Preface ........................................................ XV
List of Contributors .......................................... XIX
1. Algorithms for Quantum Systems — Quantum Algorithms
(Th. Beth, M. Grassl, D. Janzing, M. Rotteler, R.
Wocjan, and R. Zeier) ................................... 1
1.1. Introduction .......................................... 1
1.2. Fast Quantum Signal Transforms......................... 1
1.3. Quantum Error-correcting Codes ........................ 3
1.4. Efficient Decomposition of Quantum Operations into
Given One-parameter Groups ............................ 5
1.5. Simulation of Hamiltonians ............................ 8
References ................................................. 10
2. Quantum Information Processing and Error Correction with
Jump Codes
(G. Alber, M. Mussinger, and A. Delgado) ................ 14
2.1. Introduction 14
2.2. Invertible Quantum Operations and Error Correction ... 15
2.3. Quantum Error Correction by Jump Codes ............... 17
2.3.1. Spontaneous Decay and Quantum Trajectories ... 17
2.3.2. Jump Codes ................................... 19
2.4. Universal Quantum Gates in Code Spaces ............... 21
2.4.1. Universal Sets of Quantum Gates for
Qudit-Systems ................................ 21
2.4.2. Universal One-Qutrit Gates ................... 22
2.4.3. A Universal Entanglement Gate ................ 23
2.5. Summary and Outlook .................................. 25
References ................................................. 26
3. Computational Model for the One-Way Quantum Computer:
Concepts and Summary
(R. Raussendorf and H. J. Briegel) ...................... 28
3.1. Introduction ......................................... 28
3.2. The QCC as a Universal Simulator of Quantum Logic
Networks ............................................. 30
3.3. Non-Network Character of the QCC ..................... 35
3.4. Computational Model .................................. 36
3.5. Conclusion ........................................... 42
References ................................................. 42
4. Quantum Correlations as Basic Resource for Quantum Key
Distribution
(M. Curty, O. Gühne, M. Lewenstein, and
N. Lütkenhaus) ......................................... 44
4.1. Introduction ......................................... 44
4.2. Background of Classical Information Theoretic
Security ............................................. 45
4.3. Link Between Classical and Quantum ................... 46
4.4. Searching for Effective Entanglement ................. 49
4.5. Verification Sets .................................... 51
4.5.1. 6-state Protocol ............................. 51
4.5.2. 4-state Protocol ............................. 51
4.5.3. 2-state Protocol ............................. 52
4.6. Examples for Evaluation .............................. 53
4.7. Realistic Experiments ................................ 54
4.8. Conclusions .......................................... 55
References ................................................. 55
5. Increasing the Size of NMR Quantum Computers
(S.J. Glaser, R. Marx, T. Reiss, T. Schulte-
Herbrüiggen, N. Khaneja, J.M. Myers, and A.F. Fahmy) .... 58
5.1. Introduction ......................................... 58
5.2. Suitable Molecules ................................... 59
5.3. Scaling Problem for Experiments Based on Pseudo-
pure States .......................................... 62
5.4. Approaching Pure States .............................. 62
5.5. Scalable NMR Quantum Computing Based on the Thermal
Density Operator ..................................... 63
5.6. Time-optimal Implementation of Quantum Gates ......... 64
5.7. Conclusion ........................................... 67
References ................................................. 68
6. On Lossless Quantum Data Compression and Quantum Variable
-length Codes
(R. Ahlswede and N. Cai) ................................ 70
6.1. Introduction ......................................... 70
6.2. Codes, Lengths, Kraft Inequality and von Neumann
Entropy Bound ........................................ 71
6.2.1. The Codes .................................... 71
6.2.2. Length Observable and Average Length of
Codewords .................................... 72
6.2.3. Kraft Inequality and von Neumann Entropy
Bound ........................................ 72
6.2.4. Base Length .................................. 73
6.3. Construct Long Codes from Variable-length Codes ...... 73
6.4. Lossless Quantum Data Compression, if the Decoder
is Informed about the Base Lengths ................... 74
6.5. Code Analysis Based on the Base Length ............... 75
6.6. Lossless Quantum Data Compression with a Classical
Helper ............................................... 76
6.7. Lossless Quantum Data Compression for Mixed State
Sources .............................................. 79
6.8. A Result on Tradeoff between Quantum and Classical
Resources in Lossy Quantum Data Compression .......... 80
References ................................................. 81
7. Entanglement Properties of Composite Quantum Systems
(K. Eckert, O. Gühne, F. Hulpke, P. Hyllus, J.
