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Quantum entropy and its use. (English) Zbl 0891.94008
Texts and Monographs in Physics. Berlin: Springer. viii, 335 p. (1993).
Von Neumann defined quantum entropy by the expression \(\text{tr}(-D\ln D)\) where \(D\) is a density matrix; and the present book is a monograph on the development and the application of the corresponding theory. But, here, the authors start from relative quantum entropy or quantum Kullback entropy as the basic concept. It represents an applied mathematics book involving operator theory and functional analysis that is divided in 5 parts.
Part 1 (3 chapters) is an elementary introduction to quantum entropy (basic concepts and axioms), whilst Part 2 (4 chapters) is exactly Part 1 revisited in a more theoretical framework. In Part 3 (4 chapters), the authors analyze some problems related to channels, and they define the parallel of the so-called Kolmogorov entropy of dynamical systems. Part 4 (2 chapters) develops a theory of perturbed states, what is of paramount importance in quantum statistical mechanics in connection with equilibrium states and their stability. Part 5 (5 chapters) describes some examples: mainly, a central limit theorem, thermodynamics of quantum spin systems, entropic uncertainty relations.
Despite it is a theoretical book, it can be considered as a good state-of-art of the topic. For a physicist-like reader, there are no simple (even artificial) illustrative examples which would be of help to the unexperienced. The list of references on quantum entropy is incomplete. There is no news about the maximum entropy principle applied to quantum entropy. The reviewer does not see the relation between quantum entropy and quantum central limit theorem. There is one for the central limit theorem involving random variables and Shannon entropy of random variables, but the corresponding reference [Yu. V. Linnik, Theor. Probab. Appl. 4, 288–299 (1960); translation from Teor. Veroyatn. Primen. 4, 311–321 (1959; Zbl 0097.13103)] is not mentioned.

94A17 Measures of information, entropy
94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory