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.
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.
What are the information processing capabilities of physical systems?As recently as the first half of the 20 th century this question did not even have a definite meaning.What is information, and how would one process it?It took the development of theories of computing (in the 1930s) and information (late in the 1940s) for us to formulate mathematically what it means to compute or communicate.Yet these theories were abstract, based on axiomatic mathematics: what did physical systems have to do with these axioms?Rolf Landauer had the essential insight -"Information is physical" -that information is always encoded in the state of a physical system, whose dynamics on a microscopic level are well-described by quantum physics.This means that we cannot discuss information without discussing how it is represented, and how nature dictates it should behave.Wigner considered the situation from another perspective when he wrote about "the unreasonable effectiveness of mathematics in the natural sciences".Why are the computational techniques of mathematics so astonishingly useful in describing the physical world [1]?One might begin to suspect foul play in the universe's operating principles.Interesting insights into the physics of information accumulated through the 1970s and 1980s -most sensationally in the proposal for a "quantum computer".If we were to mark a particular year in which an explosion of interest took place in information physics, that year would have to be 1994, when Shor showed that a problem of practical interest (factorisation of integers) could be solved easily on a quantum computer.But the applications of information in physics -and vice versa -have been far more widespread than this popular discovery.These applications range from improved experimental technology, more sophisticated measurement techniques, methods for characterising the quantum/classical boundary, tools for quantum chaos, and deeper insight into quantum theory and nature.In this thesis I present a short review of ideas in quantum information theory.The first chapter contains introductory material, sketching the central ideas of probability and information theory.Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems.In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems.The final chapter presents a new resource: quantum information.This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe.ii This thesis is dedicated to my mother.iii This thesis was begun as an intrepid adventure.It caught
ENGLISH ABSTRACT: What are the information processing capabilities of physical systems? As recently as the first half of the 20th century this question did not even have a definite meaning. What is information, and how would one process it? It took the development of theories of computing (in the 1930s) and information (late in the 1940s) for us to formulate mathematically what it means to compute or communicate. Yet these theories were abstract, based on axiomatic mathematics: what did physical systems have to do with these axioms? Rolf Landauer had the essential insight - "Information is physical" - that information is always encoded in the state of a physical system, whose dynamics on a microscopic level are well-described by quantum physics. This means that we cannot discuss information without discussing how it is represented, and how nature dictates it should behave. Wigner considered the situation from another perspective when he wrote about "the unreasonable effectiveness of mathematics in the natural sciences". Why are the computational techniques of mathematics so astonishingly useful in describing the physical world [1]? One might begin to suspect foul play in the universe's operating principles. Interesting insights into the physics of information accumulated through the 1970s and 1980s - most sensationally in the proposal for a "quantum computer". If we were to mark a particular year in which an explosion of interest took place in information physics, that year would have to be 1994, when Shor showed that a problem of practical interest (factorisation of integers) could be solved easily on a quantum computer. But the applications of information in physics - and vice versa - have been far more widespread than this popular discovery. These applications range from improved experimental technology, more sophisticated measurement techniques, methods for characterising the quantum/classical boundary, tools for quantum chaos, and deeper insight into quantum theory and nature. In this thesis I present a short review of ideas in quantum information theory. The first chapter contains introductory material, sketching the central ideas of probability and information theory. Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems. In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems. The final chapter presents a new resource: quantum information. This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe.
Quantum cryptography is a new and exciting area that uses quantum physics to protect communication lines from being spied on or intercepted. Fundamental ideas in this field, like the uncertainty principle and the fact of quantum entanglement, are used to achieve levels of security that have never been seen before. Our in-depth study, "Quantum Cryptography: Mathematical Foundations and Practical Applications for Secure Communication Protocols," looks at both the math behind quantum cryptographic protocols and how they are used in the real world. Our study goes into great detail about the theories behind quantum cryptography. It explains ideas like quantum key distribution (QKD), quantum teleportation, and quantum secure direct communication (QSDC). One of the main ideas behind quantum cryptography is the idea of qubits, which are like regular bits but in quantum mechanics. They can be in more than one state at the same time because of superposition. Quantum cryptographic methods use this property to make sure communication is safe by putting data in quantum states and taking advantage of the fact that quantum measures are inherently unpredictable. The study we're doing looks at how to use quantum cryptography in typical everyday situations. We look at the problems that come up when you try to build infrastructure for quantum transmission, such as noise, decoherence, and scale. We make a plan for making strong and trustworthy quantum cryptographic systems by giving details about how experiments are set up and how technology is improving. Our study looks into how quantum cryptography could be used for things other than just keeping communications safe. We look into what it means for new technologies like quantum networks, quantum computing, and safe multi-party processing. We hope that by explaining the bigger effects of quantum cryptography, we can encourage more study and new ideas in this ground-breaking area.
Quantum computation and information has become a central research area in theoretical computer science. It studies how information is encoded in nature according to the laws of quantum mechanics and what this means for its computational power. On one hand, the ability of a quantum system to be in a superposition of states provides a way to encode an exponential amount of information in a quantum state. However, this information is accessible only indirectly via measurements. The goal of this thesis is precisely to investigate the fundamental question of how information is encoded and processed in quantum mechanical systems. (1) We investigate their power relative to classical encodings in the model of one-way communication complexity and show that they can be exponentially more efficient, answering a long standing open question in the area of quantum communication complexity. (2) We show how the theory developed for the study of quantum information can be employed in order to answer questions about classical coding theory and complexity. We resolve a main open question about the efficiency of Locally Decodable Codes by reducing the problem to one about quantum codes and solving the latter using tools from quantum information theory. This is the first example where quantum techniques are essential in resolving a purely classical question. (3) We investigate the power of quantum encodings in the area of cryptography. We study certain cryptographic primitives, such as Private Information Retrieval, Symmetrically-Private Information Retrieval and Coin Flipping.
Introduction Any technology that functions at the quantum level must face the issues of measurement and control. We have good reasons to believe that quantum physics enables communication and computation tasks that are either impossible or intractable in a classical world [NC00]. The security of widely used classical cryptographic systems relies upon the difficulty of certain computational tasks, such as breaking large semi-prime numbers into their two prime factors in the case of RSA encryption. By contrast, quantum cryptography can be absolutely secure, and is already a commercial reality. At the same time, the prospect of a quantum computer vastly faster than any classical computer at certain tasks is driving an international research programme to implement quantum information processing. Shor's factoring algorithm would enable a quantum computer to find factors exponentially faster than any known algorithm for classical computers, making classical encryption insecure. In this chapter, we investigate how issues of measurement and control arise in this most challenging quantum technology of all, quantum computation. The subjects of information theory and computational theory at first sight appear to belong to mathematics rather than physics. For example, communication was thought to have been captured by Shannon's abstract theory of information [SW49, Sha49]. However, physics must impact on such fundamental concepts once we acknowledge the fact that information requires a physical medium to support it.
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.