This comprehensive textbook on the rapidly advancing field introduces readers to the fundamental concepts of information theory and quantum entanglement, taking into account the current state of research and development. It thus covers all current concepts in quantum computing, both theoretical and experimental, before moving on to the latest implementations of quantum computing and communication protocols. It contains problems and exercises and is therefore ideally suited for students and lecturers in physics and informatics, as well as experimental and theoretical physicists in academia and industry who work in the field of quantum information processing.
The second edition incorporates important recent developments such as quantum metrology, quantum correlations beyond entanglement, and advances in quantum computing with solid state devices.
Table of Contents:
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 Pseudospin Systems 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
