Quantum Information Theory: Mathematical Foundation (Graduate Texts in Physics)

In spite of the relative lack of experimental data that illustrates its main concepts, quantum information theory has swelled into a field of its own, and as this book illustrates, attracts researchers from both physics and mathematics. All of this research is interesting, and time constraints force one to select only a small portion of it in order to remain current. There are many fine books on the subject that have appeared in recent years, with each one of them having “standard material” but also including topics that could be classified as new or against the mainstream. As the subtitle of the book implies, it is all mathematical, with rigorous proofs given for each result, but sometimes requiring participation from the reader through a large collection of exercises (answers to each are given at the end of each chapter). There are some typos as can be expected in such a sizeable book, and also some awkward grammatical constructions, due no doubt to the translation to English.Some of the unique features of this book that distinguish it from others on the subject include:1. The discussion on state discrimination and hypothesis testing. The author is careful to distinguish hypothesis testing, which deals with the case of two hypotheses treated asymmetrically, from state discrimination, where they are treated symmetrically. In this discussion (finite) tensor products of density matrices are viewed (atypically) as a quantum-mechanical analog of an independent, identically distributed probability distribution. For the discrimination of two states, the author uses a two-valued POVM (positive-operator valued measure) {T, 1- T} with T a Hermitian matrix, which is called a ‘test’.Using the trace norm, the author connects this notion of a test with that of a ‘likelihood test’, thus making a connection with that is usually done in ordinary statistical hypothesis testing (when the two density matrices commute the author labels this situation as “classical”). Generalizations to the discrimination of more than two states immediately follows, where (interestingly) he uses optimization theory to reduce this case to solving a very difficult single-variable minimization problem involving one Hermitian matrix. For the case of arbitrary measurements on (finite) tensor products of Hilbert space, a generalization of the (classical) Chernoff bound is given as expected in the context of asymptotic analysis of state discrimination.2. The discussion classical-quantum channel coding, in particular the treatment of superdense coding. Because of its ability to exceed the classical capacity in classical communications, the technological realization of superdense is very important. Readers will obtain a thorough understanding of this topic due to the author’s clarity of formulation and from working some of the exercises. Particularly interesting along these lines is the discussion on “pseudoclassical” channels, which addresses the conditions under which the classical-quantum capacity with individual measurements equals the classical capacity when there is quantum correlations in the measuring apparatus. A reader interested in the philosophy of quantum physics or its foundations may find this discussion provocative/thought-provoking.3. The very lengthy and detailed discussion of entanglement, which should satisfy readers who demand mathematical rigor, those readers who need to know how to determine quantitatively the presence of this important resource, and those readers who want to understand how to interpret quantum entanglement in an information-theoretic environment (which was of course not done when it was first discussed in the 1930’s). As is readily apparent from the author’s discussion, entanglement can be given a sound mathematical foundation, at least for the finite quantum systems that the author considers in this book. Readers who want to know just what happens for quantum systems with an uncountable number of degrees of freedom, such as in the context of quantum field theory, will have to look outside the covers of this book.