The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation

326 Pages·2000·10.87 MB·English

by  D. Bouwmeester, A. Zeilinger (auth.), Dr. Dirk Bouwmeester, Prof. Artur Ekert

The Physics of Quantum Information
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LIBRARY
Physics and Astronomy
http://www.springer.de/phys/
Experimental demonstration of the breathing mode (left) and the centre-of-mass motion
(right) of a string of 7 ions which form an array of 7 qubits. The figures are compilations of
a sequence of snapshots taken of the string of ions (see Chapter 5). Figures by J. Eschner,
F. Schmidt-Kaler, R. Blatt, Institut fur Experimentalphysik, Universitat Innsbruck.
Dirk Bouwmeester
Artur Ekert
Anton Zeilinger (Eds.)
The Physics
of Quantum Information
Quantum Cryptography
Quantum Teleportation
Quantum Computation
With 125 Figures
t Springer
Dr. Dirk Bouwmeester
Prof. Artur Ekert
Centre for Quantum Computation
Clarendon Laboratory
University of Oxford
Parks Road
Oxford OX. 3Pt;
United Kingdom
Prof. Anton Zeilinger
Institut fiir Expcrimentalphysik
University of Vienna
Boltzmanngasse 5
1090 Vienna
Austria
ISBN 978-3-642-08607-6 ISBN 978-3-662-04209-0 (eBook)
DOI 10.1007/978-3-662-04209-0
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The physics of quantum informat ion : quantum cryptography. quantum
teleportation. quantum computation I Dirk Bouwmeester ... (ed.).
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Originally published by Springer-Verlag Berlin Heidelberg New York in 2000
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Preface
Information is stored, transmitted and processed by physical means. Thus,
the concept of information and computation can be formulated in the con-
text of a physical theory and the study of information requires ultimately
experimentation. This sentence, innocuous at first glance, leads to non-trivial
consequences.
Following Moore's law, about every 18 months microprocessors double
their speed and, it seems, the only way to make them significantly faster
is to make them smaller. In the not too distant future they will reach the
point where the logic gates are so small that they consist of only a few
atoms each. Then quantum-mechanical effects will become important. Thus,
if computers are to continue to become faster (and therefore smaller), new,
quantum technology must replace or supplement what we have now. But
it turns out that such technology can offer much more than smaller and
faster microprocessors. Several recent theoretical results have shown that
quantum effects may be harnessed to provide qualitatively new modes of
communication and computation, in some cases much more powerful than
their classical counterparts.
This new quantum technology is being born in many laboratories. The last
two decades have witnessed experiments in which single quantum particles of
different kinds were controlled and manipulated with an unprecedented preci-
sion. Many "gedanken" experiments, so famous in the early days of quantum
mechanics, have been carried out. New experimental techniques now make
it possible to store and process information encoded in individual quantum
systems. As a result we have a new, fledgling field of quantum information
processing that represents a highly fertile synthesis of the principles of quan-
tum physics with those of computer and information science. Its scope ranges
from providing a new perspective on fundamental issues about the nature of
physical law to investigating the potential commercial exploitation by the
computing and communications industries.
As part of the worldwide effort in the field, the European Commission,
within the framework of the TMR (Training and Mobility of Researchers)
programme, is supporting a network entitled "The Physics of Quantum In-
formation" . The chapters in this book are mainly written by various members
of the network in different forms of collaboration, and they are all intended
VI Preface
to give a didactic introduction to essential, new areas. In addition, several
sections present important achievements by researchers outside the TMR net-
work. However, it was not our aim to write a monograph giving a complete
overview of the field. Research in this field has become very active, and any
comprehensive review of the field would be obsolete in a short time. The
topics that are covered by this book include theoretical and experimental
aspects of quantum entanglement, quantum cryptography, quantum telepor-
tation, quantum computation, quantum algorithms, quantum-state decoher-
ence, quantum error correction, and quantum communication.
We hope that this book will be a valuable contribution to the literature
for all those who have a modest background in quantum mechanics and a
genuine interest in the fascinating possibilities that it is offering us.
We are very grateful to Thomas Jennewein for the numerous figures that
he drew for this book.
