This thesis will be focused on the classical capacity of quantum channels, one of the first areas treated by quantum information theorists. The problem is fairly solved since some years. Nevertheless, this work will give me a reason to introduce a consistent formalism of the quantum theory, as well as to review fundamental facts about quantum non-locality and how it can be used to enhance communication. Moreover, this reflects my dwelling in the spirit of classical information theory, and it is intended to be a starting point towards a thorough study of how quantum technologies can help to shape the future of telecommunications. Whenever it was possible, heuristic reasonings were introduced instead of rigorous mathematical proofs. This finds an explanation in that I am a self-taught neophyte in the field, and just about every time I came across a new concept, physical arguments were always more compelling to me than just maths. The technical content of the thesis is twofold. On one hand, a quadratic classification based on optimization programs that I devised for distinguishing entangled states is presented in Chapter 4. In second place, a less difficult yet I hope equally interesting technical part consists of versions of some proofs throughout the text.
This thesis will be focused on the classical capacity of quantum channels, one of the first areas treated by quantum information theorists. The problem is fairly solved since some years. Nevertheless, this work will give me a reason to introduce a consistent formalism of the quantum theory, as well as to review fundamental facts about quantum non-locality and how it can be used to enhance communication. Moreover, this reflects my dwelling in the spirit of classical information theory, and it is intended to be a starting point towards a thorough study of how quantum technologies can help to shape the future of telecommunications. Whenever it was possible, heuristic reasonings were introduced instead of rigorous mathematical proofs. This finds an explanation in that I am a self-taught neophyte in the field, and just about every time I came across a new concept, physical arguments were always more compelling to me than just maths. The technical content of the thesis is twofold. On one hand, a quadratic classification based on optimization programs that I devised for distinguishing entangled states is presented in Chapter 4. In second place, a less difficult yet I hope equally interesting technical part consists of versions of some proofs throughout the text.
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 information science has grown from being a very small subfield in the 70s until being one of the most dynamic fields in physics, both in fundamentals and applications. In the theoretical section, perhaps the feature that has attracted most interest is the notion of entanglement, the ghostly relation between particles that dazzled Einstein and has provided fabulous challenges to build a coherent interpretation of quantum mechanics. While not completely solved, we have today learned enough to feel less uneasy with this fundamental problem, and the focus has shifted towards its potential powerful applications. Entanglement is now being studied from different perspectives as a resource for performing information processing tasks.\n\n\t\t\t\t With bipartite entanglement being largely understood nowadays, many questions remain unanswered in the multipartite case. The first part of this thesis deals with multipartite entanglement in different contexts. In the first chapters it is studied within the whole corresponding Hilbert space, and we investigate several entanglement measures searching for states that maximize them, including violations of Bell inequalities. Later, focus is shifted towards hamiltonians that have entangled ground states, and we investigate entanglement as a way to establish a distance between theories and we study frustration and methods to efficiently solve hamiltonians that exhibit it.\n\n\t\t\t\t In the practical section, the most promised upcoming technological advance is the advent of quantum computers. In the 90s some quantum algorithms improving the performance of all known classical algorithms for certain problems started to appear, while in the 2000s the first universal computers of few atoms began to be built, allowing implementation of those algorithms in small scales. The D-Wave machine already performs quantum annealing in thousands of qubits, although some controversy over the true quantumness of its internal workings surrounds it. Many countries in the planet are devoting large amounts of money to this field, with the recent European flagship and the involvement of the largest US technological companies giving reasons for optimism.\n\n\t\t\t\t The second part of this thesis deals with some aspects of quantum computation, starting with the creation of the field of cloud quantum computation with the appearance of the first computer available to the general public through internet, which we have used and analysed extensively. Also small incursions in quantum adiabatic computation and quantum thermodynamics are present in this second part.
