If a one-phrase summary of the subject of this thesis were required, it would be something like: miscellaneous large (but finite) dimensional phenomena in quantum information theory. That said, it could nonetheless be helpful to briefly elaborate. Starting from the observation that quantum physics unavoidably has to deal with high dimensional objects, basically two routes can be taken: either try and reduce their study to that of lower dimensional ones, or try and understand what kind of universal properties might precisely emerge in this regime. We actually do not choose which of these two attitudes to follow here, and rather oscillate between one and the other. In the first part of this manuscript, our aim is to reduce as much as possible the complexity of certain quantum processes, while of course still preserving their essential characteristics. The two types of processes we are interested in are quantum channels and quantum measurements. Oppositely, the second part of this manuscript is specifically dedicated to the analysis of high dimensional quantum systems and some of their typical features. Stress is put on multipartite systems and on entanglement-related properties of theirs. In the third part of this manuscript we eventually come back to a more dimensionality reduction state of mind. This time though, the strategy is to make use of the symmetries inherent to each particular situation we are looking at in order to derive a problem-dependent simplification.
If a one-phrase summary of the subject of this thesis were required, it would be something like: miscellaneous large (but finite) dimensional phenomena in quantum information theory. That said, it could nonetheless be helpful to briefly elaborate. Starting from the observation that quantum physics unavoidably has to deal with high dimensional objects, basically two routes can be taken: either try and reduce their study to that of lower dimensional ones, or try and understand what kind of universal properties might precisely emerge in this regime. We actually do not choose which of these two attitudes to follow here, and rather oscillate between one and the other. In the first part of this manuscript, our aim is to reduce as much as possible the complexity of certain quantum processes, while of course still preserving their essential characteristics. The two types of processes we are interested in are quantum channels and quantum measurements. Oppositely, the second part of this manuscript is specifically dedicated to the analysis of high dimensional quantum systems and some of their typical features. Stress is put on multipartite systems and on entanglement-related properties of theirs. In the third part of this manuscript we eventually come back to a more dimensionality reduction state of mind. This time though, the strategy is to make use of the symmetries inherent to each particular situation we are looking at in order to derive a problem-dependent simplification.
As Schroedinger already recognised one century ago, entanglement is at the core of quantum mechanics. Nowadays it turns out to be the fundamental notion behind many quantum phenomena, from quantum algorithms to gravity, passing by critical phenomena and topological phases of matter, triggering unexpected connections between apparently far branches of physics. At the center of all these ideas, we find the (Rényi) entanglement entropies, which are powerful entanglement measures that provide fundamental insights into the investigated system or theory. This motivates the development of techniques for determining it, such as the replica trick: its implementation via the path-integral finds a wide place in this thesis. For example, we develop an efficient strategy to compute a generalised version of the Rényi entropies for all the eigenstates of a (1+1)-dimensional conformal field theory. This represents the starting point for a simulation scheme ideal to compute the entanglement in more generic (1+1)-dimensional quantum field theories (QFTs), e.g. after a quench in the sine-Gordon field theory. The study of entanglement also intertwines with another pillar of modern physics, i.e. symmetries and how their presence influences the properties of a system. Given the interest in this connection, this thesis addresses the question of how the entanglement splits into the different sectors of an internal symmetry. We approach the problem first in the QFT context, both for the free Dirac and complex scalar fields in two-dimensional spacetime, which have an abelian conserved charge, and systems having an internal Lie group symmetry to tackle the non-abelian case. Another typical framework in which we study the symmetry resolution of entanglement is lattice models, where different techniques can be exploited in order to derive exact results, ranging from the corner transfer matrix for gapped integrable systems to the connection between quadratic lattice Hamiltonians and their two-point correlation functions. The symmetry resolution also concerns other entanglement measures, namely, we analyse the behaviour of the operator entanglement, i.e. a key quantifier of the complexity of an operator, the symmetry-resolved mutual information, the effect of symmetries on entanglement negativity. The latter quantity is a genuine measure of quantum correlations in mixed states and a consistent part of the thesis is about this subject. For example, we study its time evolution after a quench and we provide an operatorial characterisation for entanglement in mixed states, which we dub negativity Hamiltonian.
This thesis investigates universality and complexity in classical and quantum statistical physics, with a particular emphasis on nonequilibrium phenomena. While the emergence of universal behavior in equilibrium systems stands as one of the cornerstones of statistical physics, the extension of these ideas to nonequilibrium settings presents fundamental and still open challenges. The inherently diverse and system-specific nature of nonequilibrium dynamics calls for a broad spectrum of conceptual and analytical tools. Furthermore, nonequilibrium dynamics is nowadays accessible to experiments in both classical and quantum regimes. These modern experiments produce a large amount of experimental data, which calls for analysis of physics through data-driven methods, complexity theory and high-dimensional statistics. The possibility of probing and accessing correlated states of matter out of equilibrium in modern experiments raises several questions. Can we understand the output of experiments in terms of complexity theory and dimensional reduction? Is it possible to extend concepts of universality to data structures arising in simulations? Can we describe phase transitions (either classical or quantum) concepts of modern data science? How to control entanglement in quantum dynamics? Which nonequilibrium phenomena can we hope to observe in quantum simulators? The research work presented in this thesis is an attempt to answer to these questions. In this thesis, we argue that order parameters and diagnostics of criticality in nonequilibrium classical systems can be found by unsupervised learning and dimensional reduction, which allow to reduce the complexity of datasets by projections on lower-dimensional spaces, while still preserving essential universal features. We then move to the study of models of quantum dynamics relevant for quantum simulators. In particular, we show that the same concepts of dimensional reduction apply also to quantum systems, allowing for the extraction of information from outputs of quantum simulators. We then study a protocol to prepare complex, entangled states via unitary evolution and enhance entanglement via measurements in a fermionic model. Finally, we study the emergence of anomalous transport in one-dimensional chiral systems with non-Abelian symmetries.
High dimension and symmetries in quantum information theory S'il fallait résumer le sujet de cette thèse en une expression, cela pourrait être quelque chose comme: phénomènes de grande dimension (mais néanmoins finie) en théorie quantique de l'information. Cela étant dit, essayons toutefois de développer brièvement. La physique quantique a inéluctablement affaire à des objets de grande dimension. Partant de cette observation, il y a, en gros, deux stratégies qui peuvent être adoptées: ou bien essayer de ramener leur étude à celle de situations de plus petite dimension, ou bien essayer de comprendre quels sont les comportements universels précisément susceptibles d'émerger dans ce régime. Nous ne donnons ici notre préférence à aucune de ces deux attitudes, mais au contraire oscillons constamment entre l'une et l'autre. Notre but dans la première partie de ce manuscrit (Chapitres 5 et 6) est de réduire autant que possible la complexité de certains processus quantiques, tout en préservant, évidemment, leurs caractéristiques essentielles. Les deux types de processus auxquels nous nous intéressons sont les canaux quantiques et les mesures quantiques. Dans les deux cas, la complexité d'une transformation est mesurée par le nombre d'opérateurs nécessaires pour décrire son action, tandis que la proximité entre la transformation d'origine et son approximation est définie par le fait que, quel que soit l'état d'entrée, les deux états de sortie doivent être proches l'un de l'autre. Nous proposons des solutions universelles (basées sur des constructions aléatoires) à ces problèmes de compression de canaux quantiques et d'amenuisement de mesures quantiques, et nous prouvons leur optimalité. La deuxième partie de ce manuscrit (Chapitres 7, 8 et 9) est, au contraire, spécifiquement dédiée à l'analyse de systèmes quantiques de grande dimension et certains de leurs traits typiques. L'accent est mis sur les systèmes multi-partites et leurs propriétés ayant un lien avec l'intrication. Les principaux résultats auxquels nous aboutissons peuvent se résumer de la façon suivante: lorsque les dimensions des espaces sous-jacents augmentent, il est générique pour les états quantiques multi-partites d'être à peine distinguables par des observateurs locaux, et il est générique pour les relaxations de la notion de séparabilité d'en être des approximations très grossières. Sur le plan technique, ces assertions sont établies grâce à des estimations moyennes de suprema de processus gaussiens, combinées avec le phénomène de concentration de la mesure. Dans la troisième partie de ce manuscrit (Chapitres 10 et 11), nous revenons pour finir à notre état d'esprit de réduction de dimensionnalité. Cette fois pourtant, la stratégie est plutôt: pour chaque situation donnée, tenter d'utiliser au maximum les symétries qui lui sont inhérentes afin d'obtenir une simplification qui lui soit propre. En reliant de manière quantitative symétrie par permutation et indépendance, nous nous retrouvons en mesure de montrer le comportement multiplicatif de plusieurs quantités apparaissant en théorie quantique de l'information (fonctions de support d'ensembles d'états, probabilités de succès dans des jeux multi-joueurs non locaux etc.). L'outil principal que nous développons dans cette optique est un résultat de type de Finetti particulièrement malléable
The study and characterisation of quantum entanglement represents an extended effort within the field of quantum information. This thesis addresses two problems. The first regards entanglement in open systems, where it’s more difficult to define a dichotomic (two outcome) observable due to population loss. The second problem concerns genuine multipartite entanglement, that is entanglement between many particles in states that cannot be separable with respect to any possible bipartition of a system.\nThe first chapter consists of an introduction to the basic elements of quantum mechanics and quantum information theory. We briefly describe a few historical facts about quantum mechanics and the study of entanglement, to help put things into perspective.\nThe second chapter is a detailed discussion of the neutral kaon system. These particles are produced in pairs and are described as a particle-antiparticle system. This system displays some unique properties like the violation of CP-symmetry, and shares some characteristics with other mesons, such as neutral particle oscillation and decay. This makes formulating Bell-type inequalities for them more difficult, but also provides the opportunity for new tests.\nThe third chapter continues with the presentation of an already existent effective formalism for performing Bell tests on neutral kaons. The essence of the formalism is a switch to the Heisenberg picture, transferring the time evolution and dependence on measurement directions to a so-called effective operator. This has many advantages, some of the most important being a proper normalization during population loss (due to decay), an easy application to\nthe case of multiple particles and the fact that it allows one to observe entanglement within the system for a relatively long amount of time. The final part presents a way to simulate neutral kaons with atomic systems, namely with Ytterbium isotopes and discuss what are the differences between these atomic systems and neutral kaons.\nThe second part of the thesis treats the problem of proving genuine multipartite entanglement from separable two-body marginals. For this, an introduction to semidefinite programming is necessary due to the fact that the problem under study can be formulated in terms of this method. We will briefly discuss linear and semidefinite programming as well as the concept of duality and provide some basic examples.\nThe formal description of this problem and previous results are provided in the fifth chapter. Our result shows that, through a cyclic iteration between two SDPs, one obtains states with the desired properties. Using this method we managed to go up to six qubit states for various configurations. From this we can construct higher-dimensional states having the same properties.
Quantum information theory and the second quantum revolution have brought us within grasping distance of quantum computers, the building block of which is the qubit. Like their classical counter parts---the bit---they are two-dimensional, reaching their power when embedded in an network of many interacting qubits. However, many quantum systems are not two-dimensional in nature, but high dimensional. In this thesis, I explore---both experimentally and theoretically---several applications of encoding and processing information in high dimensional quantum systems. I consider information encoded in the spatial, temporal, and phase space distributions of optical systems. I further analyse the evolution of these systems using the theory of closed and open quantum systems, conditional dynamics and Fock state master equations. These results highlight the unique advantages of high dimensional quantum systems in the sub-disciplines quantum foundations, quantum simulation and quantum sensing. Our results clearly demonstrate that high dimensional quantum information presents many unique applications that could be fostered into new technology.
We especially need imagination in science. aria Mitchell Summary The many-body problem consists of two parts: the first is the non-interacting system in a materials-specific external potential; the second is the Coulomb interaction that makes the problem so hard to solve. The most straightforward idea is to use perturbation theory, with the Coulomb interaction as perturbation. This is conceptually simple, but it turns out to be difficult in practice, since the Coulomb interaction is often not small compared with typical energy differences, it is long-ranged and in the thermodynamic limit there is an infinite number of particles, contributing with an infinite number of mutual interaction processes. The present chapter outlines how one can deal with this problem. It contains an overview of facts that one can also find in many standard textbooks on the many-body problem, but that are useful to keep in mind in order to look at later chapters from a sound and well-established perspective. The many-body problem is a tough one, and it has many facets. Sorting it out is like putting together a huge puzzle. The eight introductory chapters of this book provide pieces of the puzzle, and ideas on what one might do about it. In the present chapter we choose to go in one of the possible directions, in order to arrive at something tangible. The chapter gives the general framework and the main ideas; specific approximations are the topic of Chs. 10–15. The idea is to start from an independent-particle problem and add the Coulomb interaction as a perturbation. This is not easy: first, the interaction is responsible for a rich variety of phenomena that are completely absent otherwise, such as the finite lifetime of quasiparticles, or additional structures in spectra due to the fact that a quasi-particle excitation may transfer its energy to other elementary excitations, for example plasmons. Second, because of the two-body Coulomb interaction, the problem scales badly with the number of electrons, and straightforward perturbation theory for the many-body hamiltonian with the Coulomb interaction as perturbation rapidly becomes intractable or even useless, especially in large systems. To get started, Sec. 9.1 recalls why things are not so easy. The following sections try to solve one problem after the other, starting from Sec. 9.2 where the Green's function is reformulated in a way that is appropriate for a perturbation expansion.