This is the 10th and final chapter of my book on Quantum Information, based on the course I have been teaching at Caltech since 1997. An earlier version of this chapter (originally Chapter 5) has been available on the course website since 1998, but this version is substantially revised and expanded. Topics covered include classical Shannon theory, quantum compression, quantifying entanglement, accessible information, and using the decoupling principle to derive achievable rates for quantum protocols. This is a draft, pre-publication copy of Chapter 10, which I will continue to update. See the URL on the title page for further updates and drafts of other chapters, and please send me an email if you notice errors.
This is the 10th and final chapter of my book on Quantum Information, based on the course I have been teaching at Caltech since 1997. An earlier version of this chapter (originally Chapter 5) has been available on the course website since 1998, but this version is substantially revised and expanded. Topics covered include classical Shannon theory, quantum compression, quantifying entanglement, accessible information, and using the decoupling principle to derive achievable rates for quantum protocols. This is a draft, pre-publication copy of Chapter 10, which I will continue to update. See the URL on the title page for further updates and drafts of other chapters, and please send me an email if you notice errors.
For Students Prerequisites for understanding the content in this book are a solid background in probability theory and linear algebra. If you are new to information theory, then there should be enough background in this book to get you up to speed (Chapters 2, 10, 13, and 14). However, classics on information theory such as Cover & Thomas (2006) and MacKay (2003) could be helpful as a reference. If you are new to quantum mechanics, then there should be enough material in this book (Part II) to give you the background necessary for understanding quantum Shannon theory. The book of Nielsen & Chuang (2000), sometimes affectionately known as “Mike and Ike”, has become the standard starting point for students in quantum information science and might be helpful as well. Some of the content of that book is available in the dissertation of Nielsen (1998). If you are familiar with Shannon's information theory (at the level of Cover & Thomas, 2006, for example), then the present book should be a helpful entry point into the field of quantum Shannon theory. We build on intuition developed classically to help in establishing schemes for communication over quantum channels. If you are familiar with quantum mechanics, it might still be worthwhile to review Part II because some content there might not be part of a standard course on quantum mechanics. The aim of this book is to develop “from the ground up” many of the major, exciting pre- and post-millennium developments in the general area of study known as quantum Shannon theory. As such, we spend a significant amount of time on quantum mechanics for quantum information theory (Part II), we give a careful study of the important unit protocols of teleportation, super-dense coding, and entanglement distribution (Part III), and we develop many of the tools necessary for understanding information transmission or compression (Part IV). Parts V and VI are the culmination of this book, where all of the tools developed come into play for understanding many of the important results in quantum Shannon theory.
In these first few chapters, our aim is to establish a firm grounding so that we can address some fundamental questions regarding information transmission over quantum channels. This area of study has become known as “quantum Shannon theory” in the broader quantum information community, in order to distinguish this topic from other areas of study in quantum information science. In this text, we will use the terms “quantum Shannon theory” and “quantum information theory” somewhat interchangeably. We will begin by briefly overviewing several fundamental aspects of the quantum theory. Our study of the quantum theory, in this chapter and future ones, will be at an abstract level, without giving preference to any particular physical system such as a spin- 1/2 particle or a photon. This approach will be more beneficial for the purposes of our study, but, here and there, we will make some reference to actual physical systems to ground us in reality. You may be wondering, what is quantum Shannon theory and why do we name this area of study as such? In short, quantum Shannon theory is the study of the ultimate capability of noisy physical systems, governed by the laws of quantum mechanics, to preserve information and correlations. Quantum information theorists have chosen the name quantum Shannon theory to honor Claude Shannon, who single-handedly founded the field of classical information theory with a groundbreaking paper (Shannon, 1948). In particular, the name refers to the asymptotic theory of quantum information, which is the main topic of study in this book. Information theorists since Shannon have dubbed him the “Einstein of the information age.”1 The name quantum Shannon theory is fit to capture this area of study because we often use quantum versions of Shannon's ideas to prove some of the main theorems in quantum Shannon theory. We prefer the name “quantum Shannon theory” over such names as “quantum information science” or just “quantum information.” These other names are too broad, encompassing subjects as diverse as quantum computation, quantum algorithms, quantum complexity theory, quantum communication complexity, entanglement theory, quantum key distribution, quantum error correction, and even the experimental implementation of quantum protocols.
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.
This is a preliminary version of a book in progress on the theory of quantum communication. We adopt an information-theoretic perspective throughout and give a comprehensive account of fundamental results in quantum communication theory from the past decade (and earlier), with an emphasis on the modern one-shot-to-asymptotic approach that underlies much of today's state-of-the-art research in this field. In Part I, we cover mathematical preliminaries and provide a detailed study of quantum mechanics from an information-theoretic perspective. We also provide an extensive and thorough review of quantum entropies, and we devote an entire chapter to the study of entanglement measures. Equipped with these essential tools, in Part II we study classical communication (with and without entanglement assistance), entanglement distillation, quantum communication, secret key distillation, and private communication. In Part III, we cover various developments in feedback-assisted communication tasks, such as quantum and classical feedback-assisted communication, LOCC-assisted quantum communication, and secret key agreement. This version represents an update compared to 2011.04672v2.
An impressively comprehensive textbook adopting a phenomenological approach to quantum physics. The chapters cover everything from basic definitions of key concepts to detailed discussions of the underlying theoretical framework, walking students step-by-step through the necessary mathematics and drawing clear connections between the theory and the most important modern research applications including quantum optics, fluids, nanophysics, entanglement, information, and relativity. With this book, students and researchers will have access to hundreds of real-world examples, exercises, and illustrations to support and expand their understanding. Instructors can tailor the content to suit the length and level of their course and will have access to an online solutions manual with fully worked solutions to all 300+ exercises in the book. Other online resources include Python simulations, additional exercises, and detailed appendices.
For Students Prerequisites for understanding the content in this book are a solid background in probability theory and linear algebra. If you are new to information theory, then there is enough background in this book to get you up to speed (Chapters 2, 10, 12,and 13). Though, classics on information theory such as Cover and Thomas (1991) and MacKay (2003) could be helpful as a reference. If you are new to quantum mechanics, then there should be enough material in this book (Part II) to give you the background necessary for understanding quantum Shannon theory. The book of Nielsen and Chuang (sometimes known as “Mike and Ike”) has become the standard starting point for students in quantum information science and might be helpful as well (Nielsen & Chuang, 2000). Some of the content in that book is available in Nielsen's dissertation (Nielsen, 1998). If you are familiar with Shannon's information theory (at the level of Cover and Thomas (1991), for example), then this book should be a helpful entry point into the field of quantum Shannon theory. We build on intuition developed classically to help in establishing schemes for communication over quantum channels. If you are familiar with quantum mechanics, it might still be worthwhile to review Part II because some content there might not be part of a standard course on quantum mechanics.