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De Gruyter eBooks

Quantum Systems, Channels, Information

Alexander Semenovich Holevo

Published Nov 27, 2012• 464 citations
Publisher

Abstract

The main emphasis of this work is the mathematical theory of quantum channels and their entropic and information characteristics. Quantum information theory is one of the key research areas, since it leads the way to vastly increased computing speeds by using quantum systems to store and process information. Quantum cryptography allows for secure communication of classified information. Research in the field of quantum informatics, including quantum information theory, is in progress in leading scientific centers throughout the world. The past years were marked with impressive progress made by several researchers in solution of some difficult problems, in particular, the additivity of the entropy characteristics of quantum channels. This suggests a need for a book that not only introduces the basic concepts of quantum information theory, but also presents in detail some of the latest achievements.

DOI
10.1515/9783110273403

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The main emphasis of this work is the mathematical theory of quantum channels and their entropic and information characteristics. Quantum information theory is one of the key research areas, since it leads the way to vastly increased computing speeds by using quantum systems to store and process information. Quantum cryptography allows for secure communication of classified information. Research in the field of quantum informatics, including quantum information theory, is in progress in leading scientific centers throughout the world. The past years were marked with impressive progress made by several researchers in solution of some difficult problems, in particular, the additivity of the entropy characteristics of quantum channels. This suggests a need for a book that not only introduces the basic concepts of quantum information theory, but also presents in detail some of the latest achievements.
This book explores both the state of the art and the latest developments in QKD. It describes the fundamental concepts and practical aspects of QKD from a viewpoint of information security and quantum channel efficiency improvement. The purpose of this book is to extend and update the knowledge of the readers in the dynamically changing field of QKD. The authors attempt to present in detail their results of scientific research, which is divided into two sections - Modern QKD Technologies and Quantum Channel Construction. It will be useful for researchers, engineers, graduates, and doctoral students working in quantum cryptography and information security-related areas.
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 book is intended as an introductory textbook at graduate or advanced undergraduate level. Students with diverse backgrounds, in mathematics, physics and computer science, can use it to become acquainted with the quantum theory of information. Its scope is deliberately chosen to be not too broad, while being self-consistent and usable for a semester-long course for beginners. The subject is presented using a balanced mix of intuitive arguments and rigorous formalism. The emphasis is not on the physical interpretation, but rather on the formal and mathematical features of quantum information theory, in order to be accessible from different backgrounds, the only requirement being a basic knowledge of linear algebra. In particular, a background in quantum mechanics is not needed, since quantum theory is used as a mere mathematical framework without entering into physical interpretations. The book introduces all necessary concepts from quantum theory and classical information theory, to cover the elements of quantum Shannon theory: quantum states and channels, data compression and entropy, error correcting codes and channel capacities, as well as the basics of entanglement theory and quantum cryptography. Examples are provided to help the reader in understanding the different concepts, together with exercises at the end of each chapter.
The book provides an overview of the most advanced quantum informational geometric techniques, which can help quantum communication theorists analyze quantum channels, such as security or additivity properties. Each section addresses an area of major research of quantum information theory and quantum communication networks. The authors present the fundamental theoretical results of quantum information theory, while also presenting the details of advanced quantum ccommunication protocols with clear mathematical and information theoretical background. This book bridges the gap between quantum physics, quantum information theory, and practical engineering.
Scientific progress in physics and mathematics has led to the development of efficient technology for communicating and distributing information in the 20th century. This technology forms one of the cornerstones of information society and our global economy, which are highly reliant on secure methods to transfer and distribute information quickly. To satisfy ever-increasing demands for speed and security, encryption and communication methods are constantly improved and there is an ongoing effort to develop corresponding technology further. Classical information theory is not usually part of a physics undergraduate degree course. The theory assumes simple classical properties of physical systems and based on those a largely mathematical theory of information is established. The obtained results are independent of the chosen implementation and its physical details. However, one still has to acknowledge that information is physical, that is information carriers, senders, and receivers obey the laws of physics. The notion of quantum information comes about since it turns out that the rules of quantum mechanics violate some of the basic physics assumptions in classical information theory. The consequences of this are many, ranging from improved channel capacities, the possibility of physically secure communication protocols, to invalidating some beliefs about the security of classical communication protocols. So far the approach followed in this book was to assume idealized physical systemswhich perfectly implement qubits and then demonstrate how quantum computing algorithms can be realized using perfect qubits. Error correction and fault tolerance were only mentioned briefly.