Free Delivery Over $100
|
|
|||
|
||||
OverviewRelative entropy has played a significant role in various fields of mathematics and physics as the quantum version of the Kullback–Leibler divergence in classical theory. Many variations of relative entropy have been introduced so far with applications to quantum information and related subjects. Typical examples are three different classes, called the standard, the maximal, and the measured f-divergences, all of which are defined in terms of (operator) convex functions f on (0,∞) and have respective mathematical and information theoretical backgrounds. The α-Rényi relative entropy and its new version called the sandwiched α-Rényi relative entropy have also been useful in recent developments of quantum information. In the first half of this monograph, the different types of quantum f-divergences and the Rényi-type divergences mentioned above in the general von Neumann algebra setting are presented for study. While quantum information has been developing mostly in the finite-dimensional setting, it is widely believed that von Neumann algebras provide the most suitable framework in studying quantum information and related subjects. Thus, the advance of quantum divergences in von Neumann algebras will be beneficial for further development of quantum information. Quantum divergences are functions of two states (or more generally, two positive linear functionals) on a quantum system and measure the difference between the two states. They are often utilized to address such problems as state discrimination, error correction, and reversibility of quantum operations. In the second half of the monograph, the reversibility/sufficiency theory for quantum operations (quantum channels) between von Neumann algebras via quantum f-divergences is explained, thus extending and strengthening Petz' previous work. For the convenience of the reader, an appendix including concise accounts of von Neumann algebras is provided. Full Product DetailsAuthor: Fumio HiaiPublisher: Springer Verlag, Singapore Imprint: Springer Verlag, Singapore Edition: 1st ed. 2021 Weight: 0.477kg ISBN: 9789813341982ISBN 10: 981334198 Pages: 194 Publication Date: 27 January 2021 Audience: Professional and scholarly , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: Manufactured on demand  We will order this item for you from a manufactured on demand supplier. Table of Contents1 Introduction.- 2 Standard f -Divergences.- 3 Rényi Divergences and Sandwiched Rényi Divergences.- 4 Maximal f -Divergences.- 5 Measured f -Divergences.- 6 Reversibility and Quantum Divergences.- 7 Reversibility and Measurements.- 8 Preservation of Maximal f -Divergences.- A Preliminaries on von Neumann Algebras.- B Preliminaries on Positive Self-Adjoint Operators.- C Operator Convex Functions on (0,1).- D Operator Connections of Normal Positive Functionals.ReviewsThe contents of the book presents very interesting topics for quantum information scientists ... . I would like to recommend this book for two kinds of special groups of scientists: pure mathematicians, who want to learn the power of quantum computing, and quantum information theorists, who are tired of quantum computing. ... this book will be a good guide. For the second group, including people like me, this book stimulates the dormant instincts about quantum information theory. (Kabgyun Jeong, zbMATH 1476.81004, 2022) Author InformationThe author is currently Professor Emeritus at Tohoku University. Tab Content 6Author Website:Countries AvailableAll regions |
||||
