜A is

a hermitian operator acting on HA, and IB is the identity acting on HB, the state

of the subsystem A will be screened onto an eigenstate of ˜A, while the state of the subsystem B stays unchanged.

After that, if one makes a second local measurement, one obtains on subsystem B a result which is independent of the result of the rst measurement. We can deduce that the measurement results on dierent subsystems are not correlated with each other and depend only on the states of each respective subsystem.

But in general, a pure state may be written as a superposition of pure states of the form (1.81), for example

|ψABi =

1 √

2(|ψAi ⊗ |ψBi) + (|φAi ⊗ |φBi) (1.82) with |ψA,Bi 6= |φA,Bi.

where T rAand T rB are the partial trace over subsystems A and B respectively, and

ρB = T rA|ψABi hψAB| is the reduced density matrix of the subsystem B. Because

(1.83) holds for any local operator a, the state of the subsystem A alone is given by ρA. Thus, similarly the state of the subsystem B is described by its reduced density

matrix ρB = T rA|ψABi hψAB|. However, the state of the bipartite system, is not

equal to the product of both subsystem states,

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Additionally, if one does a measurement locally on subsystem A (or B), this will lead to a state reduction of the global system state, not only of the subsystem A (or B). So, the probabilities for a result of a measurement on subsystem A are inuenced by prior measurements on the subsystem B. Thus, measurement results on subsystems A and B are correlated despite that they can be far from each other. Consequently, we can conclude that the states which may be written as a product of pure states, as in (1.81), are called separable (or factorizable) states. Otherwise, if there are no local states |ψAi ∈ HA and |ψBi ∈ HB, and the state of the system

|ψABi cannot be written as a product (as a superposition of a tensor product of

separable states):

6 ∃ |ψAi ∈ HA and |ψBi ∈ HB such that |ψABi = |ψAi ⊗ |ψBi (1.85)

then the state |ψABi is called entangled (non separable).

Now we turn out to dene entanglement in mixed states. For this aim, let ρAB be a

bipartite state shared by two parties A and B, similarly to the case of pure states, mixed product states is written as

|ρABi = ρA⊗ ρB (1.86)

where ρi

A and ρiB are the density matrices for parties A and B respectively. |ρABi

does not display correlations. A convex sum of dierent product states is written as ρAB = X i piρiAρ i B. (1.87)

with pi a probability distribution such that pi > 0 and P pi = 1. However, this sum

of product states will in general give correlated measurement results, i.e., there are local observables ˜A and ˜B such that

T r(ρ( ˜A ⊗ ˜B)) 6= T r(ρ( ˜A ⊗ IB))T r(ρ(IA⊗ ˜B)) = T rAρA A T r˜ B ρB B.˜ (1.88)

These correlations are therefore regarded as classical since they can be described in terms of the classical probabilities pi. The states of the form (1.87) are called

separable mixed states. One can dene mixed entangled states to be the states that have no decomposition into product states [82].

To summarize a mixed state is entangled if there are no local states ρi

A and ρiB, and

positive pi, such that ρAB can be expressed as a convex mixture:

6 ∃ρi

A, ρiB, pi ≥ 0, such that ρ =

X

i

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1.4. Classical information theory 37 Note that entangled (or non separable) states involve non-classical correlations of measurements on dierent subsystems that, unlike classical correlations, cannot be described in terms of classical probabilities alone.

The simplest example of entangled states are the so-calle Bell states, which are the subject of the famous Bell inequality of John Stewart Bell. They are maximally entangled states and they form an orthonormal basis of the state space (a four-dimensional space). The four Bell states are

|Φ+_{i =} _√1

2(|0iA⊗ |0iB+ |1iA⊗ |1iB) (1.90) |Φ−i = √1

2(|0iA⊗ |0iB− |1iA⊗ |1iB) (1.91) |Ψ+i = √1

2(|0iA⊗ |1iB+ |1iA⊗ |0iB) (1.92) |Ψ−i = √1

2(|0iA⊗ |1iB− |1iA⊗ |0iB) (1.93) The pairs in Bell states are called EPR pairs. The basic unit of entanglement is the "ebit" (entangled bit), dened as a quantitative measure of the entanglement contained in a Bell state.

Now, before presenting the quantum information theory, we will expose the princi-ples and the basics of the classical information theory, which will make the goal of the next section.

1.4 Classical information theory

Toward the end of the 40s, classical information theory was developed by Claude E. Shannon [69] to characterize the most fundamental aspects of communication systems. It carries on probabilistic entities the random variables which allow to associate the probabilities with the possible outcomes of a random experiment. Here, before presenting formally the concept of the classical information theory, we will dene briey the meaning of "information". Then, we will present the quantities and the properties of this classical information theory, which will serve as a model for their quantum counterparts. The classical references for information theory are [83, 84, 85, 13, 86].