Dénes Petz – författare

Matrices can be studied in different ways. They are a linear algebraic structure and have a topological/analytical aspect (for example, the normed space of matrices) and they also carry an order structure that is induced by positive semidefinite matrices. The interplay of these closely related structures is an essential feature of matrix analysis.This book explains these aspects of matrix analysis from a functional analysis point of view. After an introduction to matrices and functional analysis, it covers more advanced topics such as matrix monotone functions, matrix means, majorization and entropies. Several applications to quantum information are also included.Introduction to Matrix Analysis and Applications is appropriate for an advanced graduate course on matrix analysis, particularly aimed at studying quantum information. It can also be used as a reference for researchers in quantum information, statistics, engineering and economics.

Entropy is a concept which appears in several fields and it is in the center of interest both in mathematical and physical subjects, sometimes even at other places, for example in communication engineering. The birthplace of quantum entropy was in quantum statistical mechanics. Quantum entropy is not a single concept but rather a family of notions which started with the von Neumann entropy of a density matrix and has developed in a number of directions. The heritage of quantum entropies from quantum mechanics is their strong relation to Hilbert space. This book begins with the entropy of a state and through the entropy of coarse graining it proceeds to the quantum dynamical entropy. A bunch of topics shows the direct physical relevance of the entropy methods. The math­ ematical formalism is based on operator theory and mostly the language of operator algebras is used. The use of the concepts and methods of functional analysis not only makes the theory of quantum entropy more uniform and lu­ cid, but it essentially simplifies and extends it. A comprehensive presentation without these elements would not be possible.

This concise and readable book addresses primarily readers with a background in classical statistical physics and introduces quantum mechanical notions as required. Conceived as a primer to bridge the gap between statistical physics and quantum information, a field to which the author has contributed significantly himself, it emphasizes concepts and thorough discussions of the fundamental notions and prepares the reader for deeper studies, not the least through a selection of well chosen exercises.

This concise and readable book addresses primarily readers with a background in classical statistical physics and introduces quantum mechanical notions as required. Conceived as a primer to bridge the gap between statistical physics and quantum information, a field to which the author has contributed significantly himself, it emphasizes concepts and thorough discussions of the fundamental notions and prepares the reader for deeper studies, not the least through a selection of well chosen exercises.