Is entanglement an exclusive feature of quantum systems, or is it common to all non-classical theories? And if this is the case, how strong is quantum mechanical entanglement as compared to that exhibited by other theories? The first part of this thesis deals with these questions by considering quantum theory as part of a wider landscape of physical theories, collectively called general probabilistic theories (GPTs). Among the other things, this manuscript contains a detailed introduction to the abstract state space formalism for GPTs. We start with a comprehensive review of the proof of a famous theorem by Ludwig that constitutes one of its cornerstones (Ch. 1). After explaining the basic rules of the game, we translate our questions into precise conjectures and present our progress toward a full solution (Ch. 2). In Ch. 3 we consider entanglement at the level of measurements instead of states, focusing on one of its main implications, i.e. data hiding. We determine the maximal data hiding strength that a quantum mechanical system can exhibit, and also the maximum value among all GPTs, finding that the former scales as the square root of the latter. In the second part of this manuscript we look into quantum entanglement. In Ch. 4 we discuss the entanglement transformation properties of a class of maps that model white noise acting either locally or globally on a bipartite system. In Ch. 5 we employ matrix analysis tools to develop a unified approach to Gaussian entanglement. The third part of this thesis concerns more general forms of non-classical correlations in bipartite continuous variable systems. In Ch. 6 we devise a general scheme that allows to consistently classify correlations of bipartite Gaussian states into classical and quantum ones. Finally, Ch. 7 explores some problems connected with a certain strong subadditivity matrix inequality.
Born about a century ago, Quantum Mechanics has revolutionized the description and the interpretation of Physics at sub-microscopic level. In the last decades, due to the influence of mathematical and engineering research fields, Quantum Mechanics has given birth to related research areas like Quantum Computation, Quantum Information and Quantum Communication. With the discovery of the laser, and later the development of fiber optics and satellite networks, Quantum Communication and Quantum Optics seems to have a natural field of application in Communication Systems. Despite this, the interest in this technology and studies for communication purpose has been overshadowed by the great results in communication networks achieved in the last decades with classical paradigms. However, due to the increasing demand of communication data rate, system designers are now looking at Quantum Mechanics for new and more performanting solutions in communication purposes. Early theoretical studies on Quantum Discrimination Theory and Quantum Information predict better performance for Communication Systems that take advantage of the quantum laws. In addition, Quantum Mechanics provides the deepest description of the physical phenomena, and there are scenarios where a quantum model fits best, as in in deep space communications, where the received signal is really weak, or in a satellite networks, where we are interested in strongly reducing the power of transmitted signals, possibly without sacrificing performance significantly. However, if on one side Quantum Communication Theory promises great gains in the performance of communication systems, on the other hand it fails to describe how to implement physical devices that reach these ultimate limits. So far, only a few architectures achieving these performances are known, and only for simple modulation formats. We are interested in the scenario of optical communications, where the message transmitted is encoded in a sequence of coherent states. Transmitter devices for coherent modulation are well known and consist in laser pulse generators. Instead, receiver implementations working at the quantum limit performance limit are yet to be found. In this Thesis I deal with different topics in the quantum transmission scenario. First, I review existing classical (suboptimal) and quantum (suboptimal and optimal) receiver schemes for the binary coherent modulation. I present a new formulation of the optimal scheme known as Dolinar Receiver with the multiple copies problem, focusing on the information gained during the measurement. Second, I analyze the binary communication in a noisy environment, studying the error probability and the capacity of the binary channel induced. Given the description of the quantum channel, I optimize both the transmitted quantum states and the measurement operators employed in the communication. Third, I consider the Pulse Position Modulation, that is particularly suitable for space and satellite communication due to its simplicity of implementation and high capacity. I review some known suboptimal receivers, and I propose a receiver scheme which approaches the limit performance predicted with quantum theory outperforming the existing schemes. To sum up the results of this Thesis, in order to approach the limit performance predicted by Quantum Mechanics, an optimization is always necessary to exceed the classical effects and trigger the quantum phenomena. In particular, the information gained during the measurement plays an important role, for example in the definition of adaptive receivers. In this Thesis both these aspects have been deeply investigated.