Korbicz, J. Mompart, D. Bruß, M. Lewenstein, and
A. Sanpera) ............................................ 83
7.1. Introduction ......................................... 83
7.2. Separability of Composite Quantum Systems ............ 84
7.2.1. The Separability Problem ..................... 85
7.2.2. Results on The Separability Problem .......... 86
7.3. The Distillability Problem ........................... 88
7.3.1. Results on the Distillability Problem ........ 89
7.4. Witness Operators for the Detection of
Entanglement ......................................... 90
7.4.1. Definition and Geometrical Interpretation
of Witness Operators ......................... 90
7.4.2. Results on Witness Operators ................. 92
7.5. Quantum Correlations in Systems of Fermionic and
Bosonic States ....................................... 94
7.5.1. What is Different with Indistinguishable
Particles? ................................... 94
7.5.2. Results on Quantum Correlations for
Indistinguishable Particles .................. 95
7.5.3. Implementation of an Entangling Gate with
Bosons ....................................... 97
7.6. Summary .............................................. 97
References ................................................. 97
8. Non-Classical Gaussian States in Noisy Environments
(S. Scheel and D.-G. Welsch) ........................... 100
8.1. Introduction ........................................ 100
8.2. Gaussian States and Gaussian Operations ............. 100
8.2.1. Classicality ................................ 102
8.2.2. CP Maps and Partial Measurements ............ 102
8.2.3. Separability and Entanglement ............... 103
8.3. Entanglement Degradation ............................ 104
8.4. Quantum Teleportation in Noisy Environments ......... 106
8.4.1. Imperfect Teleportation ..................... 107
8.4.2. Teleportation Fidelity ...................... 108
8.4.3. Choice of the Coherent Displacement ......... 110
References ................................................ 111
9. Quantum Estimation with Finite Resources
(T.C. Bschorr, D.G. Fischer, H. Mack, W.P. Schleich,
and M. Freyberger) .................................... 113
9.1. Introduction ........................................ 113
9.2. Quantum Devices and Channels ........................ 114
9.3. Estimating Quantum Channels ......................... 115
9.4. Entanglement and Estimation ......................... 115
9.4.1. Estimation using Single Qubits .............. 116
9.4.2. Estimation using Entangled States ........... 118
9.5. Generalized Estimation Schemes ...................... 120
9.5.1. Estimation with Two Channels ................ 120
9.5.2. What is the Optimal Reference Channel? ...... 121
9.5.3. Estimation with Werner States ............... 122
9.6. Outlook ............................................. 123
References ................................................ 124
10. Size Scaling of Decoherence Rates
(C.S. Maierle and D. Suter)............................. 125
10.1. Introduction ........................................ 125
10.2. Decoherence Models .................................. 126
10.3. Collective and Independent Decoherence .............. 127
10.4. Average Decoherence Rate as a Measure of
Decoherence ......................................... 128
10.5. Decoherence Rate Scaling due to Partially
Correlated Fields ................................... 130
10.6. Conclusion .......................................... 134
References ................................................ 134
11. Reduced Collective Description of Spin-Ensembles
(M. Michel, H. Schmidt, F. Tonner, and G. Mahler) ...... 135
11.1. Introduction ........................................ 135
11.2. Operator Representations ............................ 135
11.3. Hamilton Models ..................................... 138
11.3.1. Symmetry-constrained Networks ............... 138
11.3.2. Topology-constrained Networks ............... 139
11.4. State Models ........................................ 140
11.4.1. Totally Permutation-symmetric Subspace ...... 140
11.4.2. Collective 1-particle Excitations ........... 140
11.4.3. 1-parameter Families of Non-pure States ..... 141
11.4.4. Families of Separable States: "Modules" ..... 141
11.5. Ensembles ........................................... 141
11.5.1. Trajectories and Ergodicity ................. 142
11.5.2. Leakage and Storage Capacity ................ 144
11.5.3. Mixing Strategies ........................... 146
11.5.4. State Construction and Separability ......... 147
11.6. Summary and Outlook ................................. 147
References ................................................ 148
12. Quantum Information Processing with Defects
(F. Jelezko and J. Wrachtrup) .......................... 150
12.1. Introduction ........................................ 150
12.2. Properties of Nitrogen-vacancy Centers in Diamond ... 150
12.3. Readout of Spin State via Site-selective
Excitation .......................................... 152
12.4. Magnetic Resonance on a Single Spin at Room
Temperature ......................................... 155
12.5. Magnetic Resonance on a Single 13C Nuclear Spin ..... 156
