Quantum Information Processing

by Beth, Thomas; Leuchs, Gerd

• ISBN13: 9783527405411
• ISBN10: 3527405410
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-05-06
• Publisher: Wiley-VCH

Summary

Quantum processing and communication is emerging as a challenging technique at the beginning of the new millennium. This is an up-to-date insight into the current research of quantum superposition, entanglement, and the quantum measurement process - the key ingredients of quantum information processing. The authors further address quantum protocols and algorithms. Complementary to similar programmes in other countries and at the European level, the German Research Foundation (DFG) started a focused research program on quantum information in 1999. The contributions - written by leading experts - bring together the latest results in quantum information as well as addressing all the relevant questions.

Author Biography

Thomas Beth studied mathematics, physics and medicine. He received his Ph.D. in 1978 and his Postdoctoral Lecturer Qualification (Dr.-Ing. habil.) in informatics in 1984. From a position as Professor of computer science at the University of London he was apppointed to a chair of informatics at the University of Karlsruhe. He also is the director of the European Institute for System Security (E.I.S.S.). In the past decade he has built up a research center for quantum information at the Institute for Algorithms and Cognitive Systems (IAKS).

Gerd Leuchs studied physics and mathematics at the University of Cologne and received his Ph.D. in 1978. After two research visits at the University of Colorado, Boulder, he headed the German Gravitational Wave Detection Group from 1985 to 1989. He then went on to be the technical director of Nanomach AG in Switzerland for four years. Since 1994 he holds the chair for optics at the Friedrich-Alexander-University of Erlangen-Nuremberg, Germany. His fields of research span the range from modern aspects of classical optics to quantum optics and quantum information.

Table of Contents

Preface	XV
List of Contributors	XIX
1 Algorithms for Quantum Systems - Quantum Algorithms
Th. Beth, M. Grassi, D. Janzing, M. Rötteler, P. Wocjan, and R. Zeier	1	(13)
1.1 Introduction	1	(1)
1.2 Fast Quantum Signal Transforms	1	(2)
1.3 Quantum Error-correcting Codes	3	(2)
1.4 Efficient Decomposition of Quantum Operations into Given One-parameter Groups	5	(3)
1.5 Simulation of Hamiltonians	8	(2)
References	10	(4)
2 Quantum Information Processing and Error Correction with Jump Codes
G. Alber, M. Mussinger, and A. Delgado	14	(14)
2.1 Introduction	14	(1)
2.2 Invertible Quantum Operations and Error Correction	15	(2)
2.3 Quantum Error Correction by Jump Codes	17	(4)
2.3.1 Spontaneous Decay and Quantum Trajectories	17	(2)
2.3.2 Jump Codes	19	(2)
2.4 Universal Quantum Gates in Code Spaces	21	(4)
2.4.1 Universal Sets of Quantum Gates for Qudit-Systems	21	(1)
2.4.2 Universal One-Qutrit Gates	22	(1)
2.4.3 A Universal Entanglement Gate	23	(2)
2.5 Summary and Outlook	25	(1)
References	26	(2)
