Lectures on quantum information / edited by Dagmar BruÇ and Gerd Leuchs.

Format:

Book

Contributor:

Bruss, Dagmar.

Leuchs, Gerd

Hazel M. Hussong Fund.

Language:

English

Subjects (All):

Quantum computers.

Information theory.

Physical Description:

xxiv, 610 pages : illustrations ; 24 cm

Place of Publication:

Weinheim ; [Chichester] : Wiley-VCH, [2007]

Summary:

Quantum Information Processing is a young and rapidly growing field of research at the intersection of physics, mathematics, and computer science. Its ultimate goal is to harness quantum physics to conceive - and ultimately build - "quantum" computers that would dramatically overtake the capabilities of today's "classical" computers. One example of the power of a quantum computer is its ability to efficiently find the prime factors of a large integer, thus shaking the supposedly secure foundations of Standard encryption schemes.

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. With its series of exercises, this is ideal reading for students and lecturers in physics and informatics, as well as experimental and theoretical physicists, and physicists in industry.

Contents:

I Classical Information Theory 1

1 Classical Information Theory and Classical Error Correction / M. Grassl 3

1.2 Basics of Classical Information Theory 3

1.2.1 Abstract communication system 3

1.2.2 The discrete noiseless channel 4

1.2.3 The discrete noisy channel 6

1.3 Linear Block Codes 9

1.3.1 Repetition code 9

1.3.2 Finite fields 11

1.3.3 Generator and parity check matrix 12

1.3.4 Hamming codes 14

1.4 Further Aspects 15

2 Computational Complexity / S. Mertens 17

2.2 Algorithms and Time Complexity 19

2.3 Tractable Trails: The Class P 20

2.4 Intractable Itineraries: The class NP 21

2.4.1 Coloring graphs 25

2.4.2 Logical truth 26

2.5 Reductions and NP-completeness 27

2.6 P vs. NP 29

2.7 Optimization 31

2.8 Complexity Zoo 34

II Foundation of Quantum Information Theory 37

3 Discrete Quantum States versus Continuous Variables / J. Eisert 39

3.2 Finite-dimensional quantum systems 40

3.2.1 Quantum states 40

3.2.2 Quantum operations 41

3.3 Continuous-variables 43

3.3.1 Phase space 43

3.3.2 Gaussian states 45

3.3.3 Gaussian unitaries 46

3.3.4 Gaussian channels 47

3.3.5 Gaussian measurements 49

3.3.6 Non-Gaussian operations 51

4 Approximate Quantum Cloning / D. Brub, C. Macchiavello 53

4.2 The No-Cloning Theorem 53

4.3 State-Dependent Cloning 54

4.4 Phase Covariant Cloning 62

4.5 Universal Cloning 63

4.5.1 The case of qubits 63

4.5.2 Higher dimensions 66

4.5.3 Entanglement structure 67

4.6 Asymmetric Cloning 67

4.7 Probabilistic Cloning 68

4.8 Experimental Quantum Cloning 68

4.9 Summary and Outlook 69

5 Channels and Maps / M. Keyl, R. F. Werner 73

5.2 Completely Positive Maps 73

5.3 The Jamiolkowski Isomorphism 76

5.4 The Stinespring Dilation Theorem 78

5.5 Classical Systems as a Special Case 82

5.6.1 The ideal quantum channel 83

5.6.2 The depolarizing channel 83

5.6.3 Entanglement breaking channels 84

5.6.4 Covariant channels 84

6 Quantum Algorithms / J. Kempe 87

6.2 Precursors 88

6.2.1 Deutsch's algorithm 89

6.2.2 Deutsch-Josza algorithm 90

6.2.3 Simon's algorithm 92

6.3 Shor's Factoring Algorithm 93

6.3.1 Reduction from factoring to period finding 93

6.3.2 Implementation of the QFT 94

6.3.3 Shor's algorithm for period finding 95

6.4 Grover's Algorithm 96

6.5 Other Algorithms 97

6.5.1 The hidden subgroup problem 97

6.5.2 Search algorithms 98

6.5.3 Other algorithms 99