12.6. Two-qubit Gate with Electron Spin and 13C Nuclear
Spin of Single NV Defect ............................ 158
12.7. Outlook: Towards Scalable NV Based Quantum
Processor ........................................... 160
References ................................................ 160
13. Quantum Dynamics of Vortices and Vortex Qubits
(A. Wallraff, A. Kemp, and A.V. Ustinov) ............... 162
13.1. Introduction ........................................ 162
13.2. Macroscopic Quantum Effects with Single Vortices .... 163
13.2.1. Quantum Tunneling ........................... 163
13.2.2. Energy Level Quantization ................... 165
13.3. Vortex-Antivortex Pairs ............................. 167
13.3.1. Thermal and Quantum Dissociation ............ 167
13.3.2. Energy Levels of a Bound Vortex-Antivortex
Pair ........................................ 171
13.4. The Josephson Vortex Qubit .......................... 173
13.4.1. Principle of the Vortex Qubit ............... 174
13.4.2. Model ....................................... 175
13.4.3. Perturbative Calculation of Vortex
Potential ................................... 177
13.4.4. Quantum Mechanics of a Vortex in a Double
Well ........................................ 179
13.4.5. Depinning Current and Qubit Readout ......... 180
13.5. Conclusions ......................................... 182
References ................................................ 183
14. Decoherence in Resonantly Driven Bistable Systems
(S. Kohler and P. Hänggi) .............................. 186
14.1. Introduction ........................................ 186
14.2. The Model and its Symmetries ........................ 186
14.3. Coherent Tunneling .................................. 188
14.4. Dissipative Tunneling ............................... 192
14.5. Conclusions ......................................... 196
References ................................................ 197
15. Entanglement and Decoherence in Cavity QED with a
Trapped Ion
(W. Vogel and Ch. DiFidio) ............................. 198
15.1. Introduction ........................................ 198
15.2. Decoherence Effects ................................. 199
15.3. Greenberger-Horne-Zeilinger State ................... 201
15.4. Photon-number Control ............................... 203
15.5. Entanglement of Separated Atoms ..................... 205
15.6. Summary ............................................. 207
References ................................................ 207
16. Quantum Information Processing with Ions
Deterministically Coupled to an Optical Cavity
(M. Keller, B. Lange, K. Hayasaka, W. Lange, and
H. Walther) ........................................... 209
16.1 Introduction ......................................... 209
16.2 Deterministic Coupling of Ions and Cavity Field ...... 210
16.3 Single-ion Mapping of Cavity-Modes ................... 212
16.4 Atom-Photon Interface ................................ 215
16.5 Single-Photon Source ................................. 217
16.6 Cavity-mediated Two-Ion Coupling ..................... 219
References ................................................ 221
17. Strongly Coupled Atom-Cavity Systems
(A. Kuhn, M. Hennrich, and G. Rempe) ................... 223
17.1. Introduction ........................................ 223
17.2. Atoms, Cavities and Light ........................... 223
17.2.1. Field Quantization in a Fabry-Perot
Cavity ...................................... 223
17.2.2. Two-Level Atom .............................. 224
17.2.3. Three-Level Atom ............................ 225
17.2.4. Adiabatic Passage ........................... 227
17.3. Single-Photon Sources ............................... 228
17.3.1 Vacuum-Stimulated Raman Scattering ........... 229
17.3.2 Deterministic Single-Photon Sequences ........ 230
17.4. Summary and Outlook ................................. 233
References ................................................ 233
18. A Relaxation-free Verification of the Quantum Zeno
Paradox on an Individual Atom
(Ch. Balzer, Th. Hannemann, D. Reiß, Ch. Wunderlich,
W. Neuhauser, and P.E. Toschek) ....................... 237
18.1. Introduction ........................................ 237
18.2. The Hardware and Basic Procedure .................... 238
18.3. First Scheme: Statistics of the Sequences of
Equal Results ....................................... 241
18.4. Second Scheme: Driving the Ion by Fractionated
π-Pulses ............................................ 243
18.5. Conclusions ......................................... 246
18.6. Survey of Related Work .............................. 247
References ................................................ 249
19. Spin Resonance with Trapped Ions: Experiments and New
Concepts
(K. Abich, Ch. Balzer, T. Hannemann, F. Mintert,
W. Neuhauser, D. Reiß, R E. Toschek, and
Ch. Wunderlich) ....................................... 251
19.1. Introduction ........................................ 251
19.2. Self-learning Estimation of Quantum States .......... 252
19.3. Experimental Realization of Quantum Channels ........ 254
19.4. New Concepts for QIP with Trapped Ions .............. 256
19.4.1. Spin Resonance with Trapped Ions ............ 257
