# Quantum Information Theory

This is a massively comprehensive work that can be used as lecture notes or for self study. Quantum information theory is not the easiest topic and it is still in full expansion. So besides many other books devoted to the subject (I. Chuang and M. Nielsen: *Quantum Computation and Quantum Information*, Cambridge University Press, 2000, has become a kind of standard) this book has the advantage that it includes the more recent developments of the field. What I have in front of me is the first edition of 2013. A second edition comes out in 2017. In fact a preprint version of 2016 is available on arxiv.org/pdf/1106.1445 and can be modified by teachers under the Creative Commons non-commercial licence. Besides little changes throughout, and additional exercises, the text has been extended and modified at several places with modified proofs and extra results and it has an extra chapter on *Quantum Entropy Inequalities and Recoverability*, there is a different appendix and of course extra references.

Despite the many pages, the author does not claim completeness or to have covered all possible protocols. Much has been achieved, yet much has to be investigated before we will be able to deal with a quantum version of the Internet. He has chosen to concentrate on channels. So channel capacity is the key concept here. It expresses how much information can be sent over it and most importantly whether we are able to reach this optimal capacity.

The book is self-contained in the sense that, besides assuming a solid background in probability and linear algebra, it starts with some basic elements from classical Shannon information theory as well as from quantum theory, although some prior knowledge is advisable. Consider these introductory parts I and II more as an executive summary, rather than a basic course on these topics, even though they fill already some 150 pages. The second part on quantum information theory covers some aspects that may not be found in a classical book on quantum mechanics (quantum channels, channel capacity, purification systems to simulate noisy quantum systems etc.). Basic unit protocols of teleportation, super-dense coding and entanglement distribution assuming a noiseless channel are the core subjects of part III.

The bulk of the preparatory work comes with part IV defining the basic tools for quantum information theory where the practical, but much more complicated situation of noise channels are taken into account. Distance measures are needed, entropy is revisited, different measures for quantum information are needed, both in a static and in a dynamic (i.e., for channels) situation and these measures should be such that asymptotically as the number of samples in typical sequences goes to infinity, the error or uncertainty should vanish (quantum typicality). Finally the packing lemma and its dual the covering lemma are the last tools introduced. In the first, one party packs its messages in a subspace so that it is easily recognizable for another party, and in the second the goal is the opposite, namely to make the messages indistinguishable.

With all these tools prepared, the results are developed first in part V for noiseless channels and the main results of the book in part VI for the noisy situation. In the first case this is the Schumacher compression (based on von Neumann entropy) and the related, yet incompatible entanglement concentration (based on entropy of entanglement). Part VI is the present state of the noisy quantum Shannon theory is the culmination of the whole book. It amply illustrates that quantum information theory is so much richer than classical information theory and not everything is fully understood. Channel capacity and optimal communication of classical information over a noisy quantum channel, without or with entanglement, and coherent communication, private or public. Some of this is combined and refined in a chapter on quantum communication and in the subsequent one on communicating classical and quantum information, which unifies all the channel coding theorems seen before. The book concludes with a brief survey and outlook.

The many possibilities to be considered make it easy to get lost. So the author takes care to proceed step by step, starting from the situation closest to the classical one to proceed to more involved cases. Each chapter starts with an introductory text explaining what this chapter is about and how it fits into the rest of the book with backward and forward references. Many of the chapters also end with a short summary to recapitulate the material just introduced. Each chapter has also a short section with some historical remarks and references for further reading. Exercises are sprinkled throughout the text, often it is a way to shortcut the exposition.

Conclusion: this is an excellent textbook for graduate students to be introduced to the topic of quantum information theory. I trust this qualification will be prolonged in the second edition. It is probably too much to be dealt with in just one course, but if the students are really graduate, they may be familiar with either or both of quantum theory and information theory, otherwise it may be tough to assimilate the whole theory. There are suggestions for students and teachers on how to use the book and make selections.

**Submitted by Adhemar Bultheel |

**19 / Feb / 2017