6.6 Recent Developments 99

6.6.1 Quantum walks 99

6.6.2 Adiabatic quantum algorithms 100

7 Quantum Error Correction / M. Grassl 105

7.2 Quantum Channels 105

7.3 Using Classical Error-Correcting Codes 109

7.3.1 Negative results: the quantum repetition code 109

7.3.2 Positive results: a simple three-qubit code 110

7.3.3 Shor's nine-qubit code 112

7.3.4 Steane's seven-qubit code and CSS codes 114

7.3.5 The five-qubit code and stabilizer codes 116

7.4 Further Aspects 118

III Theory of Entanglement 121

8 The Separability versus Entanglement Problem / A. Sen(De), U. Sen, M. Lewenstein, A. Sanpera 123

8.2 Bipartite Pure States: Schmidt Decomposition 123

8.3 Bipartite Mixed States: Separable and Entangled States 124

8.4 Operational Entanglement Criteria 125

8.4.1 Partial transposition 125

8.4.2 Majorization 127

8.5 Nonoperational Entanglement Criteria 128

8.5.1 Entanglement witnesses 128

8.5.2 Positive maps 131

8.6 Bell Inequalities 135

8.7 Classification of Bipartite States with Respect to Quantum Dense Coding 138

8.7.1 The Holevo bound 139

8.7.2 Capacity of quantum dense coding 140

8.8 Further Reading: Multipartite States 142

9 Entanglement Theory with Continuous Variables / P. van Loock 147

9.2 Phase-Space Description 149

9.3 Entanglement of Gaussian States 149

9.3.1 Gaussian states 150

9.3.2 Gaussian operations 151

9.3.3 Pure entangled Gaussian states 152

9.3.4 Mixed entangled Gaussian states and inseparability criteria 154

9.4 More on Gaussian Entanglement 157

10 Entanglement Measures / M. B. Plenio, S. S. Virmani 161

10.2 Manipulation of Single Systems 163

10.3 Manipulation in the Asymptotic Limit 164

10.4 Postulates for Axiomatic Entanglement Measures: Uniqueness and Extremality Theorems 166

10.5 Examples of Axiomatic Entanglement Measures 169

11 Purification and Distillation / W. Dur, H.-J. Briegel 177

11.2 Pure States 179

11.2.1 Bipartite systems 179

11.2.2 Multipartite systems 180

11.3 Distillability and Bound Entanglement in Bipartite Systems 181

11.3.1 Distillable entanglement and yield 181

11.3.2 Criteria for entanglement distillation 182

11.4 Bipartite Entanglement Distillation Protocols 184

11.4.1 Filtering protocol 184

11.4.2 Recurrence protocols 185

11.4.3 N [RightArrow] M protocols, hashing, and breeding 190

11.5 Distillability and Bound Entanglement in Multipartite systems 192

11.5.1 n-party distillability 192

11.5.2 m-party distillability with respect to coarser partitions 192

11.5.3 Bound entanglement in multipartite systems 193

11.6 Entanglement Purification Protocols in Multipartite Systems 193

11.6.1 Graph states 194

11.6.2 Recurrence protocol 194

11.6.3 Hashing protocol 196

11.6.4 Entanglement purification of nonstabilizer states 197

11.7 Distillability with Noisy Apparatus 197

11.7.1 Distillable entanglement and yield 197

11.7.2 Error model 198

11.7.3 Bipartite recurrence protocols 199

11.7.4 Multipartite recurrence protocols 200

11.7.5 Hashing protocols 201

11.8 Applications of Entanglement Purification 202

11.8.1 Quantum communication and cryptography 202

11.8.2 Secure state distribution 203

11.8.3 Quantum error correction 203

11.8.4 Quantum computation 204

12 Bound Entanglement / Pawel Horodecki 209

12.2 Distillation of Quantum Entanglement: Repetition 209

12.2.1 Bipartite entanglement distillation 209

12.2.2 Multipartite entanglement distillation 212

12.3 Bound Entanglement-Bipartite Case 213

12.3.1 Bound entanglement-the phenomenon 213

12.3.2 Bound entanglement and entanglement measures. Asymptotic irreversibility 215