Quantum information theory studies the fundamental limits that physical laws impose on information processing tasks such as data compression and data transmission on noisy channels. This thesis presents general techniques that allow one to solve many fundamental problems of quantum information theory in a unified framework. The central theorem of this thesis proves the existence of a protocol that transmits quantum data that is partially known to the receiver through a single use of an arbitrary noisy quantum channel. In addition to the intrinsic interest of this problem, this theorem has as immediate corollaries several central theorems of quantum information theory. The following chapters use this theorem to prove the existence of new protocols for two other types of quantum channels, namely quantum broadcast channels and quantum channels with side information at the transmitter. These protocols also involve sending quantum information partially known by the receiver with a single use of the channel, and have as corollaries entanglement-assisted and unassisted asymptotic coding theorems. The entanglement-assisted asymptotic versions can, in both cases, be considered as quantum versions of the best coding theorems known for the classical versions of these problems. The last chapter deals with a purely quantum phenomenon called locking. We demonstrate that it is possible to encode a classical message into a quantum state such that, by removing a subsystem of logarithmic size with respect to its total size, no measurement can have significant correlations with the message. The message is therefore "locked" by a logarithmic-size key. This thesis presents the first locking protocol for which the success criterion is that the trace distance between the joint distribution of the message and the measurement result and the product of their marginals be sufficiently small.
T his thesis studies quantum communication complexity and quantum error correction, two fundamental areas in the theory of quantum computation, and is divided into two parts.The first part focuses on generalising three central problems in quantum communication complexity, often by the introduction of more general Boolean functions.The first studied problem is the one of approximating the Hamming distance in the simultaneous message passing model in communication complexity -a natural generalisation of the Equality problem.An efficient quantum communication protocol is established and later used as the basis for approximating other useful distance measures, e.g.graph distance and ℓ 1 -distance.The second generalised problem is the Boolean Hidden Matching problem, the first proposed one to exhibit an exponential quantum-classical communication separation in the one-way model.Originally defined as a task of matching bits using the Parity function, the problem is generalised by replacing the Parity function with any Boolean function.The hardness of the resulting problem, both classical and quantumly, is characterised in terms of two properties of the Boolean function used, its sign degree and pure high degree.The first part of the thesis is completed by studying the generalisation of random access codes where the parties are no longer interested in recovering any initial bit, but the value of a fixed Boolean function on any subset of the initial bits with constant size.Different random access codes are defined by means of different accessible classical and quantum resources and their efficiency analysed and compared.The second part of the thesis is devoted to the study of quantum error correction and the toric code.Even though unrealistic for some physical platforms, most decoders proposed for the toric code assume that stabilizer measurements can always be performed deterministically.This second part aims to study the effect of probabilistic stabilizer measurements, i.e., asynchronism, on the toric code.A few different decoders, based on gradually more refined approximations, are proposed in order to handle the asynchronism and it is numerically shown that they are able to maintain a high threshold even in the limit where stabilizer measurements are performed continuously.i Dedication and acknowledgements M y deepest thanks goes to my supervisor, Ashley Montanaro, who supported and motivated me during these last three years.Coming from a physics background, I am grateful to Ashley for introducing me to the field of Quantum Computation, and also for helping me to mature both academically and intellectually since my Masters.I also thank Ashley for incentivising my taste for literature by lending me his copy of John Milton's Paradise Lost.I thank my co-authors, Naomi Nickerson and Hugo Cable, for all the help and discussions behind Chapter 6 about quantum error correction and the toric code, and for their kind welcome at the time of my visit at PsiQuantum.I also thank Ryan Mann for many useful discussions, most of them about concepts from Chapter 6 and self-avoiding walks; Ronald de Wolf for having reliably improved all my papers so far with valuable suggestions and comments, for suggesting the block-encoding scheme in Chapter 4, originally with the 2, 3 → 1 QRACs, for pointing out Refs. [119,121,177] and for the precise reviewing of the thesis; Noah Linden for the precise reviewing of the thesis and for testing me throughout my PhD with interesting questions; Raphaël Clifford for helpful discussions related to Chapter 2; Makrand Sinha for useful discussions about the hypercontractive inequality; Máté Farkas and Mark Howard for pointing out Refs. [4,72,73, 190] and [69], respectively.These last four years would not be the same without all the support from QE-CDT (Center for Doctoral Training in Quantum Engineering) and QET Labs (Quantum Engineering Technology Labs) and all the people behind them, both the management and the academic teams.And not less important, my time in Bristol would definitely not be the same without my cohort: