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The theory of quantum information
Watrous J., Cambridge University Press, New York, NY, 2018. 598 pp. Type: Book (978-1-107180-56-7)
Date Reviewed: Nov 21 2019

Information theory has its modern origins in the 1948 work [1] of Claude Shannon, who gave results that addressed two key problems: the extent to which information can be compressed without loss, and the rate at which information can be transmitted over a channel. These seminal results laid the groundwork for the discipline of information theory and thence, in conjunction with advances in computing technology, for the communications revolution. Information theory continues to be a highly mathematical discipline in spite of its applied focus.

Based on a landmark 1982 paper [2] by the physicist Richard Feynman, there has been a push in recent decades to build a new type of computing device based on the quantum states and interactions of elementary physical particles--a quantum computer. Such a quantum computer is not yet a reality on a practically significant scale (and it is not completely certain when it will be so), but small demonstrations have been attempted in laboratory settings; there is a significant body of theoretical work describing and analyzing quantum computation.

Quantum information theory, the topic of this book, lies at the confluence of these two major scholarly disciplines. It deals with results concerning information theory in a quantum computing context, for example, for determining the channel capacity when a sender and a receiver share an entangled state (which is double the capacity of a classical channel in the sense of Shannon).

The book is laid out over eight chapters, with the first being devoted to graduate-level mathematical preliminaries such as probability theory and linear algebra. The second chapter is on the quantum-theoretic analogs of basic information theory. Later chapters deal with advanced topics and require significant mathematical depth, something probably not common outside graduate classrooms. While the author notes that quantum mechanics, per se, is not covered in the book, significant exposure to and understanding of it as well as classical information theory is surely a desideratum. This book will delight theoreticians with an interest in studying this purely theoretical topic with no expectation that it will correlate with any real-life purpose, but not so much for engineers or practitioners seeking to apply insights to real problems (or even to theoretical topics outside this domain).

This book is highly mathematical, as befits its subject, following the usual definition-theorem-proof routine, and is suitable for theoretical computer scientists, mathematicians, and mathematical physicists who have significant depth of understanding in information theory and some background in quantum computation (or quantum physics). It provides an up-to-date, organized presentation of many mathematical results that have been published over the past few decades. The author has not only presented a quality theoretical treatise, but has taken care to provide exercises and bibliographic remarks at the end of each chapter, which graduate students and others may find useful, as well as a topic index and a detailed bibliography.

My main quibble would be with the publisher’s claim on the back cover that the book is “largely self-contained” and “accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory.” In my view, a reader with just basic mathematical concepts being able to study it in significant detail would be rather extraordinary. The author’s own remark in the preface, that the book is “primarily intended for graduate students and researchers having some familiarity with quantum information and computation,” is far more sensible.

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Reviewer:  Shrisha Rao Review #: CR146793 (2004-0067)
1) Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, (1948), 379–423.
2) Feynman, R. P. Simulating physics with computers. International Journal of Theoretical Physics 21, (1982), 467–488.
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