12.3.3 Which states are bound entangled? 216

12.3.4 Applications in single copy case 219

12.3.5 Applications in asymptotic regime 221

12.4 Bound Entanglement: Multipartite Case 225

12.4.1 Which multipartite states are bound entangled? 225

12.4.2 Activation effects 227

12.4.3 Remote quantum information concentration 228

12.4.4 Violation of Bell inequalities and communication complexity reduction 228

12.4.5 Feedback to classical theory: multipartite bound information and its activation 229

12.4.6 Bound entanglement and multiparty quantum channels 230

12.5 Further Reading: Continuous Variables 230

13 Multiparticle Entanglement / J. Eisert, D. Gross 237

13.2 Pure States 238

13.2.1 Classifying entanglement of single specimens 238

13.2.2 Asymptotic manipulation of multiparticle quantum states 241

13.3 Mixed States 243

13.3.1 Classifying mixed state entanglement 243

13.3.2 Methods of detection 245

13.4 Quantifying Multiparticle Entanglement 246

13.5 Stabilizer States and Graph States 247

13.6 Applications of Multiparticle Entangled States 249

IV Quantum Communication 253

14 Quantum Teleportation / L. C. Davila Romero, N. Korolkova 255

14.1.1 Setting up the problem and the role of entanglement 255

14.1.2 A template for quantum teleportation 257

14.1.3 Efficiency and fidelity 259

14.2 Experimental Realization 260

14.2.1 The first quantum teleportation experiment 261

14.2.2 Further experiments 262

14.3 Continuous Variables-Concept and Extension 263

15 Theory of Quantum Key Distribution (QKD) / N.

Lutkenhaus 271

15.2 Classical Background to QKD 271

15.3 Ideal QKD 272

15.4 Idealized QKD in noisy environment 275

15.5 Realistic QKD in noisy and lossy environment 277

15.6 Improved Schemes 280

15.7 Improvements in Public Discussion 282

16 Quantum Communication Experiments with Discrete Variables / H. Weinfurter 285

16.1 Aunt Martha 285

16.2 Quantum Cryptography 286

16.2.1 Faint pulse QKD 286

16.2.2 Entanglement-Based QKD-Single Photon QKD 290

16.3 Entanglement-Based Quantum Communication 292

16.3.1 Quantum Dense Coding 292

16.3.2 Error Correction 293

17 Continuous Variable Quantum Communication / U. L. Andersen, G. Leuchs 297

17.2 Continuous Variable Quantum Systems 297

17.3 Tools for State Manipulation 300

17.3.1 Gaussian transformations 300

17.3.2 Homodyne detection and feed forward 303

17.3.3 Non-Gaussian transformations 303

17.4 Quantum Communication Protocols 304

17.4.1 Quantum dense coding 305

17.4.2 Quantum key distribution 306

17.4.3 Long distance communication 308

V Quantum Computing: Concepts 313

18 Requirements for a Quantum Computer / A. Ekert, A. Kay 315

18.1 Classical World of Bits and Probabilities 315

18.1.1 Parallel composition = tensor products 318

18.1.2 Sequential composition = matrix products 319

18.2 Logically Impossible Operations? 320

18.3 Quantum World of Probability Amplitudes 323

18.4 Interference Revisited 326

18.5 Tools of the Trade 328

18.5.1 Quantum states 328

18.5.2 Unitary operations 331

18.5.3 Quantum measurements 334

18.6 Composite Systems 335

18.6.1 Density operators 340

18.7 Quantum Circuits 341

18.7.1 Economy of resources 342

18.7.2 Computations 344

19 Probabilistic Quantum Computation and Linear Optical Realizations / N. Lutkenhaus 349

19.2 Gottesman/Chuang Trick 349

19.3 Optical Background 351

19.3.1 Optical qubits 351

19.3.2 Linear Optics Framework 352

19.4 Knill-Laflamme-Milburn (KLM) scheme 353

19.4.1 Extension of Gottesman-Chuang trick 353

