From Classical to Quantum Shannon Information Theory

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The aim of this book is to develop from the ground up" all of the major exciting preand post millenium developments in the general area of study known as quantum Shannon theory As such we spend a signicant amount of time on quantum mechanics for quantum information theory Part II we give a careful study of the important unit protocols of teleportation super dense coding and entanglement distribution Part III and we develop many of the tools necessary for understanding information transmission or compression Part IV Parts V and VI are the culmination of this book where all of the tools developed come into play for understanding many of the important results in quantum Shannon theory">The aim of this book is to develop from the ground up" all of the major exciting preand post millenium developments in the general area of study known as quantum Shannon theory As such we spend a signicant amount of time on quantum mechanics for quantum information theory Part II we giv [更多]
Contents I Introduction 1 Concepts in Quantum Shannon Theory 1.1 Overview of the Quantum Theory 1.2The Emergence of Quantum Shannon Theory O 2 Classical Shannon Theory 39 2.1 Data Compression 3 9 2.2 Channel Capacity 2.3 Summary I The Quantum Theory 65 3 The Noiseless Quantum Theory 3.1 Overview 68 3.2 Quantum Bits 8 3.3 Reversible evolution 3.4 Measurement 3.5 Composite Quantum Systems 3.6 Summary and Extensions to Qudit States 3. 7 History and further reading 106 4 The Noisy Quantum Theory 7 4.1 Noisy Quantum States 4.2 Measurement in the Noisy Quantum Theory 4.3 Composite Noisy Quantum Systems ..[21 4.4 Noisy Evolution 130 4.5 Summary I47 4.6 History and Further reading 148 CONTENTS 5 The Purified Quantun Theory 互49 5.1 Purification 5 5.2 Isometric evolution 5 5.3 Coherent Quantum Instrument 161 5.4 Coherent measurement 162 5.5 History and Further reading III Unit Quantum Protocols 165 6 Three Unit Quantum Protocols I67 6.1 Nonlocal unit resources 68 6.2 Protocols 170 6. 3 Optimality of the Three Unit PI roLOColS 6.4 Extensions for Quantum Shannon Theory I81 6.5 Three Unit Qudit Protocols 6.6 History and further reading I88 7 Co t protocols 89 7. 1 Definition of Coherent Communication 190 7.2 Implementations of a Coherent bit Channel 192 7.3 Coherent Dense Coding 193 7.4 Coherent teleportation ∏95 7.5 The Coherent Communication Identity 197 7.6 History and Further Reading 198 8 The Unit Resource Capacity Region 199 8.1 The Unit Resource Achievable region 8.2 The Direct Coding Theorem 8. 3 The Converse Theorem 204 8.4 History and Further reading 208 Iv Tools of Quantum Shannon Theory 209 9 Distance measures 9.1 Trace Distance 2L2 9.2 Fidelity 220 9.3 Relationships bet ween Trace Distance and Fidelity p27 9.4 Gentle Measurement 231 9.5 Fidelity of a Noisy Quantum Channel ...[34 9. 6 The hilbert-Schmidt Distance Measure 237 9. 7 History and further reading 238 92011 Mark M. Wilde This work is licensed under a Creative Commons Attribution-Non Commercial-ShareAlike 3.0 Unported License CONTENTS 10 Classical Infornation and Entropy 239 10.1 Entropy of a random variable 24O 10.2 Conditional Entropy 244 10.3 Joint Entropy 25 10.4 Mutual Information 246 10.5 Relative Entropy 247 10.6 Conditional Mutual Information 248 10.7 Information Inequalities 249 10.8 Classical Information and Entropy of Quantum Systems D55 10.9 History and Further Reading P57 ll Quantum Information and Entropy 259 11.1 Quantum Entropy 260 11.2 Joint Quantun Entropy 26 11.3 Potential yet Unsatisfactory Definitions of Conditional Quantum Entropy..267 11.4 Conditional Quantum Entropy 268 11.5 Coherent Information ②7山 11.6 Quantum Mutual Information 273 11.7 Conditional Quantum Mutual Infornation 27 11.8 Quantum Relative Entrop 11.9 Quantum Information Inequalities 280 11.10History and Further Reading 29)1 12 The Information of Quantum Channels 293 12.1 Mutual information of a classical channel 12.2 Private Information of a Wiretap Channel 294 1300l 12.3 Holevo Infornation of a Quantum Channel B03 12.4 Mutual Information of a Quantum Channel B08 12.5 Coherent Information of a Quantum Channel 33 12.6 Private Information of a Quantum Channel 12.7 Summary B24 12.8 History and Further Reading 325 13 Classical Typicality 327 13.1 An Example of Typicality B28 13.2 Weak Typicality B29 13.3 Properties of the Typical Set 331 13.4 Application of Typical Sequences: Shannon Compression B33 13.5 Weak Joint Typicality 33 13.6 Weak Conditional Typicality B38 13.7 Strong Typicality ·· B40 13. 