Title Details

Preface to the New Edition xvii

Preface to Lectures on Quantum Information (2006) xix

Part I Classical Information Theory 1

1 Classical Information Theory and Classical Error Correction 3
Markus Grassl

1.1 Introduction 3

1.2 Basics of Classical Information Theory 3

1.3 Linear Block Codes 10

1.4 Further Aspects 16

References 16

2 Computational Complexity 19
Stephan Mertens

2.1 Basics 19

2.2 Algorithms and Time Complexity 21

2.3 Tractable Trails: The Class P 22

2.4 Intractable Itineraries: The Class NP 24

2.5 Reductions and NP-Completeness 29

2.6 P Versus NP 31

2.7 Optimization 34

2.8 Complexity Zoo 37

References 37

Part II Foundations of Quantum Information Theory 39

3 Discrete Quantum States versus Continuous Variables 41
Jens Eisert

3.1 Introduction 41

3.2 Finite-Dimensional Quantum Systems 42

3.3 Continuous-Variables 45

References 53

4 Approximate Quantum Cloning 55
Dagmar Bruß and Chiara Macchiavello

4.1 Introduction 55

4.2 The No-Cloning Theorem 56

4.3 State-Dependent Cloning 57

4.4 Phase-Covariant Cloning 63

4.5 Universal Cloning 65

4.6 Asymmetric Cloning 69

4.7 Probabilistic Cloning 70

4.8 Experimental Quantum Cloning 70

4.9 Summary and Outlook 71

Exercises 72

References 73

5 Channels and Maps 75
M. Keyl and R. F.Werner

5.1 Introduction 75

5.2 Completely Positive Maps 75

5.3 The Choi–Jamiolkowski Isomorphism 78

5.4 The Stinespring Dilation Theorem 80

5.5 Classical Systems as a Special Case 83

5.6 Channels with Memory 84

5.7 Examples 86

Problems 89

References 90

6 Quantum Algorithms 91
Julia Kempe

6.1 Introduction 91

6.2 Precursors 93

6.3 Shor’s Factoring Algorithm 97

6.4 Grover’s Algorithm 100

6.5 Other Algorithms 101

6.6 Recent Developments 103

Exercises 105

References 106

7 Quantum Error Correction 111
Markus Grassl

7.1 Introduction 111

7.2 Quantum Channels 111

7.3 Using Classical Error-Correcting Codes 115

7.4 Further Aspects 124

References 124

Part III Theory of Entanglement 127

8 The Separability versus Entanglement Problem 129
Sreetama Das, Titas Chanda,Maciej Lewenstein, Anna Sanpera, Aditi Sen De, and Ujjwal Sen

8.1 Introduction 129

8.2 Bipartite Pure States: Schmidt Decomposition 130

8.3 Bipartite Mixed States: Separable and Entangled States 131

8.4 Operational Entanglement Criteria 132

8.5 Non-operational Entanglement Criteria 141

8.5.1 Technical Preface 141

8.6 Bell Inequalities 149

8.7 Quantification of Entanglement 152

8.8 Classification of Bipartite States with Respect to Quantum Dense Coding 158

8.9 Multipartite States 162

Exercises 167

Acknowledgments 168

References 169

9 Quantum Discord and Nonclassical Correlations Beyond Entanglement 175
Gerardo Adesso, Marco Cianciaruso, and Thomas R. Bromley

9.1 Introduction 175

9.2 Quantumness Versus Classicality (of Correlations) 176

9.3 Quantifying Quantum Correlations – Quantum Discord 180

9.4 Interpreting Quantum Correlations – Local Broadcasting 184

9.5 Alternative Characterizations of Quantum Correlations 186

9.6 General Desiderata for Measures of Quantum Correlations 190

9.7 Outlook 191

Exercises 191

References 192

10 Entanglement Theory with Continuous Variables 195
Peter van Loock and Evgeny Shchukin

10.1 Introduction 195

10.2 Phase-Space Description 197

10.3 Entanglement of Gaussian States 197

10.4 More on Gaussian Entanglement 209

Exercises 211

References 212

11 Entanglement Measures 215
Martin B. Plenio and Shashank S. Virmani

11.1 Introduction 215

11.2 Manipulation of Single Systems 217

11.3 Manipulation in the Asymptotic Limit 218

11.4 Postulates for Axiomatic Entanglement Measures: Uniqueness and Extremality Theorems 221

11.5 Examples of Axiomatic Entanglement Measures 224

Acknowledgments 228

References 228

12 Purification and Distillation 231
Wolfgang Dür and Hans-J. Briegel

12.1 Introduction 231

12.2 Pure States 233

12.3 Distillability and Bound Entanglement in Bipartite Systems 235

12.4 Bipartite Entanglement Distillation Protocols 239

12.5 Distillability and Bound Entanglement in Multipartite Systems 247

12.6 Entanglement Purification Protocols in Multipartite Systems 248

12.7 Distillability with Noisy Apparatus 252

12.8 Applications of Entanglement Purification 257

12.9 Summary and Conclusions 260

Acknowledgments 261

References 261

13 Bound Entanglement 265
Paweł Horodecki

13.1 Introduction 265

13.2 Distillation of Quantum Entanglement: Repetition 265

13.3 Bound Entanglement – Bipartite Case 269

13.4 Bound Entanglement: Multipartite Case 282

13.5 Further Reading: Continuous Variables 287

Exercises 287

References 288

14 Multipartite Entanglement 293
Michael Walter, David Gross, and Jens Eisert

14.1 Introduction 293

14.2 General Theory 294

14.3 Important Classes of Multipartite states 310

14.4 Specialized Topics 316

Acknowledgments 321

References 321

Part IV Quantum Communication 331

15 Quantum Teleportation 333
Natalia Korolkova

15.1 Introduction 333

15.2 Quantum Teleportation Protocol 334

15.3 Implementations 340

References 349

16 Theory of Quantum Key Distribution (QKD) 353
Norbert Lütkenhaus

16.1 Introduction 353

16.2 Classical Background to QKD 353

16.3 Ideal QKD 354

16.4 Idealized QKD in Noisy Environment 357

16.5 Realistic QKD in Noisy and Lossy Environment 360

16.6 Improved Schemes 363

16.7 Improvements in Public Discussion 364

16.8 Conclusion 365

References 365

17 Quantum Communication Experiments with Discrete Variables 369
Harald Weinfurter

17.1 Aunt Martha 369

17.2 Quantum Cryptography 369

17.3 Entanglement-Based Quantum Communication 375

17.4 Conclusion 379

References 379

18 Continuous Variable Quantum Communication with Gaussian States 383
Ulrik L. Andersen and Gerd Leuchs

18.1 Introduction 383

18.2 Continuous-Variable Quantum Systems 384

18.3 Tools for State Manipulation 386

18.4 Quantum Communication Protocols 391

Exercises 397

References 397

Part V Quantum Computing: Concepts 401

19 Requirements for a Quantum Computer 403
Artur Ekert and Alastair Kay

19.1 Classical World of Bits and Probabilities 403

19.2 Logically Impossible Operations? 408

19.3 Quantum World of Probability Amplitudes 410

19.4 Interference Revisited 414

19.5 Tools of the Trade 416

19.6 Composite Systems 422

19.7 Quantum Circuits 428

19.8 Summary 433

Exercises 433

20 Probabilistic Quantum Computation and Linear Optical Realizations 437
Norbert Lütkenhaus

20.1 Introduction 437

20.2 Gottesman/Chuang Trick 438

20.3 Optical Background 439

20.4 Knill–Laflamme–Milburn (KLM) Scheme 441

References 446

21 One-Way Quantum Computation 449
Dan Browne and Hans Briegel

21.1 Introduction 449

21.2 Simple Examples 451

21.3 Beyond Quantum Circuit Simulation 455

21.4 Implementations 465

21.5 Recent Developments 466

21.6 Outlook 469

Acknowledgments 469

Exercises 469

References 470

22 Holonomic Quantum Computation 475
Angelo C. M. Carollo and Vlatko Vedral

22.1 Geometric Phase and Holonomy 475

22.2 Application to Quantum Computation 479

References 480

Part VI Quantum Computing: Implementations 483

23 Quantum Computing with Cold Ions and Atoms: Theory 485
Dieter Jaksch, Juan José García-Ripoll, Juan Ignacio Cirac, and Peter Zoller

23.1 Introduction 485

23.2 Trapped Ions 485

23.3 Trapped Neutral Atoms 495

References 515

24 Quantum Computing Experiments with Cold Trapped Ions 519
Ferdinand Schmidt-Kaler and Ulrich Poschinger

24.1 Introduction to Trapped-Ion Quantum Computing 519

24.2 Paul Traps 522

24.3 Ion Crystals and Normal Modes 526

24.4 Trap Technology 529

Acknowledgements 547

References 547

25 Quantum Computing with Solid-State Systems 553
Guido Burkard and Daniel Loss

25.1 Introduction 553

25.2 Concepts 554

25.3 Electron Spin Qubits 563

25.4 Superconducting Qubits 575

References 583

26 Time-Multiplexed Networks for Quantum Optics 587
Sonja Barkhofen, Linda Sansoni and Christine Silberhorn

26.1 Introduction 587

26.2 Multiplexing 588

26.3 Photon-Number-Resolving Detection with Time Multiplexing 589

26.4 Quantum Walks in Time 592

26.5 Conclusion 600

References 601

27 A Brief on Quantum Systems Theory and Control Engineering 607
Thomas Schulte-Herbrüggen, Robert Zeier,Michael Keyl, and Gunther Dirr