3 Computational Model for the One-Way Quantum Computer: Concepts and Summary
R. Raussendorf and H.J. Briegel	28	(16)
3.1 Introduction	28	(2)
3.2 The QCc as a Universal Simulator of Quantum Logic Networks	30	(5)
3.3 Non-Network Character of the QCc	35	(1)
3.4 Computational Model	36	(6)
3.5 Conclusion	42	(1)
References	42	(2)
4 Quantum Correlations as Basic Resource for Quantum Key Distribution
M. Curly, O. Gühne, M. Lewenstein, and N. Lütkenhaus	44	(14)
4.1 Introduction	44	(1)
4.2 Background of Classical Information Theoretic Security	45	(1)
4.3 Link Between Classical and Quantum	46	(3)
4.4 Searching for Effective Entanglement	49	(2)
4.5 Verification Sets	51	(2)
4.5.1 6-state Protocol	51	(1)
4.5.2 4-state Protocol	51	(1)
4.5.3 2-state Protocol	52	(1)
4.6 Examples for Evaluation	53	(1)
4.7 Realistic Experiments	54	(1)
4.8 Conclusions	55	(1)
References	55	(3)
5 Increasing the Size of NMR Quantum Computers
S.J. Glaser, R. Marx, T. Reiss, T. Schulte-Herbrüggen, N. Khaneja, J.M. Myers, and A.F. Fahmy	58	(12)
5.1 Introduction	58	(1)
5.2 Suitable Molecules	59	(3)
5.3 Scaling Problem for Experiments Based on Pseudo-pure States	62	(1)
5.4 Approaching Pure States	62	(1)
5.5 Scalable NMR Quantum Computing Based on the Thermal Density Operator	63	(1)
5.6 Time-optimal Implementation of Quantum Gates	64	(3)
5.7 Conclusion	67	(1)
References	68	(2)
6 On Lossless Quantum Data Compression and Quantum Variable-length Codes
R. Ahlswede and N. Cai	70	(13)
6.1 Introduction	70	(1)
6.2 Codes, Lengths, Kraft Inequality and von Neumann Entropy Bound	71	(2)
6.2.1 The Codes	71	(1)
6.2.2 Length Observable and Average Length of Codewords	72	(1)
6.2.3 Kraft Inequality and von Neumann Entropy Bound	72	(1)
6.2.4 Base Length	73	(1)
6.3 Construct Long Codes from Variable-length Codes	73	(1)
6.4 Lossless Quantum Data Compression, if the Decoder is Informed about the Base Lengths	74	(1)
6.5 Code Analysis Based on the Base Length	75	(1)
6.6 Lossless Quantum Data Compression with a Classical Helper	76	(3)
6.7 Lossless Quantum Data Compression for Mixed State Sources	79	(1)
6.8 A Result on Tradeoff between Quantum and Classical Resources in Lossy Quantum Data Compression	80	(1)
References	81	(2)
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	(17)
7.1 Introduction	83	(1)
7.2 Separability of Composite Quantum Systems	84	(4)
7.2.1 The Separability Problem	85	(1)
7.2.2 Results on The Separability Problem	86	(2)
7.3 The Distillability Problem	88	(2)
7.3.1 Results on the Distillability Problem	89	(1)
7.4 Witness Operators for the Detection of Entanglement	90	(4)
7.4.1 Definition and Geometrical Interpretation of Witness Operators	90	(2)
7.4.2 Results on Witness Operators	92	(2)
7.5 Quantum Correlations in Systems of Fermionic and Bosonic States	94	(3)
7.5.1 What is Different with Indistinguishable Particles?	94	(1)
7.5.2 Results on Quantum Correlations for Indistinguishable Particles	95	(2)
7.5.3 Implementation of an Entangling Gate with Bosons	97	(1)
7.6 Summary	97	(1)
References	97	(3)
8 Non-Classical Gaussian States in Noisy Environments
S. Scheel and D.-G. Welsch	100	(13)