8 Strong Joint Typicality B49 13.9 Strong Conditional Typicality 350 92011 Mark M. Wilde This work is licensed under a Creative Commons Attribution-Non Commercial-ShareAlike 3.0 Unported License CONTENTS 13.10 Concluding Remarks B56 13. 11 History and Further reading B57 14 Quantun Typicality 359 14.1 The Typical Subspace 360 14.2 Conditional Quantum Typicalit 369 14.3 The Method of Types for Quantum Systems B78 14.4 Concluding remarks 1380 14.5 History and Further reading 381 15 The Packing Lemma 383 15.1 Introductory Example 382 15. 2 The Setting of the Packing Lemma 15.3 Statenent of the Packing lemna B86 15.4 Proof of the Packing Lemma 88 15.5 Derandomization and Expurgation B93 15.6 History and Further reading 395 16 The Covering lemma 397 16.1 Introductory Example 16.2 Setting and Statement of the Covering Lemma B98 16.3 Proof of the covering lemma 02 16.4 History and Further Reading v Noiseless Quantum Shannon Theory 409 17 Schumacher Compression 啁11 17.1 The inforrnation processing. task 4T2 17.2 The Quantum Data Compression Theorem 414 17.3 Quantum Compression Example 17.4 Variations on the Schumacher Theme 419 17.5 Concluding remarks 420 17.6 History and Further Reading 18 Entanglement Concentration 23 18.1 An Example of Entanglement Concentration A24 18.2 The information Processing task 427 18. 3 The Entanglement Concentration Theorem A27 18.4 Common randomness Concentration 18.5 Schumacher Compression versus Entanglement Concentration A35 18.6 Concluding remarks 438 18.7 History and Further reading 438 92011 Mark M. Wilde This work is licensed under a Creative Commons Attribution-Non Commercial-ShareAlike 3.0 Unported License CONTENTS 7 Vi Noisy Quantum Shannon Theory 1 9 Classical Communication 445 19.1 Naive Approach: Product Measurernents at the Decoder 447 19.2 The Information processing task A50 19.3 The Classical Capacity Theorem .A52 19.4 Examples of channels ...④57 19.5 Superadditivity of the Holevo Information 19.6 Concluding Remarks 图68 19.7 History and Further Reading 69 20 Entanglement-Assisted Classical Communication 啁71 20. 1 The Information Processing task 47 20.2 A Preliminary Example /4 20.3 The Entanglement-Assisted Classical Capacity Thcorem ④78 20.4 The direct coding theorem 20.5 The Converse Theorem 87 20.6 Examples of Channels 495 20.7 Concluding remarks 5)00 20.8 History and Further read 21 Coherent Communication with Noisy Resources 503 21.1 Entanglement-Assisted Quantum Communication 504 21.2 Quantum Communication 21.3 Noisy Super-Dense Coding 21.4 State transfer 21.5 Trade-off Coding 516 21.6 Concluding remarks 525 21.7 History and Further reading 525 22 Private Classical Communication 27 22. 1 The Information Processing task b28 22.2 The Private Classical Capacity Theorem 30 22.3 The Direct Coding Theorem b3工 22.4 The Converse Theorem 5 22.5 Discussion of private Classical Capacity 22.6 History and Further reading 23 Quant,urn Communication 54 23.1 The Information Processing task 546 23.2 The No-cloning noren an d Quantum Communication D48 23.3 The Quantum Capacity Theorem b48 23.4 The Direct Coding Theorem 549 92011 Mark M. Wilde This work is licensed under a Creative Commons Attribution-Non Commercial-ShareAlike 3.0 Unported License CONTENTS 23.5 Converse theorem 556 23.6 Example Channels .⑤58 23.7 Discussion of Quantum Capacity 562 23.8 Entanglement Distillation 567 23.9 History and Further Reading 24 Trading resources for Communication 573 24.1 The Information processing task 574 24.2 The Quantum Dynamic Capacity Th horen ⑤76 24.3 The direct coding theorem 1581 24.4 The Converse Theorem 583 24.5 Examples of Channels b94 24.6 History and Further reading 604 25 Summary and outlook 25.1 Unit Protocols 68 25.2 Noiseless Quantum Shannon Theory 609 25.3 Noisy Quantum Shannon Theory 609 25.4 Protocols not covered in this book ⑥612 25.5 Network Quantum Shannon Theory 613 25.6 Future directions ⑥B3 a Miscellaneous mathematics 65 A.1 The Operator Chernoff Bound 617 B Monotonicity of Quantum Relative Entropy 623 92011 Mark M. Wilde This work is licensed under a Creative Commons Attribution-Non Commercial-ShareAlike 3.0 Unported License How to use this book For students Prerequisites for understanding the content in this book are a solid background in probability theory and linear algebra. If you are new to information theory, then there is enough background in this book to get you up to speed( Chapters 2 10 12 and 13 Thougn, lassics on information theory such Cover and Thomas 53 and Mackay 179 could be helpful as a reference. If you