19.4.2. Simultaneous Cooling of Axial Vibrational
Modes ....................................... 260
19.5. Raman Cooling of two Trapped Ions ................... 261
References ................................................ 263
20. Controlled Single Neutral Atoms as Qubits
(V. Gomer, W. Alt, S. Kuhr, D. Schrader, and
D. Meschede) .......................................... 265
20.1. Introduction ........................................ 265
20.2. Cavity QED for QIP .................................. 265
20.3. Single Atom Controlled Manipulation ................. 266
20.4. How to Prepare Exactly 2 Atoms in a Dipole Trap? .... 267
20.5. Optical Dipole Trap ................................. 267
20.6. Relaxation and Decoherence .......................... 268
20.7. Qubit Conveyor Belt ................................. 269
20.8. Outlook ............................................. 270
References ................................................ 270
21. Towards Quantum Logic with Cold Atoms in a C02 Laser
Optical Lattice
(G. Cennini, G. Ritt, C. Geckeler, R. Scheunemann,
and M. Weitz) ......................................... 275
21.1. Introduction ........................................ 275
21.2. Entanglement and Beyond ............................. 276
21.3. Quantum Logic and Far-detuned Optical Lattices ...... 277
21.4. Resolving and Addressing Cold Atoms in Single
Lattice Sites ....................................... 279
21.5. Recent Work ......................................... 282
References ................................................ 284
22. Quantum Information Processing with Atoms in Optical
Micro-Structures
(R. Dumke, M. Volk, T. Müther, F.B.J. Buchkremer,
W. Ertmer, and G. Birkl) .............................. 287
22.1. Introduction 287
22.2. Microoptical Elements for Quantum Information
Processing .......................................... 288
22.3. Experimental Setup 289
22.4. Scalable Qubit Registers Based on Arrays of Dipole
Traps ............................................... 290
22.5. Initialization, Manipulation and Readout ............ 291
22.6. Variation of Trap Separation ........................ 292
22.7. Implementation of Qubit Gates ....................... 293
References ................................................ 296
23. Quantum Information Processing with Neutral Atoms on
Atom Chips
(R. Krüger, A. Haase, M. Andersson, and J.
Schmiedmayer) ......................................... 298
23.1. Introduction ........................................ 298
23.2. The Atom Chip ....................................... 298
23.2.1. Combined Magneto-Electric Traps ........... 299
23.2.2. RF-induced Adiabatic Potentials for
Manipulating Atoms ........................ 300
23.2.3. Imperfections in the Atom Chip: Disorder
Potentials ................................ 301
23.3. The Qubit ........................................... 302
23.4. Entangling Qubits ................................... 303
23.4.1. Quantum Gate via Cold Controlled
Collisions .................................. 303
23.4.2. Motional Qubit Gates with Controlled
Collisions .................................. 305
23.5. Input/Output ........................................ 305
23.5.1. Qubit Detection ............................. 305
23.5.2. Quantum Input/Output ........................ 307
23.6. Noise and Decoherence ............................... 307
23.7. Summary and Conclusion .............................. 308
References ................................................ 309
24. Quantum Gates and Algorithms Operating on Molecular
Vibrations
(U. Troppmann, C.M. Tesch, and R. de Vivie-Riedle) ..... 312
24.1. Introduction ........................................ 312
24.2. Qubit States Encoded in Molecular Vibrations ........ 313
24.3. Optimal Control Theory for Molecular Dynamics ....... 313
24.3.1. Local Quantum Gates ......................... 315
24.4. Multi-target OCT for Global Quantum Gates ........... 317
24.4.1. Global Quantum Gates for Molecular
Vibrational Qubits .......................... 317
24.5. Basis Set Independence and Quantum Algorithms ....... 318
24.6. Towards More Complex Molecular Systems .............. 321
24.7. Outlook ............................................. 324
References ................................................ 325
25. Fabrication and Measurement of Aluminum and Niobium
Based Single-Electron Transistors and Charge Qubits
(W. Krech, D. Born, M. Mihalik, and M. Grajcar) ........ 327
25.1. Introduction ........................................ 327
25.2. Motivation for this Work ............................ 328
25.3. Sample Preparation .................................. 329
25.3.1. Scheme of the Junction Preparation
Technique ................................... 329
25.3.2. Fabrication of Tunnel Devices: SET and
Charge Qubit Structures ..................... 330
25.4. Experimental Results ................................ 331
25.5. Conclusions ......................................... 333
References ................................................ 335
26. Quantum Dot Circuits for Quantum Computation
(R.H. Blick, A.K. Hüttel, A.W. Holleitner,
L. Pescini, and H. Lorenz) ............................ 338
26.1. Introduction ........................................ 338