8.1 Introduction	100	(1)
8.2 Gaussian States and Gaussian Operations	100	(4)
8.2.1 Classicality	102	(1)
8.2.2 CP Maps and Partial Measurements	102	(1)
8.2.3 Separability and Entanglement	103	(1)
8.3 Entanglement Degradation	104	(2)
8.4 Quantum Teleportation in Noisy Environments	106	(5)
8.4.1 Imperfect Teleportation	107	(1)
8.4.2 Teleportation Fidelity	108	(2)
8.4.3 Choice of the Coherent Displacement	110	(1)
References	111	(2)
9 Quantum Estimation with Finite Resources
T.C. Bschorr, D.G. Fischer, H. Mack, W.P. Schleich, and M. Freyberger	113	(12)
9.1 Introduction	113	(1)
9.2 Quantum Devices and Channels	114	(1)
9.3 Estimating Quantum Channels	115	(1)
9.4 Entanglement and Estimation	115	(5)
9.4.1 Estimation using Single Qubits	116	(2)
9.4.2 Estimation using Entangled States	118	(2)
9.5 Generalized Estimation Schemes	120	(3)
9.5.1 Estimation with Two Channels	120	(1)
9.5.2 What is the Optimal Reference Channel?	121	(1)
9.5.3 Estimation with Werner States	122	(1)
9.6 Outlook	123	(1)
References	124	(1)
10 Size Scaling of Decoherence Rates
C.S. Maierle and D. Suter	125	(10)
10.1 Introduction	125	(1)
10.2 Decoherence Models	126	(1)
10.3 Collective and Independent Decoherence	127	(1)
10.4 Average Decoherence Rate as a Measure of Decoherence	128	(2)
10.5 Decoherence Rate Scaling due to Partially Correlated Fields	130	(4)
10.6 Conclusion	134	(1)
References	134	(1)
11 Reduced Collective Description of Spin-Ensembles
M. Michel, H. Schmidt, F. Tanner, and G. Mahler	135	(15)
11.1 Introduction	135	(1)
11.2 Operator Representations	135	(3)
11.3 Hamilton Models	138	(2)
11.3.1 Symmetry-constrained Networks	138	(1)
11.3.2 Topology-constrained Networks	139	(1)
11.4 State Models	140	(1)
11.4.1 Totally Permutation-symmetric Subspace	140	(1)
11.4.2 Collective 1-particle Excitations	140	(1)
11.4.3 1-parameter Families of Non-pure States	141	(1)
11.4.4 Families of Separable States: "Modules"	141	(1)
11.5 Ensembles	141	(6)
11.5.1 Trajectories and Ergodicity	142	(2)
11.5.2 Leakage and Storage Capacity	144	(2)
11.5.3 Mixing Strategies	146	(1)
11.5.4 State Construction and Separability	147	(1)
11.6 Summary and Outlook	147	(1)
References	148	(2)
12 Quantum Information Processing with Defects
F. Jelezko and J. Wrachtrup	150	(12)
12.1 Introduction	150	(1)
12.2 Properties of Nitrogen-vacancy Centers in Diamond	150	(2)
12.3 Readout of Spin State via Site-selective Excitation	152	(3)
12.4 Magnetic Resonance on a Single Spin at Room Temperature	155	(1)
12.5 Magnetic Resonance on a Single ¹³c Nuclear Spin	156	(2)
12.6 Two-qubit Gate with Electron Spin and ¹³C Nuclear Spin of Single NV Defect	158	(2)
12.7 Outlook: Towards Scalable NV Based Quantum Processor	160	(1)
References	160	(2)
13 Quantum Dynamics of Vortices and Vortex Qubits
A. Wallraff, A. Kemp, and A.V. Ustinov	162	(24)
13.1 Introduction	162	(1)
13.2 Macroscopic Quantum Effects with Single Vortices	163	(4)
13.2.1 Quantum Tunneling	163	(2)
13.2.2 Energy Level Quantization	165	(2)
13.3 Vortex-Antivortex Pairs	167	(6)
13.3.1 Thermal and Quantum Dissociation	167	(4)