are new to quantum mechanics, then there should be enough material in this book(Part ii)to give the background necessary for understanding quantum hannon theory. The book of Nielsen and Chuang(sometimes known as "Mike and Ike) has become the standard starting point for students in quantum information science and night be helpful as well 187. Some of the content in that book is available in Nielsens dissertation [184. If you are familiar with Shannon's information theory (at the level of over and t nomas 53, for example), then this book should be a helpful entry point into the field of quantum Shannon theory. We build on intuition developed classically to help in determining schemes for communication over quantum channels. If you are familiar with quantum mechanics, it might still be worthwhile to review Part Ii because some content there might not be part of a standard course on quantum mechanics The aim of this book is to develop "from the ground up"all of the major, exciting, pre and post-millenium developments in the general area of study known as quantum Shannon theory. As such, we spend a significant amount of time on quantum mechanics for quan- tum information theory(Part II), we give a careful study of the important unit protocols of teleportation, super-dense coding, and entanglement distribution(Part III), and we de velop many of the tools necessary for understanding information transmission or compression Part IV). Parts V and VI are the culmination of this book, where all of the tools developed come into play for understanding many of the important results in quantum Shannon theory For instructors This book could be useful for self-learning or as a, reference, but one of the main goals is for it to be employed as an instructional aid for the classroom. To aid instructors in designing a course to suit their own needs. this book is available under a creative Commons Attribution NonCommercial-ShareAlike license. This means that you can modify and redistribute the CONTENTS book as you wish, as long you attribute the author of this book and as long as you do not use itforcommericalpurposes(seehttp://creativecommons.org/licenses/by-nc-sa/3.0/ for a readable summary of the terms of the license). These requirements can be waived if you obtain permission from the present author. By releasing the book under this license, I expect and encourage instructors to modify this book for their own needs. This will allow for the addition of new exercises, new developments in the theory, and the latest open problems. It might also be a helpful starting point for a book on a related topic, such as network quantum Shannon theory I used an earlier version of this book in a one-semester course on quantum Shannon theory at Mc Gill University during Winter semester 2011(in many parts of the US, this semester is typically called " Spring semester"). We almost went through the entire book, but it might also be possible to spread the content over two semesters instead. here is the order in which we proceeded 1. Introduction in Part i 2. Quantum mechanics in Part II 3. Unit protocols in Part III 4. Chapterg on distance measures, Chapter[LO on classical information and entropy,and Chapter[1lon quantum information and entropy 5. The first part of Chapter[ on classical typicality and Shannon compression 6. The first part of Chapter 14 on quantum typicality 7. ChapterI7on Schumacher compression 8. Back to Chapters 13 and 14 for the method of types 9. Chapte er18 on entanglement concentration 10. Chapter 19 on classical communication 11. Chapter on entanglement-assisted classical communication 12. The final explosion of results in Chapter 21(one of which is a route to proving the achievability part of the quantum capacity theorem) The above order is just a particular order that suited the needs for the class at Mcgi but other orders are of course possible. One could sacrifice the last part of Part Iii on the unit resource capacity region if there is no desire to cover the quantum dynamic capacity theorem. One could also focus on going from classical communication to private classical communication to quantum communication in order to develop some more intuition behind the quantum capacity theorem 92011 Mark M. Wilde This work is licensed under a Creative Commons Attribution-Non Commercial-ShareAlike 3.0 Unported License

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lq52las 非常好的资源
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xieros 很清晰,很好的教材,谢谢
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