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Quantum f-Divergences in von Neumann Algebras: Reversibility of Quantum Operations
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Librairie Eyrolles - Paris 5e
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Quantum f-Divergences in von Neumann Algebras: Reversibility of Quantum Operations

Quantum f-Divergences in von Neumann Algebras: Reversibility of Quantum Operations

Fumio Hiai

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194 pages, parution le 26/01/2021
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Retrait à la librairie - Paris 5e

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Résumé

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 -Renyi relative entropy and its new version called the sandwiched -Renyi 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 Renyi-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.1 Introduction.- 2 Standard f -Divergences.- 3 Renyi Divergences and Sandwiched Renyi 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.
The author is currently Professor Emeritus at Tohoku University.

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  PAPIER
Éditeur(s) Apress
Auteur(s) Fumio Hiai
Parution 26/01/2021
Nb. de pages 194
EAN13 9789813341982

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