The fundamental properties of quantum information and its applications to computing and cryptography have been greatly illuminated by considering information-theoretic tasks that are provably possible or impossible within non-relativistic quantum mechanics. I describe here a general framework for defining tasks within (special) relativistic quantum theory and illustrate it with examples from relativistic quantum cryptography and relativistic distributed quantum computation. The framework gives a unified description of all tasks previously considered and also defines a large class of new questions about the properties of quantum information in relation to Minkowski causality. It offers a way of exploring interesting new fundamental tasks and applications, and also highlights the scope for a more systematic understanding of the fundamental information-theoretic properties of relativistic quantum theory.
The fundamental properties of quantum information and its applications to computing and cryptography have been greatly illuminated by considering information-theoretic tasks that are provably possible or impossible within non-relativistic quantum mechanics. I describe here a general framework for defining tasks within (special) relativistic quantum theory and illustrate it with examples from relativistic quantum cryptography and relativistic distributed quantum computation. The framework gives a unified description of all tasks previously considered and also defines a large class of new questions about the properties of quantum information in relation to Minkowski causality. It offers a way of exploring interesting new fundamental tasks and applications, and also highlights the scope for a more systematic understanding of the fundamental information-theoretic properties of relativistic quantum theory.
Quantum mechanics, which describes the behaviour of matter and energy on very small scales, is one of the most successful theories in the history of science. Einstein's theory of special relativity, which describes the relationship between space and time, is likewise a highly successful and widely accepted theory. And yet there is a well-documented tension between the two theories, to the extent that it is still not clear that the two can ever be reconciled. This thesis is concerned with furthering the current understanding of the relationship between quantum mechanics and special relativity. In the first part of the thesis we study the behaviour of quantum information in relativistic spacetime. The field of quantum information arose from the realisation that quantum information has a number of crucial properties that distinguish it from classical information, such as the no-cloning property, quantum contextuality, and quantum discord. More recently, it has been realised that placing quantum information under relativistic constraints leads to the emergence of further unique features which are not exhibited by either non-relativistic quantum information or relativistic classical information; as part of this ongoing research programme we develop a new relativistic quantum `paradox' which puts pressure on conventional views about the spatiotemporal persistence of quantum states over time. We then study a new set of relativistic quantum protocols which involve the distribution of entangled states over spacetime, defining one task involving the distribution of the two halves of a known entangled state, and another task involving the distribution of the two halves of an unknown entangled state. The second part of the thesis deals with relativistic quantum cryptography, a field which first began attracting serious attention when it was realised that a cryptographic task known as `bit commitment,' can be implemented with perfect security under relativistic constraints. This result was highly significant, since it is provably impossible to implement bit commitment with perfect security in a purely classical or purely quantum context, and hence bit commitment is an ideal starting point for probing the power of relativistic quantum cryptography. In this thesis we propose several new relativistic quantum bit commitment protocols which have notable advantages over previously known protocols. We then move to a related task, a generalization of zero-knowledge proving which we refer to as knowledge-concealing evidencing of knowledge of a quantum state; we prove no-go theorems concerning the possibility of implementing this task with perfect security, and then set out a relativistic protocol for the task which is asymptotically secure as the dimension of the state in question becomes large. These results have interesting foundational significance above and beyond their applications in the field of cryptography, providing a new perspective on the connections between knowledge, realism and quantum states.
Quantum entanglement, a fundamental phenomenon in quantum mechanics, presents a unique challenge to classical information theory. This research delves into the analysis and manipulation of quantum entanglement, specifically exploring its potential for instantaneous information transfer and quantum communication. We propose a novel framework for measuring and manipulating entanglement, leveraging its non-local properties to achieve a paradigm shift in information processing. This investigation challenges traditional information theory by demonstrating a potential for bypassing the limitations imposed by locality and causality. The study investigates the implications of this approach on the foundations of information theory, considering its impact on cryptography and distributed quantum computing. This work explores the potential for achieving truly instantaneous communication, a concept previously considered purely theoretical, and examines the impact on our understanding of information's fundamental nature.
Quantum information has dramatically changed information science and technology, looking at the quantum nature of the information carrier as a resource for building new information protocols, designing radically new communication and computation algorithms, and ultra-sensitive measurements in metrology, with a wealth of applications. From a fundamental perspective, this new discipline has led us to regard quantum theory itself as a special theory of information, and has opened routes for exploring solutions to the tension with general relativity, based, for example, on the holographic principle, on non-causal variations of the theory, or else on the powerful algorithm of the quantum cellular automaton, which has revealed new routes for exploring quantum fields theory, both as a new microscopic mechanism on the fundamental side, and as a tool for efficient physical quantum simulations for practical purposes. In this golden age of foundations, an astonishing number of new ideas, frameworks, and results, spawned by the quantum information theory experience, have revolutionized the way we think about the subject, with a new research community emerging worldwide, including scientists from computer science and mathematics.
Relativistic quantum information investigates the interplay between quantum mechanics and relativity and is becoming increasingly relevant for future quantum technologies operating over large distances or in gravitational fields. This dissertation studies how relativistic effects influence the storage, transmission, and interference of quantum states of light, with particular emphasis on quantum memories and frequency-entangled photons. A theoretical framework is developed to describe relativistic transformations of photonic quantum states and their interaction with realistic quantum memories. Special attention is given to the influence of gravitational redshift and relativistic motion on quantum coherence, entanglement, and Hong–Ou–Mandel interference. The resulting models provide quantitative predictions for experimentally accessible scenarios and identify conditions under which relativistic effects become relevant for quantum communication and distributed quantum information processing. The presented results contribute to the emerging field of relativistic quantum information by connecting concepts from quantum optics, quantum communication, and general relativity. Beyond their fundamental significance, they provide theoretical tools for the development of future space-based quantum networks and relativistic quantum technologies. Beyond the results presented in this dissertation, the developed framework points toward a broader research program centered on the **Nonlinear Separability Problem**. This problem is proposed as a unifying mathematical perspective for investigating the limits of distinguishability, classification, and information processing in both classical and quantum systems. Its formulation naturally connects to a wide range of contemporary research areas, including computational complexity theory (e.g., P versus NP), quantum information processing, machine learning, optimization, information theory, quantum foundations, and mathematical physics. In particular, the framework suggests new approaches to studying the role of nonlinearity in quantum state discrimination, quantum communication protocols, and the mathematical structure of high-dimensional information spaces. While many of these directions remain to be explored, the present work establishes a conceptual and mathematical basis for future investigations. It is anticipated that further development of the Nonlinear Separability Problem may contribute to a deeper understanding of fundamental questions across mathematics, computer science, and quantum technologies, thereby opening new interdisciplinary avenues of research.
This thesis is a study of how information theory can be used to analyse and interpret quantum mechanical systems, specifically, systems that are being used for information processing and communication tasks. Quantum information science is a relatively new field, and it has only been recently that we have started to look at using quantum mechanical systems to accomplish information processing tasks that are not possible using classical systems. As this method of information processing is a relatively new paradigm, there are many possibilities for new algorithms and discoveries in this area of computing. The application of information theory towards the analysis and description of quantum mechanical systems yields new interesting results - it can even be argued that information theory provides a natural and intuitive way to intepret quantum mechanics, as correlations and measurement outcomes in quantum processes can be described using information theory. Chapter 1 is an introduction to the strange new world of quantum mechanics and takes a brief look at one of the applications of quantum information science in quantum cryptography. A mathematical overview of the postulates and framework that quantum mechanics is built on is presented in Chapter 2. Chapter 3 is a brief study of classical information theory and some of the fundamental concepts and ideas that form the crux of the subject. In Chapter 4, we present the main results of the thesis by taking a look at an optimal measurement scheme that improves on previous schemes by allowing us to extract the maximal amount of information from states in a quantum channel of specific geometry. The findings have an impact on various other areas of the field, including security analysis of quantum cryptographic protocols and building better quantum cloning machines.
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. \n \n \nA 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. \n \n \nAs 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.