Oxford, Vienna, March 2000 Dzrk Bouwmeester
Artur Ekert
Anton Zeilinger
Contents
1. The Physics of Quantum Information:
Basic Concepts ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Quantum Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Qubits................................................ 3
1.3 Single-Qubit Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Entanglement.......................................... 7
1.5 Entanglement and Quantum Indistinguishability. . . . . . . . . . . . 9
1.6 The Controlled NOT Gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
1.7 The EPR Argument and Bell's Inequality ................. 12
1.8 Comments............................................. 14
2. Quantum Cryptography.................................. 15
2.1 What is Wrong with Classical Cryptography? . . . . . . . . . . . . .. 15
2.1.1 From SCYTALE to ENIGMA. . . . . . . . . . . . . . . . . . . . .. 15
2.1.2 Keys and Their Distribution. . . . . . . . . . . . . . . . . . . . . .. 16
2.1.3 Public Keys and Quantum Cryptography. . . . . . . . . . .. 19
2.1.4 Authentication: How to Recognise Cinderella? . . . . . .. 21
2.2 Quantum Key Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22
2.2.1 Preliminaria..................................... 22
2.2.2 Security in Non-orthogonal States: No-Cloning Theorem 22
2.2.3 Security in Entanglement. . . . . . . . . . . . . . . . . . . . . . . . .. 24
2.2.4 What About Noisy Quantum Channels? . . . . . . . . . . . .. 25
2.2.5 Practicalities..................................... 26
2.3 Quantum Key Distribution with Single Particles. . . . . . . . . . .. 27
2.3.1 Polarised Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
2.3.2 Phase Encoded Systems. . . . . . . . . . . . . . . . . . . . . . . . . .. 31
2.4 Quantum Key Distribution with Entangled States . . . . . . . . .. 33
2.4.1 Transmission of the Raw Key . . . . . . . . . . . . . . . . . . . . .. 33
2.4.2 Security Criteria ................................. 34
2.5 Quantum Eavesdropping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
2.5.1 Error Correction ................................. 36
2.5.2 Privacy Amplification. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37
2.6 Experimental Realisations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43
2.6.1 Polarisation Encoding. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43
VIII Contents
2.6.2 Phase Encoding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44
2.6.3 Entanglement-Based Quantum Cryptography. . . . . . .. 46
2.7 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
3. Quantum Dense Coding
and Quantum Teleportation .............................. 49
3.1 Introduction........................................... 49
3.2 Quantum Dense Coding Protocol. . . . . . . . . . . . . . . . . . . . . . . .. 50
3.3 Quantum Teleportation Protocol. . . . . . . . . . . . . . . . . . . . . . . .. 51
3.4 Sources of Entangled Photons. . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
3.4.1 Parametric Down-Conversion ..... . . . . . . . . . . . . . . . .. 53
3.4.2 Time Entanglement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54
3.4.3 Momentum Entanglement . . . . . . . . . . . . . . . . . . . . . . . .. 57
3.4.4 Polarisation Entanglement. . . . . . . . . . . . . . . . . . . . . . . .. 58
3.5 Bell-State Analyser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60
3.5.1 Photon Statistics at a Beamsplitter . . . . . . . . . . . . . . . .. 60
3.6 Experimental Dense Coding with Qubits .................. 62
3.7 Experimental Quantum Teleportation of Qubits . . . . . . . . . . .. 67
3.7.1 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
3.7.2 Teleportation of Entanglement. . . . . . . . . . . . . . . . . . . .. 72
3.7.3 Concluding Remarks and Prospects. . . . . . . . . . . . . . . .. 72
3.8 A Two-Particle Scheme for Quantum Teleportation . . . . . . . .. 74
3.9 Teleportation of Continuous Quantum Variables. . . . . . . . . . .. 77
3.9.1 Employing Position and Momentum Entanglement ... 77
3.9.2 Quantum Optical Implementation. . . . . . . . . . . . . . . . .. 79
3.10 Entanglement Swapping: Teleportation of Entanglement. . . .. 84
3.11 Applications of Entanglement Swapping. . . . . . . . . . . . . . . . . .. 88
3.1l.1 Quantum Telephone Exchange. . . . . . . . . . . . . . . . . . . .. 88
3.1l.2 Speeding up the Distribution of Entanglement. . . . . .. 89
3.1l.3 Correction of Amplitude Errors Developed