19.4.2 Implementation with linear optics 355

19.4.3 Offline probabilistic gates 356

20 One-way Quantum Computation / D.E. Browne, H.J. Briegel 359

20.1.1 Cluster states and graph states 360

20.1.2 Single-qubit measurements and rotations 360

20.2.1 Connecting one-way patterns - arbitrary single-qubit operations 362

20.2.2 Graph states as a resource 364

20.2.3 Two-qubit gates 364

20.2.4 Cluster-state quantum computing 364

20.3 Beyond quantum circuit simulation 365

20.3.1 Stabilizer formalism 365

20.3.2 A logical Heisenberg picture 366

20.3.3 Dynamical variables on a stabilizer sub-space 367

20.3.4 One-way patterns in the stabilizer formalism 368

20.3.5 Pauli measurements 368

20.3.6 Pauli measurements and the Clifford group 370

20.3.7 Non-Pauli measurements 371

20.3.8 Diagonal unitaries 371

20.3.9 Gate patterns beyond the standard network model -CD-decomposition 373

20.4 Implementations 374

20.4.1 Optical lattices 374

20.4.2 Linear optics and cavity QED 375

20.5 Recent developments 376

20.6 Outlook 376

21 Holonomic Quantum Computation / A.C.M. Carollo, Vlatko Vedral 381

21.1 Geometric Phase and Holonomy 381

21.1.1 Adiabatic implementation of holonomies 382

21.2 Application to Quantum Computation 384

VI Quantum Computing: Implementations 389

22 Quantum Computing with Cold Ions and Atoms: Theory / D. Jaksch, J.J. Garcia-Ripoll, J.I. Cirac, Peter Zoller 391

22.2 Trapped Ions 391

22.2.1 Motional degrees of freedom 392

22.2.2 Internal degrees of freedom and atom-laser interaction 393

22.2.3 Lamb-Dicke limit and sideband transitions 393

22.2.4 Single-qubit operations and state measurement 394

22.2.5 The gate Cirac-Zoller '95 395

22.2.6 Optimal gates based on quantum control 397

22.3 Trapped Neutral Atoms 401

22.3.1 Optical lattices 401

22.3.2 The (Bose) Hubbard Hamiltonian 406

22.3.3 Loading schemes 408

22.3.4 Quantum computing in optical lattices 408

23 Quantum Computing Experiments with Cold Trapped Ions / F. Schmidt-Kaler 423

23.2 Paul Traps 425

23.2.1 Stability diagram of dynamic trapping 426

23.2.2 3D confinement in a linear Paul trap 427

23.3 Ion crystals and their normal modes 428

23.3.1 Lagrangian of the ion motion in the trap 428

23.3.2 Eigenmodes 430

23.4 Ion-light interaction 432

23.5 Levels and Transitions for Typical Qubit Candidates 433

23.6 Various Two-Qubit Gates 434

23.6.1 The Cirac and Zoller scheme 1995 434

23.6.2 Experimental realization of the Cirac and Zoller gate 435

23.6.3 The Sorensen and Molmer scheme 436

23.6.4 The Jonathan, Plenio, and Knight scheme 439

23.6.5 Geometric phase shift gates 440

23.6.6 The Mintert and Wunderlich gate proposal 442

23.6.7 Gate proposals based on the interaction of ions with a common optical mode 442

23.7 Teleportation 443

23.8 Segmented Traps and Future Directions 444

24 Quantum Computing with Solid State Systems / G. Burkard, D. Loss 451

24.2.1 The exchange coupling 452

24.2.2 Anisotropic exchange 454

24.2.3 Universal QC with the exchange coupling 456

24.2.4 Adiabaticity 458

24.3 Electron Spin Qubits 458

24.3.1 Quantum dots 459

24.3.2 Exchange in laterally coupled QDs 459

24.3.3 Semiconductor microcavities 466

24.3.4 Decoherence 467

24.4 Superconducting Qubits 469

24.4.1 Regimes of operation 469

24.4.2 Decoherence, visibility, and leakage 470

25 Quantum Computing Implemented via Optimal Control: Theory and Application to Spin and Pseudo-Spin Systems / T. Schulte-Herbruggen, A. K. Sporl, R. Marx, N. Khaneja, J. M. Myers, A. F. Fahmy, S. J. Glaser 481