26.2. Realizing Quantum Bits in Double Quantum Dots ....... 339
26.3. Controlling the Electron Spin in Single Dots ........ 346
26.4. Summary ............................................. 351
References ................................................ 351
27. Manipulation and Control of Individual Photons and
Distant Atoms via Linear Optical Elements
(X.-B. Zou and W. Mathis) .............................. 353
27.1. Introduction ........................................ 353
27.2. Manipulation and Control of Individual Photons
via Linear Optical Elements ......................... 354
27.2.1. Teleportation Implementation of Non-
deterministic NLS Gate and Single-mode
Photon Filter ............................... 354
27.2.2. Implementation of Non-deterministic NLS
Gate via Parametric Amplifiers .............. 359
27.2.3. Phase Measurement of Light and Generation
of Superposition of Fock States ............. 360
27.2.4. Joint Measurement of Photon Number Sum and
Phase Difference Operators on a Two-mode
Field ....................................... 365
27.2.5. Remark ...................................... 370
27.3. Quantum Entanglement Between Distant Atoms Trapped
in Different Optical Cavities ....................... 370
27.3.1. Generation of W States, GHZ States and
Cluster States Based on Single-photon
Detectors ................................... 370
27.3.2. Generation of W States and GHZ States
Based on Four-photon Coincidence
Detection ................................... 376
27.4. Conclusion .......................................... 379
References ................................................ 379
28. Conditional Linear Optical Networks
(S. Scheel) ............................................ 382
28.1. Introduction ........................................ 382
28.2. Measurement-induced Nonlinearities .................. 383
28.2.1. Beam Splitters and Networks ................. 384
28.2.2. Post-processing of Single-Photon Sources
and Number-Resolving Detectors .............. 385
28.3. Probability of Success and Permanents ............... 386
28.4. Upper Bounds on Success Probabilities ............... 388
28.5. Extension Using Weak Nonlinearities ................. 390
References ................................................ 391
29. Multiphoton Entanglement
(M. Bourennane, M. Eibl, S. Gaertner, N. Kiesel, Ch.
Kurtsiefer, M. ßukowski, and H. Weinfurter) ........... 393
29.1. Introduction ........................................ 393
29.2. Entangled Multiphoton State Preparation ............. 394
29.3. Experiment .......................................... 395
29.4. Quantum Correlations ................................ 396
29.5. Bell Inequality ..................................... 398
29.6. Genuine Four-photon Entanglement .................... 400
29.7. Entanglement Persistence ............................ 400
29.8. Conclusions ......................................... 401
References ................................................ 403
30. Quantum Polarization for Continuous Variable Information
Processing
(N. Korolkova) ......................................... 405
30.1. Introduction ........................................ 405
30.2. Nonseparability and Squeezing ....................... 406
30.2.1. Polarization Squeezing ...................... 406
30.2.2. Continuous Variable Polarization
Entanglement ................................ 407
30.3. Applications ........................................ 410
30.4. Stokes Operators Questioned: Degree of
Polarization in Quantum Optics ...................... 413
References ................................................ 416
31. A Quantum Optical XOR Gate
(H. Becker, K. Schmid, W. Dultz, W. Martienssen, and
H. Roskos) ............................................ 418
31.1. Introduction ........................................ 418
31.2. Double Bump Photons ................................. 418
31.3. The XOR Gate ........................................ 420
31.4. Quad Bump Photons ................................... 423
31.5. Outlook ............................................. 424
References ................................................ 424
32. Quantum Fiber Solitons — Generation, Entanglement,
and Detection
(G. Leuchs, N. Kowlkova, O. Glöckl, St. Lorenz,
J. Heersink, Ch. Silberhorn, Ch. Marquardt,
and U.L. Andersen) .................................... 425
32.1. Introduction ........................................ 425
32.2. Quantum Correlations and Entanglement ............... 426
32.3. Multimode Quantum Correlations ...................... 428
32.4. Generation of Bright Entangled Beams ................ 431
32.5. Detection of Entanglement of Bright Beams ........... 432
32.5.1. Sub-shot-noise Phase Quadrature
Measurements on Intense Beams ............... 432
32.5.2. Direct Experimental Test of Non-
Separability ................................ 434
32.6. Entanglement Swapping ............................... 435
32.7. Polarization Variables .............................. 437
References ................................................ 439
Index ......................................................... 443
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