Quantum Information , Stephen M. Barnett Oxford U. Press, New York, 2009. $100.00 (300 pp.). ISBN 978-0-19-852762-6

From the earliest proposals by Richard Feynman and David Deutsch almost 30 years ago, quantum information has emerged as a well-defined discipline. Since then, many novel quantum-computation and secure-communication proposals have been made. For example, quantum computing procedures would permit the search for an object in an unsorted database of *N* objects in $ N $ steps; and quantum cryptography allows for the secure exchange of an encryption key on a public channel, which ensures that the exponentially fast Shor algorithm for factoring a prime number does *not* put the present-day communication security protocols in jeopardy of being breached.

Such proposals have helped to lay the foundation of an exciting, active field that spans physics, mathematics, computer science, and electrical engineering. Experimental progress has been modest, due to the inherent problems associated with decoherence—the inevitable loss of information that results from the system’s interactions with its surroundings. Nonetheless, the field is still evolving and holds great promise.

Stephen Barnett’s * Quantum Information * is a concise and remarkably readable account of most of the developments in the field. Barnett is a leading researcher who has made wide-ranging and fundamental contributions to quantum optics and quantum information science. His book touches on almost all aspects of quantum information and quantum computing, including communication and measurement theory, entanglement, and computing algorithms.

* Quantum Information * includes 21 appendices on topics such as number theory for cryptography, quantum copying, and the Araki–Lieb inequalities. The strategy is successful: a text accessible to an advanced undergraduate student with a decent background in elementary quantum mechanics followed by a set of sophisticated appendices for the more advanced reader. Also, at the end of each chapter are 30–40 exercise problems; most of them consolidate the topics discussed in the text and help to expand understanding.

But how much of quantum information can one grasp armed with just the basic postulates of quantum mechanics? It turns out that no knowledge of the Schrödinger equation and its solutions is required to understand the basic algorithms of quantum computing and quantum information. For example, a basic comprehension of light polarization and complementarity is enough to understand quantum cryptography’s Bennett-Brassard protocol, which guarantees completely secure channels of communication.

The required level of background is provided in * Quantum Information *, which opens with chapters on the basics of probability, information, and quantum mechanics. Although self-contained, those chapters also introduce related concepts such as entropy and its relation to information, provide a workable knowledge of the density matrix, and set the stage for discussions on quantum aspects of measurement, communication, and computing. In addition to the traditional topics on quantum computing and quantum communication, the chapters contain nuggets that are usually not provided in texts. For example, the chapter on quantum processing has a section on cluster states, which are crucial to the novel notion of one-way computing. The chapter on generalized measurement includes a detailed discussion of probability operator measure. However, most of the details are saved for the appendices.

For an evolving subject like quantum information, it is natural that a text will contain topics that reflect the author’s preference and omit others. Barnett’s text is strong on formalism, but avoids any discussion of practical systems, whose inclusion would have required detailed discussions and a much longer book. But there are other topics whose absence I felt more acutely. A relatively detailed discussion of quantum decoherence, which Barnett does not provide, is necessary to understand the challenges in designing quantum computing devices. Also conspicuously absent is a treatment of concurrence, the most widely used measure of entanglement.

Those quibbles aside, * Quantum Information * is an impressive book. The engaging introductory chapters, extensive problem sets, and exhaustive appendices result in a textbook that I highly recommend for a one-semester course on quantum information at the advanced undergraduate or graduate level.