27.1 Introduction 607

27.2 Systems Theory of Closed Quantum Systems 609

27.3 Toward a Systems Theory for Open Quantum Systems 620

27.4 Relation to Numerical Optimal Control 624

27.5 Outlook on Infinite-Dimensional Systems 626

27.6 Conclusion 633

Acknowledgments 633

Exercises 634

References 635

28 Quantum Computing Implemented via Optimal Control: Application to Spin and PseudospinSystems 643
Thomas Schulte-Herbrüggen, Andreas Spörl, Raimund Marx, Navin Khaneja, JohnMyers, Amr Fahmy,Samuel Lomonaco, Louis Kauffman, and Steffen Glaser

28.1 Introduction 643

28.2 From Controllable Spin Systems to Suitable Molecules 645

28.3 Scalability 647

28.4 Algorithmic Platform for Quantum Control Systems 649

28.5 Applied Quantum Control 651

28.6 Worked Example: Unitary Controls for Classifying Knots by NMR 656

28.7 Conclusions 661

Acknowledgments 662

Exercises 662

References 663

Part VII Quantum Interfaces and Memories 669

29 Cavity Quantum Electrodynamics: Quantum Information Processing with Atoms and Photons 671
Jean-Michel Raimond and Gerhard Rempe

29.1 Introduction 671

29.2 Microwave Cavity Quantum Electrodynamics 672

29.3 Optical Cavity Quantum Electrodynamics 677

29.4 Conclusions and Outlook 683

References 684

30 Quantum Repeater 691
Wolfgang Dür, Hans-J. Briegel, Peter Zoller, and Peter v Loock

30.1 Introduction 691

30.2 Concept of the Quantum Repeater 693

30.3 Proposals for Experimental Realization 697

30.4 Summary and Conclusions 699

Acknowledgments 699

References 699

31 Quantum Interface Between Light and Atomic Ensembles 701
Eugene S. Polzik and Jaromír Fiurášek

31.1 Introduction 701

31.2 Off-Resonant Interaction of Light with Atomic Ensemble 702

31.3 Entanglement of Two Atomic Clouds 711

31.4 Quantum Memory for Light 712

31.5 Multiple Passage Protocols 715

31.6 Atoms-Light Teleportation and Entanglement Swapping 718

31.7 Quantum Cloning into Atomic Memory 720

31.8 Summary 721

Acknowledgment 721

References 721

32 Echo-Based Quantum Memory 723
G. T. Campbell, K. R. Ferguson, M. J. Sellars, B. C. Buchler, and P. K. Lam

32.1 Overview of Photon Echo Techniques 724

32.2 Platforms for Echo-Based Quantum Memory 728

32.3 Characterization 731

32.4 Demonstrations 734

32.5 Outlook 736

References 737

33 Quantum Electrodynamics of a Qubit 741
Gernot Alber and Georgios M. Nikolopoulos

33.1 Quantum Electrodynamics of a Qubit in a Spherical Cavity 742

33.2 Suppression of Radiative Decay of a Qubit in a Photonic Crystal 750

Exercises 755

References 756

34 Elementary Multiphoton Processes in Multimode Scenarios 759
Nils Trautmann and Gernot Alber

34.1 A Generic Quantum Electrodynamical Model 761

34.2 The Multiphoton Path Representation 761

34.3 Examples 767

34.4 Conclusion 772

Appendix A: Evaluation of the Field Commutator 773

References 774

Part VIII Towards Quantum Technology Applications 777

35 Quantum Interferometry with Gaussian States 779
Ulrik L. Andersen, Oliver Glöckl, Tobias Gehring, and Gerd Leuchs

35.1 Introduction 779

35.2 The Interferometer 780

35.3 Interferometer with Coherent States of Light 783

35.4 Interferometer with Squeezed States of Light 786

35.5 Fundamental Limits 792

35.6 Summary and Discussion 793

Problems 795

References 796

36 Quantum Logic-Enabled Spectroscopy 799
Piet O. Schmidt

36.1 Introduction 799

36.2 Trapping and Doppler Cooling of a Two-Ion Crystal 800

36.3 Coherent Atom–Light Interaction and State Manipulation 802

36.4 Quantum Logic Spectroscopy for Optical Clocks 805

36.5 Photon Recoil Spectroscopy 809

36.6 Quantum Logic with Molecular Ions 815

36.7 Nonclassical States for Spectroscopy 819

36.8 Future Directions 821

Acknowledgments 822

References 822

37 Quantum Imaging 827
Claude Fabre and Nicolas Treps

37.1 Introduction 827

37.2 The Quantum Laser Pointer 828

37.3 Manipulation of Spatial Quantum Noise 830

37.4 Two-Photon Imaging 832

37.5 Other Topics in Quantum Imaging 833

37.6 Conclusion and Perspectives 834

Acknowledgment 835

References 835

38 Quantum Frequency Combs 837
Claude Fabre and Nicolas Treps

38.1 Introduction 837

38.2 Parametric Down Conversion of a Frequency Comb 839

38.3 Experiment 840

38.4 Experimental Results 843

38.5 Application to Quantum Information Processing 849

38.6 Application to Quantum Metrology 853

38.7 Conclusion 854

Acknowledgment 855

References 855

Index 859