Information theory : Poincare Seminar 2018 / Bertrand Duplantier, Vincent Rivasseau, editors.

Format:

Book

Contributor:

Duplantier, Bertrand, editor.

Rivasseau, Vincent, editor.

Series:

Progress in mathematical physics ; 78.

Progress in mathematical physics ; 78

Language:

English

Subjects (All):

Information theory--Congresses.

Information theory.

Physical Description:

1 online resource (222 pages)

Place of Publication:

Cham, Switzerland : Birkhäuser, [2021]

Summary:

This eighteenth volume in the Poincaré Seminar Series provides a thorough description of Information Theory and some of its most active areas, in particular, its relation to thermodynamics at the nanoscale and the Maxwell Demon, and the emergence of quantum computation and of its counterpart, quantum verification. It also includes two introductory tutorials, one on the fundamental relation between thermodynamics and information theory, and a primer on Shannon's entropy and information theory. The book offers a unique and manifold perspective on recent mathematical and physical developments in this field.

Contents:

Intro

Contents

Foreword

Thermodynamics and Information Theory

1. Introduction

2. Thermodynamics: A Brief Review

2.1. The Two Principles of Thermodynamics

2.2. Molecular Theory of Heat and the Framework of Statistical Mechanics

2.3. Brownian Motion: Equilibrium Is Dynamical

2.4. Universality of Brownian Motion: Feynman's Ratchet and Pawl

2.4.1. Application to molecular motors

3. Equilibrium and Non-equilibrium Dynamics

3.1. Markovian Dynamics

3.2. Connection to Thermodynamics

3.3. Time-Reversal Invariance and Detailed Balance

3.4. Physical Iinterpretation of Detailed Balance

3.5. Entropy Production in Markovian Systems

4. The Gallavotti{Cohen Fluctuation Theorem for Markovian Thermodynamics

4.1. Generalised Detailed Balance

4.2. Time Reversal and the Gallavotti{Cohen Symmetry

5. Non-equilibrium Work Identities

5.1. Jarzynski's Work Theorem

5.2. Crooks' Relation

6. Information Theory

7. Thermodynamics and Information: The Maxwell Demon

8. Conclusion

Appendix A. Large Deviations and Cumulant Generating Functions

Appendix B. Proof of the Jarzynski Formula for Hamiltonian Dynamics

Acknowledgments

References

This is IT: A Primer on Shannon's Entropy and Information

1. Shannon's Life as a Child

2. A Noble Prize Laureate

3. Intelligence or Information?

4. Probabilistic, not Semantic

5. The Celebrated 1948 Paper

6. Shannon, not Weaver

7. Shannon, not Wiener

8. Shannon's Bandwagon

9. An Axiomatic Approach to Entropy

10. Units of Information

11. H or ^Eta?

12. No One Knows What Entropy Really Is

13. How Does Entropy Arise Naturally?

14. Shannon's Source Coding Theorem

15. Continuous Entropy

16. Change of Variable in the Entropy

17. Discrete vs. Continuous Entropy

18. Most Beautiful Equation

19. Entropy Power.

20. A Fundamental Information Inequality

21. The MaxEnt Principle

22. Relative Entropy or Divergence

23. Generalized Entropies and Divergences

24. How Does Relative Entropy Arise Naturally?

25. Cherno Information

26. Fisher Information

27. Kolmogorov Information

28. Shannon's Mutual Information

29. Conditional Entropy or Equivocation

30. Knowledge Reduces Uncertainty - Mixing Increases Entropy

31. A Suggestive Venn Diagram

32. Shannon's Channel Coding Theorem

33. Shannon's Capacity Formula

34. The Entropy Power Inequality and a Saddle Point Property

35. MaxEnt vs. MinEnt Principles

36. A Simple Proof of the Entropy Power Inequality

37. Conclusion

Landauer's Bound and Maxwell's Demon

1.1. Maxwell's Demon and Szilard's Engine

1.2. Landauer's Principle and Bennett's Resolution

2. Experimental Implementations

2.1. Experiments on Maxwell's Demon

2.1.1. The Szilard engine: work production from information

2.1.2. The autonomous Maxwell demon improves cooling

2.2. Experiments on Landauer's Principle

2.3. Other Experiments on the Physics of Information

3. Extensions to the Quantum Regime

3.1. Experiments on Quantum Maxwell's Demon

3.2. Experiments on Quantum Landauer's Principle

4. Applications

Appendix A. Stochastic Thermodynamics and Information Energy Cost

A.1. Estimate the Free Energy Difference from Work Fluctuations

A.2. Landauer Bound and the Jarzynski Equality

A.2.1. Experimental test of the generalized Jarzynski equality

Appendix B. Set-up Used in the Experiment Presented in Section 2.2

B.1. The One-Bit Memory System

B.2. Heat Measurements

Verification of Quantum Computation: An Overview of Existing Approaches

1. Introduction.

1.1. Blind Quantum Computing

1.1.1. Quantum one-time pad

1.1.2. Childs' protocol for blind computation

1.1.3. Universal Blind Quantum Computation (UBQC)

2. Prepare-and-Send Protocols

2.1. Quantum Authentication-Based Veri cation

2.1.1. Clifford-QAS VQC

2.1.2. Poly-QAS VQC

2.2. Trap-Based Verification

2.3. Veri cation Based on Repeated Runs

2.4. Summary of Prepare-and-Send Protocols

3. Receive-and-Measure Protocols

3.1. Measurement-only Verification

3.2. Post-hoc Verification

3.3. Summary of receive-and-measure protocols

4. Entanglement-based Protocols

4.1. Verification Based on CHSH Rigidity

4.1.1. RUV protocol

4.1.2. GKW protocol

4.1.3. HPDF protocol

4.2. Verification Based on Self-testing Graph States

4.3. Post-hoc Verifi cation

4.3.1. FH protocol

4.3.2. NV protocol

4.4. Summary of Entanglement-based Protocols

5. Outlook

5.1. Sub-universal Protocols

5.2. Fault Tolerance

5.3. Experiments and Implementations

6. Conclusions

7. Appendix

7.1. Quantum Information and Computation

7.1.1. Basics of quantum mechanics

7.1.2. Density matrices

7.1.3. Puri cation

7.1.4. CPTP maps

7.1.5. Trace distance

7.1.6. Quantum computation

7.1.7. Bloch sphere

7.1.8. Quantum error correction

7.2. Measurement-based Quantum Computation

7.3. Complexity Theory

References.

Notes:

Description based on print version record.

ISBN:

3-030-81480-7