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Approximate quantum Markov chains / David Sutter.

Sutter, David, author.
2018
QA274.7
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Title
Approximate quantum Markov chains / David Sutter.
Author
Sutter, David, author.
ISBN
9783319787329 (electronic book)
3319787322 (electronic book)
9783319787312
3319787314
DOI
https://doi.org/10.1007/978-3-319-78732-9
Published
Cham, Switzerland : Springer, [2018]
Copyright
©2018
Language
English
Description
1 online resource.
Item Number
10.1007/978-3-319-78732-9 doi
Call Number
QA274.7
Dewey Decimal Classification
519.2/33
Summary
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.
Bibliography, etc. Note
Includes bibliographical references and index.
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Access limited to authorized users.
Digital File Characteristics
text file PDF
Source of Description
Online resource; title from PDF title page (viewed April 25, 2018).
Series
SpringerBriefs in mathematical physics ; v. 28.
Available in Other Form
Approximate quantum Markov chains.
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Intro; Acknowledgements; Contents; 1 Introduction; 1.1 Classical Markov Chains; 1.1.1 Robustness of Classical Markov Chains; 1.2 Quantum Markov Chains; 1.2.1 Robustness of Quantum Markov Chains; 1.3 Outline; References; 2 Preliminaries; 2.1 Notation; 2.2 Schatten Norms; 2.3 Functions on Hermitian Operators; 2.4 Quantum Channels; 2.5 Entropy Measures; 2.5.1 Fidelity; 2.5.2 Relative Entropy; 2.5.3 Measured Relative Entropy; 2.5.4 Rényi Relative Entropy; 2.6 Background and Further Reading; References; 3 Tools for Non-commuting Operators; 3.1 Pinching; 3.1.1 Spectral Pinching.

3.1.2 Smooth Spectral Pinching3.1.3 Asymptotic Spectral Pinching; 3.2 Complex Interpolation Theory; 3.3 Background and Further Reading; References; 4 Multivariate Trace Inequalities; 4.1 Motivation; 4.2 Multivariate Araki-Lieb-Thirring Inequality; 4.3 Multivariate Golden-Thompson Inequality; 4.4 Multivariate Logarithmic Trace Inequality; 4.5 Background and Further Reading; References; 5 Approximate Quantum Markov Chains; 5.1 Quantum Markov Chains; 5.2 Sufficient Criterion for Approximate Recoverability; 5.2.1 Approximate Markov Chains are not Necessarily Close to Markov Chains.

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