Quantum f-divergences in von Neumann algebras : reversibility of quantum operations / Fumio Hiai.

Format:

Book

Author/Creator:

Hiai, Fumio, 1948- author.

Series:

Mathematical physics studies 0921-3767

Mathematical physics studies, 0921-3767

Language:

English

Subjects (All):

Von Neumann algebras.

Quantum theory--Mathematics.

Quantum theory.

Reverse mathematics.

Functional analysis.

Genre:

Electronic books.

Physical Description:

1 online resource (x, 194 pages) : illustrations.

Place of Publication:

Singapore : Springer, [2021]

System Details:

text file

Summary:

Relative 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.

Contents:

Introduction

Standard f-Divergences

Rényi Divergences and Sandwiched Rényi Divergences

Maximal f-Divergences

Measured f-Divergences

Reversibility and Quantum Divergences

Reversibility and Measurements

Preservation of Maximal f-Divergences

Preliminaries on von Neumann Algebras

Preliminaries on Positive Self-Adjoint Operators

Operator Convex Functions on (0,1)

Operator Connections of Normal Positive Functionals.

Notes:

Includes bibliographical references and index.

Online resource; title from PDF title page (SpringerLink, viewed March 8, 2021).

Other Format:

Print version:

ISBN:

9789813341999

9813341998

OCLC:

1239997950

Access Restriction:

Restricted for use by site license.