Open Systems, Entanglement and Quantum Optics

The subject of this book is a presentation of some aspects of modern theory of open quantum systems. It introduces several up-to- date topics, such as detecting quantum entanglement, modeling of quantum noise, quantum communication processes, and computational complexity in the analysis of quantum operations. Also discussed are light propagation in optically dressed media, as well as entropy and information measure for quantized electromagnetic fields media. This book is intended for researchers and students interested in the theory of open quantum systems, quantum information theory and quantum systems interacting with electromagnetic fields

A few steps more towards NPT bound entanglement

We consider the problem of existence of bound entangled states with non-positive partial transpose (NPT). As one knows, existence of such states would in particular imply nonadditivity of distillable entanglement. Moreover it would rule out a simple mathematical description of the set of distillable states. Distillability is equivalent to so called n-copy distillability for some n. We consider a particular state, known to be 1-copy nondistillable, which is supposed to be bound entangled. We study the problem of its two-copy distillability, which boils down to show that maximal overlap of some projector Q with Schmidt rank two states does not exceed 1/2. Such property we call the the half-property. We first show that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. We then attack the problem in twofold way: a) prove the half-property for some classes of Schmidt rank two states b) bound the required overlap from above for all Schmidt rank two states. We have succeeded to prove the half-property for wide classes of states, and to bound the overlap from above by c<3/4. Moreover, we translate the problem into the following matrix analysis problem: bound the sum of the squares of the two largest singular values of matrix A \otimes I + I \otimes B with A,B traceless 4x4 matrices, and Tr A^\dagger A + Tr B^\dagger B = 1/4.Comment: 15 pages, Final version for IEEE Trans. Inf. Theor

Causal Models for a Quantum World

Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. A first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. As these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models

A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing

The motivation for this thesis was to recast quantum self-testing [MY98,MY04] in operational terms. The result is a category-theoretic framework for discussing the following general question: How do different implementations of the same input-output process compare to each other? In the proposed framework, an input-output process is modelled by a causally structured channel in some fixed theory, and its implementations are modelled by causally structured dilations formalising hidden side-computations. These dilations compare through a pre-order formalising relative strength of side-computations. Chapter 1 reviews a mathematical model for physical theories as semicartesian symmetric monoidal categories. Many concrete examples are discussed, in particular quantum and classical information theory. The key feature is that the model facilitates the notion of dilations. Chapter 2 is devoted to the study of dilations. It introduces a handful of simple yet potent axioms about dilations, one of which (resembling the Purification Postulate [CDP10]) entails a duality theorem encompassing a large number of classic no-go results for quantum theory. Chapter 3 considers metric structure on physical theories, introducing in particular a new metric for quantum channels, the purified diamond distance, which generalises the purified distance [TCR10,Tom12] and relates to the Bures distance [KSW08a]. Chapter 4 presents a category-theoretic formalism for causality in terms of '(constructible) causal channels' and 'contractions'. It simplifies aspects of the formalisms [CDP09,KU17] and relates to traces in monoidal categories [JSV96]. The formalism allows for the definition of 'causal dilations' and the establishment of a non-trivial theory of such dilations. Chapter 5 realises quantum self-testing from the perspective of chapter 4, thus pointing towards the first known operational foundation for self-testing.Comment: PhD thesis submitted to the University of Copenhagen (ISBN 978-87-7125-039-8). Advised by prof. Matthias Christandl, submitted 1st of December 2020, defended 11th of February 2021. Keywords: dilations, applied category theory, quantum foundations, causal structure, quantum self-testing. 242 pages, 1 figure. Comments are welcom

Assumptions in Quantum Cryptography

Quantum cryptography uses techniques and ideas from physics and computer science. The combination of these ideas makes the security proofs of quantum cryptography a complicated task. To prove that a quantum-cryptography protocol is secure, assumptions are made about the protocol and its devices. If these assumptions are not justified in an implementation then an eavesdropper may break the security of the protocol. Therefore, security is crucially dependent on which assumptions are made and how justified the assumptions are in an implementation of the protocol. This thesis is primarily a review that analyzes and clarifies the connection between the security proofs of quantum-cryptography protocols and their experimental implementations. In particular, we focus on quantum key distribution: the task of distributing a secret random key between two parties. We provide a comprehensive introduction to several concepts: quantum mechanics using the density operator formalism, quantum cryptography, and quantum key distribution. We define security for quantum key distribution and outline several mathematical techniques that can either be used to prove security or simplify security proofs. In addition, we analyze the assumptions made in quantum cryptography and how they may or may not be justified in implementations. Along with the review, we propose a framework that decomposes quantum-key-distribution protocols and their assumptions into several classes. Protocol classes can be used to clarify which proof techniques apply to which kinds of protocols. Assumption classes can be used to specify which assumptions are justified in implementations and which could be exploited by an eavesdropper. Two contributions of the author are discussed: the security proofs of two two-way quantum-key-distribution protocols and an intuitive proof of the data-processing inequality.Comment: PhD Thesis, 221 page

Practical and reliable error bars for quantum process tomography

Current techniques in quantum process tomography typically return a single point estimate of an unknown process based on a finite albeit large amount of measurement data. Due to statistical fluctuations, however, other processes close to the point estimate can also produce the observed data with near certainty. Unless appropriate error bars can be constructed, the point estimate does not carry any sound operational interpretation. Here, we provide a solution to this problem by constructing a confidence region estimator for quantum processes. Our method enables reliable estimation of essentially any figure of merit for quantum processes on few qubits, including the diamond distance to a specific noise model, the entanglement fidelity, and the worst-case entanglement fidelity, by identifying error regions which contain the true state with high probability. We also provide a software package, QPtomographer, implementing our estimator for the diamond norm and the worst-case entanglement fidelity. We illustrate its usage and performance with several simulated examples. Our tools can be used to reliably certify the performance of, e.g., error correction codes, implementations of unitary gates, or more generally any noise process affecting a quantum system