13.3.2 Energy Levels of a Bound Vortex-Antivortex Pair	171	(2)
13.4 The Josephson Vortex Qubit	173	(9)
13.4.1 Principle of the Vortex Qubit	174	(1)
13.4.2 Model	175	(2)
13.4.3 Perturbative Calculation of Vortex Potential	177	(2)
13.4.4 Quantum Mechanics of a Vortex in a Double Well	179	(1)
13.4.5 Depinning Current and Qubit Readout	180	(2)
13.5 Conclusions	182	(1)
References	183	(3)
14 Decoherence in Resonantly Driven Bistable Systems
S. Kohler and P. Hänggi	186	(12)
14.1 Introduction	186	(1)
14.2 The Model and its Symmetries	186	(2)
14.3 Coherent Tunneling	188	(4)
14.4 Dissipative Tunneling	192	(4)
14.5 Conclusions	196	(1)
References	197	(1)
15 Entanglement and Decoherence in Cavity QED with a Trapped Ion
W. Vogel and Ch. DiFidio	198	(11)
15.1 Introduction	198	(1)
15.2 Decoherence Effects	199	(2)
15.3 Greenberger-Horne-Zeilinger State	201	(2)
15.4 Photon-number Control	203	(2)
15.5 Entanglement of Separated Atoms	205	(2)
15.6 Summary	207	(1)
References	207	(2)
16 Quantum Information Processing with Ions Deterministically Coupled to an Optical Cavity
M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther	209	(14)
16.1 Introduction	209	(1)
16.2 Deterministic Coupling of Ions and Cavity Field	210	(2)
16.3 Single-ion Mapping of Cavity-Modes	212	(3)
16.4 Atom-Photon Interface	215	(2)
16.5 Single-Photon Source	217	(2)
16.6 Cavity-mediated Two-Ion Coupling	219	(2)
References	221	(2)
17 Strongly Coupled Atom-Cavity Systems
A. Kuhn, M. Hennrich, and G. Rempe	223	(14)
17.1 Introduction	223	(1)
17.2 Atoms, Cavities and Light	223	(5)
17.2.1 Field Quantization in a Fabry-Perot Cavity	223	(1)
17.2.2 Two-Level Atom	224	(1)
17.2.3 Three-Level Atom	225	(2)
17.2.4 Adiabatic Passage	227	(1)
17.3 Single-Photon Sources	228	(5)
17.3.1 Vacuum-Stimulated Raman Scattering	229	(1)
17.3.2 Deterministic Single-Photon Sequences	230	(3)
17.4 Summary and Outlook	233	(1)
References	233	(4)
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	(14)
18.1 Introduction	237	(1)
18.2 The Hardware and Basic Procedure	238	(3)
18.3 First Scheme: Statistics of the Sequences of Equal Results	241	(2)
18.4 Second Scheme: Driving the Ion by Fractionated π-Pulses	243	(3)
18.5 Conclusions	246	(1)
18.6 Survey of Related Work	247	(2)
References	249	(2)
19 Spin Resonance with Trapped Ions: Experiments and New Concepts
K. Abich, Ch. Balzer, T. Hannemann, F. Mintert, W. Neuhauser, D. Reiß, P.E. Toschek, and Ch. Wunderlich	251	(14)
19.1 Introduction	251	(1)
19.2 Self-learning Estimation of Quantum States	252	(2)
19.3 Experimental Realization of Quantum Channels	254	(2)
19.4 New Concepts for QIP with Trapped Ions	256	(5)
19.4.1 Spin Resonance with Trapped Ions	257	(3)
19.4.2 Simultaneous Cooling of Axial Vibrational Modes	260	(1)
19.5 Raman Cooling of two Trapped Ions	261	(2)
References	263	(2)
20 Controlled Single Neutral Atoms as Qubits
V. Gomer, W. Alt, S. Kuhr, D. Schrader, and D. Meschede	265	(10)
20.1 Introduction	265	(1)
20.2 Cavity QED for QIP	265	(1)
20.3 Single Atom Controlled Manipulation	266	(1)
20.4 How to Prepare Exactly 2 Atoms in a Dipole Trap?	267	(1)
20.5 Optical Dipole Trap	267	(1)
