https://lib.uliege.be https://matheo.uliege.be Quantum Entanglement: a Study of Recent Separability Criteria Auteur : Hansenne, Kiara Promoteur(s) : Bastin, Thierry Faculté : Faculté des Sciences Diplôme : Master en sciences physiques, à finalité approfondie Année académique : 2019-2020 URI/URL : http://hdl.handle.net/2268.2/9319 Avertissement à l'attention des usagers : Tous les documents placés en accès ouvert sur le site le site MatheO sont protégés par le droit d'auteur. Conformément aux principes énoncés par la "Budapest Open Access Initiative"(BOAI, 2002), l'utilisateur du site peut lire, télécharger, copier, transmettre, imprimer, chercher ou faire un lien vers le texte intégral de ces documents, les disséquer pour les indexer, s'en servir de données pour un logiciel, ou s'en servir à toute autre fin légale (ou prévue par la réglementation relative au droit d'auteur). Toute utilisation du document à des fins commerciales est strictement interdite. 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Faculty of Sciences Department of Physics Quantum Entanglement A Study of Recent Separability Criteria Kiara Hansenne Work presented for the degree of Master in Physics Reading committee Supervisor Prof. John Martin Prof. Alejandro Silhanek Prof. Thierry Bastin Prof. Geoffroy Lumay Prof. Pierre Mathonet Academic Year 2019-2020 Acknowledgements I would first like to thank Professor Thierry Bastin, my supervisor. I am deeply grateful for his time, implication and interest that were present form the beginning to the very last days of the construction of this work. I also would like to thank Professors John Martin, Alejandro Silhanek, Geoffroy Lumay and Pierre Mathonet for agreeing to be part of my reading committee. I hope they will appreciate reading this manuscript. Finally, I wish to thank my family and friends, for their unfailing support during all my years of study, and particularly these last few months. i Table of contents Acknowledgements i Introduction 1 Notations 4 1 Basic notions of quantum mechanics 5 1.1 Quantum states and density operators . .5 1.1.1 Pure states . .5 1.1.2 Mixed states . .6 1.1.3 Bloch representation of quantum states . .8 1.2 Bipartite systems . 13 1.2.1 Partial trace . 14 1.2.2 Bipartite Bloch representation . 15 1.2.3 Schmidt decomposition . 18 1.2.4 Bipartite entanglement . 19 1.3 Multipartite systems . 20 1.3.1 Multipartite entanglement . 21 2 First separability criteria 22 2.1 Positive partial transpose criterion . 22 2.2 Entanglement witnesses . 24 2.2.1 Definitions and properties . 24 2.2.2 Link with positive maps . 28 2.2.3 Geometric entanglement witnesses . 28 2.2.4 Multipartite entanglement . 29 2.3 Entanglement measures . 30 2.3.1 Entropy of entanglement . 31 2.3.2 Negativity . 32 2.3.3 Convex roof construction . 33 2.4 Concurrence criterion . 33 2.4.1 Two-qubit concurrence . 33 2.4.2 Generalised concurrences . 34 2.5 Computable cross-norm or realignment criterion . 36 ii 3 Recent separability criteria 39 3.1 Correlation matrix criterion . 39 3.2 Covariance matrix criterion . 42 3.3 Enhanced CCNR criterion . 45 3.4 LWFL family of criteria . 45 3.5 Li-Qiao criterion . 46 3.6 Symmetric informationally complete measures criterion . 50 3.7 SSC family of criteria . 52 3.8 Comparing the criteria . 56 Conclusion 58 A Appendix 60 A.1 Singular value decomposition . 60 A.2 Special unitary group . 61 A.3 From the basis decomposition to the Bloch representation . 67 Bibliography 69 iii Introduction Quantum entanglement is a feature of quantum mechanics that has risen numerous philo- sophical, physical and mathematical questions since the early days of the quantum theory. It can be seen as the most non-classical feature of quantum mechanics and has absolutely no classical equivalent. Consequently, debates took place when it has been described in 1935 by Einstein, Podolsky and Rosen in Ref. [1] and by Schr¨odingerin Ref. [2] for the first time. In the latter paper, Schr¨odingerwrote the now famous citation `I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.' Indeed, Schr¨odinger acknowledged the existence of global states of bipartite systems (systems made of two subsystems) that cannot be factorized, i.e. that cannot be written as a tensor product of states of the subsystems. Therefore, for entangled states, only a common description of the subsystems exists. Einstein, Podolsky and Rosen proposed in Ref. [1] a thought experiment known today as the EPR paradox, with the aim of proving the incompleteness of quantum mechanics. The thought experiment goes as follows: consider two particles prepared in an entangled state. The particles are then spatially separated and one performs a