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Approximate quantum Markov chains

Sutter, David.

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Approximate quantum Markov chains

Record Type:	Electronic resources : Monograph/item
Title/Author:	Approximate quantum Markov chains/ by David Sutter.
Author:	Sutter, David.
Published:	Cham :Springer International Publishing : : 2018.,
Description:	viii, 118 p. :ill., digital ;24 cm.
[NT 15003449]:	Introduction -- Classical Markov chains -- Quantum Markov chains -- Outline -- Preliminaries -- Notation -- Schatten norms -- Functions on Hermitian operators -- Quantum channels -- Entropy measures -- Background and further reading -- Tools for non-commuting operators -- Pinching -- Complex interpolation theory -- Background and further reading -- Multivariate trace inequalities -- Motivation -- Multivariate Araki-Lieb-Thirring inequality -- Multivariate Golden-Thompson inequality -- Multivariate logarithmic trace inequality -- Background and further reading -- Approximate quantum Markov chains -- Quantum Markov chains -- Sufficient criterion for approximate recoverability -- Necessary criterion for approximate recoverability -- Strengthened entropy inequalities -- Background and further reading -- A A large conditional mutual information does not imply bad recovery -- B Example showing the optimality of the Lmax-term -- C Solutions to exercises -- References -- Index.
Contained By:	Springer eBooks
Subject:	Markov processes. -
Online resource:	http://dx.doi.org/10.1007/978-3-319-78732-9
ISBN:	9783319787329

Approximate quantum Markov chains
Sutter, David.

Approximate quantum Markov chains[electronic resource] /by David Sutter. - Cham :Springer International Publishing :2018. - viii, 118 p. :ill., digital ;24 cm. - SpringerBriefs in mathematical physics,v.282197-1757 ;. - SpringerBriefs in mathematical physics ;v.28..

Introduction -- Classical Markov chains -- Quantum Markov chains -- Outline -- Preliminaries -- Notation -- Schatten norms -- Functions on Hermitian operators -- Quantum channels -- Entropy measures -- Background and further reading -- Tools for non-commuting operators -- Pinching -- Complex interpolation theory -- Background and further reading -- Multivariate trace inequalities -- Motivation -- Multivariate Araki-Lieb-Thirring inequality -- Multivariate Golden-Thompson inequality -- Multivariate logarithmic trace inequality -- Background and further reading -- Approximate quantum Markov chains -- Quantum Markov chains -- Sufficient criterion for approximate recoverability -- Necessary criterion for approximate recoverability -- Strengthened entropy inequalities -- Background and further reading -- A A large conditional mutual information does not imply bad recovery -- B Example showing the optimality of the Lmax-term -- C Solutions to exercises -- References -- Index.

This book is an introduction to quantum Markov chains and explains how this concept is connected to the question of how well a lost quantum mechanical system can be recovered from a correlated subsystem. To achieve this goal, we strengthen the data-processing inequality such that it reveals a statement about the reconstruction of lost information. The main difficulty in order to understand the behavior of quantum Markov chains arises from the fact that quantum mechanical operators do not commute in general. As a result we start by explaining two techniques of how to deal with non-commuting matrices: the spectral pinching method and complex interpolation theory. Once the reader is familiar with these techniques a novel inequality is presented that extends the celebrated Golden-Thompson inequality to arbitrarily many matrices. This inequality is the key ingredient in understanding approximate quantum Markov chains and it answers a question from matrix analysis that was open since 1973, i.e., if Lieb's triple matrix inequality can be extended to more than three matrices. Finally, we carefully discuss the properties of approximate quantum Markov chains and their implications. The book is aimed to graduate students who want to learn about approximate quantum Markov chains as well as more experienced scientists who want to enter this field. Mathematical majority is necessary, but no prior knowledge of quantum mechanics is required.

ISBN: 9783319787329

Standard No.: 10.1007/978-3-319-78732-9doiSubjects--Topical Terms:

532104
Markov processes.

LC Class. No.: QA274.7

Dewey Class. No.: 519.233

Approximate quantum Markov chains
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100 1 $a Sutter, David. $3 3321833
245 1 0 $a Approximate quantum Markov chains $h [electronic resource] / $c by David Sutter.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2018.
300 $a viii, 118 p. : $b ill., digital ; $c 24 cm.
490 1 $a SpringerBriefs in mathematical physics, $x 2197-1757 ; $v v.28
505 0 $a Introduction -- Classical Markov chains -- Quantum Markov chains -- Outline -- Preliminaries -- Notation -- Schatten norms -- Functions on Hermitian operators -- Quantum channels -- Entropy measures -- Background and further reading -- Tools for non-commuting operators -- Pinching -- Complex interpolation theory -- Background and further reading -- Multivariate trace inequalities -- Motivation -- Multivariate Araki-Lieb-Thirring inequality -- Multivariate Golden-Thompson inequality -- Multivariate logarithmic trace inequality -- Background and further reading -- Approximate quantum Markov chains -- Quantum Markov chains -- Sufficient criterion for approximate recoverability -- Necessary criterion for approximate recoverability -- Strengthened entropy inequalities -- Background and further reading -- A A large conditional mutual information does not imply bad recovery -- B Example showing the optimality of the Lmax-term -- C Solutions to exercises -- References -- Index.
520 $a This book is an introduction to quantum Markov chains and explains how this concept is connected to the question of how well a lost quantum mechanical system can be recovered from a correlated subsystem. To achieve this goal, we strengthen the data-processing inequality such that it reveals a statement about the reconstruction of lost information. The main difficulty in order to understand the behavior of quantum Markov chains arises from the fact that quantum mechanical operators do not commute in general. As a result we start by explaining two techniques of how to deal with non-commuting matrices: the spectral pinching method and complex interpolation theory. Once the reader is familiar with these techniques a novel inequality is presented that extends the celebrated Golden-Thompson inequality to arbitrarily many matrices. This inequality is the key ingredient in understanding approximate quantum Markov chains and it answers a question from matrix analysis that was open since 1973, i.e., if Lieb's triple matrix inequality can be extended to more than three matrices. Finally, we carefully discuss the properties of approximate quantum Markov chains and their implications. The book is aimed to graduate students who want to learn about approximate quantum Markov chains as well as more experienced scientists who want to enter this field. Mathematical majority is necessary, but no prior knowledge of quantum mechanics is required.
650 0 $a Markov processes. $3 532104
650 1 4 $a Physics. $3 516296
650 2 4 $a Quantum Physics. $3 893952
650 2 4 $a Mathematical Physics. $3 1542352
650 2 4 $a Condensed Matter Physics. $3 1067080
650 2 4 $a Statistical Physics and Dynamical Systems. $3 3135115
650 2 4 $a Quantum Information Technology, Spintronics. $3 1067172
710 2 $a SpringerLink (Online service) $3 836513
773 0 $t Springer eBooks
830 0 $a SpringerBriefs in mathematical physics ; $v v.28. $3 3321834
856 4 0 $u http://dx.doi.org/10.1007/978-3-319-78732-9
950 $a Physics and Astronomy (Springer-11651)