20.6 Relaxation and Decoherence	268	(1)
20.7 Qubit Conveyor Belt	269	(1)
20.8 Outlook	270	(1)
References	270	(5)
21 Towards Quantum Logic with Cold Atoms in a CO2 Laser Optical Lattice
G. Cennini, G. Ritt, C. Geckeler, R. Scheunemann, and M. Weitz	275	(12)
21.1 Introduction	275	(1)
21.2 Entanglement and Beyond	276	(1)
21.3 Quantum Logic and Far-detuned Optical Lattices	277	(2)
21.4 Resolving and Addressing Cold Atoms in Single Lattice Sites	279	(3)
21.5 Recent Work	282	(2)
References	284	(3)
22 Quantum Information Processing with Atoms in Optical Micro-Structures
R. Dumke, M. Volk, T. Müther, F.B.J. Buchkremer, W. Earner, and G. Birkl	287	(11)
22.1 Introduction	287	(1)
22.2 Microoptical Elements for Quantum Information Processing	288	(1)
22.3 Experimental Setup	289	(1)
22.4 Scalable Qubit Registers Based on Arrays of Dipole Traps	290	(1)
22.5 Initialization, Manipulation and Readout	291	(1)
22.6 Variation of Trap Separation	292	(1)
22.7 Implementation of Qubit Gates	293	(3)
References	296	(2)
23 Quantum Information Processing with Neutral Atoms on Atom Chips
P. Krüger, A. Haase, MAndersson, and J. Schmiedmayer	298	(14)
23.1 Introduction	298	(1)
23.2 The Atom Chip	298	(4)
23.2.1 Combined Magneto-Electric Traps	299	(1)
23.2.2 RF-induced Adiabatic Potentials for Manipulating Atoms	300	(1)
23.2.3 Imperfections in the Atom Chip: Disorder Potentials	301	(1)
23.3 The Qubit	302	(1)
23.4 Entangling Qubits	303	(2)
23.4.1 Quantum Gate via Cold Controlled Collisions	303	(2)
23.4.2 Motional Qubit Gates with Controlled Collisions	305	(1)
23.5 Input/Output	305	(2)
23.5.1 Qubit Detection	305	(2)
23.5.2 Quantum Input/Output	307	(1)
23.6 Noise and Decoherence	307	(1)
23.7 Summary and Conclusion	308	(1)
References	309	(3)
24 Quantum Gates and Algorithms Operating on Molecular Vibrations
U. Troppmann, C.M. Tesch, and R. de Vivie-Riedle	312	(15)
24.1 Introduction	312	(1)
24.2 Qubit States Encoded in Molecular Vibrations	313	(1)
24.3 Optimal Control Theory for Molecular Dynamics	313	(4)
24.3.1 Local Quantum Gates	315	(2)
24.4 Multi-target OCT for Global Quantum Gates	317	(1)
24.4.1 Global Quantum Gates for Molecular Vibrational Qubits	317	(1)
24.5 Basis Set Independence and Quantum Algorithms	318	(3)
24.6 Towards More Complex Molecular Systems	321	(3)
24.7 Outlook	324	(1)
References	325	(2)
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	(11)
25.1 Introduction	327	(1)
25.2 Motivation for this Work	328	(1)
25.3 Sample Preparation	329	(2)
25.3.1 Scheme of the Junction Preparation Technique	329	(1)
25.3.2 Fabrication of Tunnel Devices: SET and Charge Qubit Structures	330	(1)
25.4 Experimental Results	331	(2)
25.5 Conclusions	333	(2)
References	335	(3)
26 Quantum Dot Circuits for Quantum Computation
R.H. Blick, A.K. Hüttel, A.W. Holleitner, L. Pescini, and H. Lorenz	338	(15)
26.1 Introduction	338	(1)
26.2 Realizing Quantum Bits in Double Quantum Dots	339	(7)
26.3 Controlling the Electron Spin in Single Dots	346	(5)
26.4 Summary	351	(1)
References	351	(2)
27 Manipulation and Control of Individual Photons and Distant Atoms via Linear Optical Elements
