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E-grāmata: Approximate Quantum Markov Chains

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Approximate Quantum Markov ChainsPalielināt
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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.


Recenzijas

This book is mainly written in a noncommutative framework, and focuses on the robustness of the quantum Markov property under approximations. The monograph is clearly written and can serve as a useful introduction to the Markov property in a noncommutative setting. (Nicolas Privault, zbMATH 1407.81002, 2019)

1 Introduction
1(10)
1.1 Classical Markov Chains
2(2)
1.1.1 Robustness of Classical Markov Chains
3(1)
1.2 Quantum Markov Chains
4(2)
1.2.1 Robustness of Quantum Markov Chains
5(1)
1.3 Outline
6(5)
References
9(2)
2 Preliminaries
11(34)
2.1 Notation
11(2)
2.2 Schatten Norms
13(4)
2.3 Functions on Hermitian Operators
17(4)
2.4 Quantum Channels
21(3)
2.5 Entropy Measures
24(16)
2.5.1 Fidelity
25(3)
2.5.2 Relative Entropy
28(3)
2.5.3 Measured Relative Entropy
31(7)
2.5.4 Renyi Relative Entropy
38(2)
2.6 Background and Further Reading
40(5)
References
40(5)
3 Tools for Non-commuting Operators
45(16)
3.1 Pinching
46(8)
3.1.1 Spectral Pinching
46(4)
3.1.2 Smooth Spectral Pinching
50(2)
3.1.3 Asymptotic Spectral Pinching
52(2)
3.2 Complex Interpolation Theory
54(4)
3.3 Background and Further Reading
58(3)
References
59(2)
4 Multivariate Trace Inequalities
61(14)
4.1 Motivation
61(4)
4.2 Multivariate Araki-Lieb-Thirring Inequality
65(1)
4.3 Multivariate Golden-Thompson Inequality
66(4)
4.4 Multivariate Logarithmic Trace Inequality
70(2)
4.5 Background and Further Reading
72(3)
References
72(3)
5 Approximate Quantum Markov Chains
75(26)
5.1 Quantum Markov Chains
75(4)
5.2 Sufficient Criterion for Approximate Recoverability
79(3)
5.2.1 Approximate Markov Chains are not Necessarily Close to Markov Chains
81(1)
5.3 Necessary Criterion for Approximate Recoverability
82(7)
5.3.1 Tightness of the Necessary Criterion
86(3)
5.4 Strengthened Entropy Inequalities
89(7)
5.4.1 Data Processing Inequality
89(5)
5.4.2 Concavity of Conditional Entropy
94(1)
5.4.3 Joint Convexity of Relative Entropy
95(1)
5.5 Background and Further Reading
96(5)
References
97(4)
Appendix A A Large Conditional Mutual Information Does Not Imply Bad Recovery 101(4)
Appendix B Example Showing the Optimality of the Amax-Term 105(4)
Appendix C Solutions to Exercises 109(8)
Index 117
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