measurement on one of them, say the first particle. Given the laws of quantum mechanics, the state of the other particle collapses in a state dictated by both the outcome of the measurement on the first particle and the initial state of the composite system. Then, it appears that the correlations between both particles are stronger then the correlations one could classically expect. So, EPR argued that the measurement outcomes of both particles were determined at the creation of the pair and that quantum mechanics missed some (local) hidden variable that should make the theory a causal and local one. This was in contradiction with Bohr and Heisenberg's interpretation of quantum mechanics, known as Copenhagen interpretation. This paradox induced many debates within the scientific community but even outside of it. Indeed, a daily newspaper also reported the EPR paper [3]. The EPR paradox was resolved in 1964 by Bell, by showing that a local hidden variable theory is incompatible with the statistical predictions of quantum mechanics [4]. In order to do this, he proved an inequality (known as Bell's inequality) that all local theories have to verify. However, he also showed that the predictions of quantum mechanics violate this inequality. It meant that the quantumly correlations between entangled states are impossible to obtain within a classical theory. Then, in 1982, Aspect, Grangier and Roger carried out the first experimental violation of this inequality [5]. This experiment, called Aspect's experiment, confirmed the predictions of quantum mechanics, and thus confirmed its incompatibility with local theories. Aspect's experiment helped the transition of quantum entanglement from purely the- 1 oretical considerations and philosophical debates to practical experiments, and it began to be considered as a powerful resource that enables tasks not permitted by classical resources. For instance, one can mention quantum cryptography [6], quantum teleporta- tion [7] and quantum computing [8]. Additionally to the fundamental reason, all these practical perspectives were a reason why a strong entanglement theory was needed. However, determining whether a given state is entangled or not is still an open problem today, both from the theoretical and experimental points of view. This is known as the separability problem (states with no entanglement are called separable and form a convex subset of all the quantum states). Theoretically, entanglement is defined by a mathematical property of quantum states that are described by density operators acting on Hilbert spaces. Although a general solution is still lacking, the separability problem has been solved for pure states [9], and for 2 × 2 and 2 × 3 systems [10]. We also note that from a philosophical point of view, characterising the set of separable states could answer the question whether the world is more quantum or more classical, i.e. does the set of states contain more quantum correlated (i.e. entangled) or classically correlated states? The aim of this work is to give a selective but up to date review of the separability problem. Indeed, the two most-cited reviews on this topic were published in 2009 [11, 12]. Keeping this goal in mind, we present and analyse several separability criteria that appeared relevant to us. We focus on the theoretical perspective of the problem. The manuscript is structured as follows. In the first chapter, we present some basic notions of quantum mechanics used through- out the Chapters 2 and 3. We first introduce quantum states and make the distinction between pure states (described by state vectors) and mixed states (described by density operators). Then, we present several concepts used to treat bipartite systems, namely partial trace, Bloch representation and Schmidt decomposition of quantum states. In the same section, we get to the heart of the matter by giving the mathematical defini- tions of entanglement for bipartite systems. Finally, we generalize these definitions to multipartite systems (systems composed of two or more subsystems). The second chapter is dedicated to the first separability criteria that have historically been presented in literature. We begin with the celebrated positive partial transpose cri- terion, then introduce criteria based on entanglement witnesses and entanglement mea- sures. The latter are used to detect entanglement, but also to quantify it. The next section of this chapter is devoted to concurrences, which, as we will see, solve the separa- bility problem for multipartite pure states. We close the second chapter by introducing another celebrated criterion, namely the computable cross-norm or realignment criterion.