X.-B. Zou and W. Mathis	353	(29)
27.1 Introduction	353	(1)
27.2 Manipulation and Control of Individual Photons via Linear Optical Elements	354	(16)
27.2.1 Teleportation Implementation of Non-deterministic NLS Gate and Single-mode Photon Filter	354	(5)
27.2.2 Implementation of Non-deterministic NLS Gate via Parametric Amplifiers	359	(1)
27.2.3 Phase Measurement of Light and Generation of Superposition of Fock States	360	(5)
27.2.4 Joint Measurement of Photon Number Sum and Phase Difference Operators on a Two-mode Field	365	(5)
27.2.5 Remark	370	(1)
27.3 Quantum Entanglement Between Distant Atoms Trapped in Different Optical Cavities	370	(9)
27.3.1 Generation of W States, GHZ States and Cluster States Based on Single-photon Detectors	370	(6)
27.3.2 Generation of W States and GHZ States Based on Four-photon Coincidence Detection	376	(3)
27.4 Conclusion	379	(1)
References	379	(3)
28 Conditional Linear Optical Networks
S. Scheel	382	(11)
28.1 Introduction	382	(1)
28.2 Measurement-induced Nonlinearities	383	(3)
28.2.1 Beam Splitters and Networks	384	(1)
28.2.2 Post-processing of Single-Photon Sources and Number-Resolving Detectors	385	(1)
28.3 Probability of Success and Permanents	386	(2)
28.4 Upper Bounds on Success Probabilities	388	(2)
28.5 Extension Using Weak Nonlinearities	390	(1)
References	391	(2)
29 Multiphoton Entanglement
M. Bourennane, M. Eibl, S. Gaertner, N. Kiesel, Ch. Kurtsiefer, M. Zukowski, and H. Weinfurter	393	(12)
29.1 Introduction	393	(1)
29.2 Entangled Multiphoton State Preparation	394	(1)
29.3 Experiment	395	(1)
29.4 Quantum Correlations	396	(2)
29.5 Bell Inequality	398	(2)
29.6 Genuine Four-photon Entanglement	400	(1)
29.7 Entanglement Persistence	400	(1)
29.8 Conclusions	401	(2)
References	403	(2)
30 Quantum Polarization for Continuous Variable Information Processing
N. Korolkova	405	(13)
30.1 Introduction	405	(1)
30.2 Nonseparability and Squeezing	406	(4)
30.2.1 Polarization Squeezing	406	(1)
30.2.2 Continuous Variable Polarization Entanglement	407	(3)
30.3 Applications	410	(3)
30.4 Stokes Operators Questioned: Degree of Polarization in Quantum Optics	413	(3)
References	416	(2)
31 A Quantum Optical XOR Gate
H. Becker, K. Schmid, W. Dultz, W. Martienssen, and H. Roskos	418	(7)
31.1 Introduction	418	(1)
31.2 Double Bump Photons	418	(2)
31.3 The XOR Gate	420	(3)
31.4 Quad Bump Photons	423	(1)
31.5 Outlook	424	(1)
References	424	(1)
32 Quantum Fiber Solitons - Generation, Entanglement, and Detection
G. Leuchs, N. Korolkova, O. Glöckl, St. Lorenz, T. Heersink, Ch. Silberhorn, Ch. Marquardt, and U.L. Andersen	425	(18)
32.1 Introduction	425	(1)
32.2 Quantum Correlations and Entanglement	426	(2)
32.3 Multimode Quantum Correlations	428	(3)
32.4 Generation of Bright Entangled Beams	431	(1)
32.5 Detection of Entanglement of Bright Beams	432	(3)
32.5.1 Sub-shot-noise Phase Quadrature Measurements on Intense Beams	432	(2)
32.5.2 Direct Experimental Test of Non-Separability	434	(1)
32.6 Entanglement Swapping	435	(2)
32.7 Polarization Variables	437	(2)
References	439	(4)
Index	443