25.2 From Controllable Spin Systems to Suitable Molecules 483

25.2.1 Reachability and controllability 483

25.2.2 Molecular hardware for quantum computation 483

25.3 Scalability 485

25.3.1 Scaling problem with pseudo-pure states 485

25.3.2 Approaching pure states 485

25.3.3 Scalable quantum computing on thermal ensembles 486

25.4 Control Theory for Spin- and Pseudo-Spin Systems 487

25.5 Applied Quantum Control 492

25.5.1 Regime of fast local controls: the NMR limit 492

25.5.2 Regime of finite local controls: beyond NMR 494

25.6.1 Ensemble quantum computing 495

25.6.2 From gate-complexity to time-complexity by optimal control 495

25.6.3 Beyond NMR spin systems 496

VII Transfer of Quantum Information Between Different Types of Implementations 503

26 Quantum Repeater / W. Dur, H.-J. Briegel, P. Zoller 505

26.2 Concept of the quantum repeater 507

26.2.1 Entanglement purification 507

26.2.2 Connection of elementary pairs 507

26.2.3 Nested purification loops 508

26.3 Proposals for Experimental Realization 511

26.3.1 Photons and cavities 512

26.3.2 Atomic ensembles 512

26.3.3 Quantum dots 512

27 Quantum Interface Between Light and Atomic Ensembles / E. S. Polzik, J. Fiurasek 515

27.2 Off-Resonant Interaction of Light with Atomic Ensemble 516

27.3 Entanglement of Two Atomic Clouds 524

27.4 Quantum Memory for Light 526

27.5 Multiple Passage Protocols 528

27.6 Atoms-light teleportation and entanglement swapping 531

27.7 Quantum Cloning into Atomic Memory 532

28 Cavity Quantum Electrodynamics: Quantum Information Processing with Atoms and Photons / J.-M. Raimond, G. Rempe 537

28.2 Microwave Cavity Quantum Electrodynamics 538

28.3 Optical Cavity Quantum Electrodynamics 543

28.4 Conclusions and Outlook 549

29 Quantum Electrodynamics of a Qubit / G. Alber, G. M. Nikolopoulos 555

29.1 Quantum Electrodynamics of a Qubit in a Spherical Cavity 556

29.1.1 The model 556

29.1.2 Mode structure of the free radiation field in a spherical cavity 558

29.1.3 Dynamics of spontaneous photon emission 559

29.2 Suppression of Radiative Decay of a Qubit in a Photonic Crystal 564

29.2.1 Photonic crystals and associated density of states 564

29.2.2 "Photon + atom" bound states 566

29.2.3 Beyond the two-level approximation 567

VIII Towards Quantum Technology Applications 573

30 Quantum Interferometry / O. Glockl, U. L. Andersen, G.

Leuchs 575

30.2 The Interferometer 576

30.2.1 Sensitivity 577

30.3 Interferometer with Coherent States of Light 579

30.3.1 Geometrical visualization 579

30.4 Interferometer with Squeezed States of Light 581

30.4.1 Interferometer operating with a coherent state and a squeezed vacuum state 581

30.4.2 Interferometer operating with two bright squeezed states 584

30.4.3 Interferometer operating with a bright squeezed state and a squeezed vacuum state 585

31 Quantum Imaging / C. Fabre, N. Treps 591

31.2 The Quantum Laser Pointer 592

31.3 Manipulation of Spatial Quantum Noise 593

31.3.1 Observation of pure spatial quantum correlations in parametric down conversion 594

31.3.2 Noiseless image parametric amplification 595

31.4 Two-Photon Imaging 595

31.5 Other Topics in Quantum Imaging 597.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Hazel M. Hussong Fund.

ISBN:

9783527405275

3527405275

OCLC:

80331225