due to Propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90
3.11.4 Entangled States of Increasing Numbers of Particles.. 91
4. Concepts of Quantum Computation ...................... 93
4.1 Introduction to Quantum Computation ................... 93
4.l.1 A New Way of Harnessing Nature. . . . . . . . . . . . . . . . .. 93
4.l.2 From Bits to Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94
4.l.3 Quantum Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98
4.l.4 Building Quantum Computers ..................... 100
4.l.5 Deeper Implications .............................. 101
4.l.6 Concluding Remarks .............................. 103
4.2 Quantum Algorithms ................................... 104
4.2.1 Introduction ..................................... 104
4.2.2 Quantum Parallel Computation .................... 105
4.2.3 The Principle of Local Operations .................. 107
Contents IX
4.2.4 Oracles and Deutsch's Algorithm ................... 109
4.2.5 The Fourier Transform and Periodicities ............. 114
4.2.6 Shor's Quantum Algorithm for Factorisation ....... " 119
4.2.7 Quantum Searching and NP ....................... 121
4.3 Quantum Gates and Quantum Computation
with Trapped Ions ...................................... 126
4.3.1 Introduction ..................................... 126
4.3.2 Quantum Gates with Trapped Ions ................. 126
4.3.3 N Cold Ions Interacting with Laser Light ............ 128
4.3.4 Quantum Gates at Non-zero Temperature ........... 130
5. Experiments Leading Towards
Quantum Computation ................................... 133
5.1 Introduction ........................................... 133
5.2 Cavity QED-Experiments:
Atoms in Cavities and Trapped Ions .................... " 134
5.2.1 A Two-Level System Coupled to a Quantum Oscillator 134
5.2.2 Cavity QED with Atoms and Cavities ............... 135
5.2.3 Resonant Coupling: Rabi Oscillations
and Entangled Atoms ............................. 137
5.2.4 Dispersive Coupling: Schrodinger's Cat
and Decoherence ................................. 143
5.2.5 Trapped-Ion Experiments ......................... 147
5.2.6 Choice of Ions and Doppler Cooling ................. 148
5.2.7 Sideband Cooling ................................. 150
5.2.8 Electron Shelving and Detection of Vibrational Motion 153
5.2.9 Coherent States of Motion ......................... 154
5.2.10 Wigner Function of the One-Phonon State ........... 157
5.2.11 Squeezed States and Schrodinger Cats with Ions ...... 159
5.2.12 Quantum Logic with a Single Trapped 9Be+ Ion ..... 160
5.2.13 Comparison and Perspectives ...................... 161
5.3 Linear Ion Traps for Quantum Computation ............... 163
5.3.1 Introduction ..................................... 163
5.3.2 Ion Confinement in a Linear Paul Trap .............. 164
5.3.:~ Laser Cooling and Quantum Motion ................ 167
5.3.4 Ion Strings and Normal Modes ..................... 169
5.3.5 Ions as Quantum Register ......................... 171
5.3.6 Single-Qubit Preparation and Manipulation .......... 172
5.3.7 Vibrational Mode as a Quantum Data Bus .......... 173
5.3.8 Two-Bit Gates in an Ion-Trap Quantum Computer ... 174
5.3.9 Readout of the Qubits ............................ 175
5.3.10 Conclusion ...................................... 175
5.4 Nuclear Magnetic Resonance Experiments ................. 177
5.4.1 Introduction ..................................... 177
5.4.2 The NMR Hamiltonian ........................... 177
X Contents
5.4.3 Building an NMR Quantum Computer .............. 179
5.4.4 Deutsch's Problem ............................... 181
5.4.5 Quantum Searching and Other Algorithms .......... 184
5.4.6 Prospects for the Future .......................... 185
5.4.7 Entanglement and Mixed States. . . . . . . . . . . . . . . . . . .. 188
5.4.8 The Next Few Years .............................. 188
6. Quantum Networks
and Multi-Particle Entanglement . ........................ 191
6.1 Introduction ........................................... 191
6.2 Quantum Networks I:
Entangling Particles at Separate Locations ................ 192
6.2.1 Interfacing Atoms and Photons .................... 192
6.2.2 Model of Quantum State Transmission .............. 193
6.2.3 Laser Pulses for Ideal Transmission ................. 195
6.2.4 Imperfect Operations and Error Correction .......... 197
6.3 Multi-Particle Entanglement ............................. 197
6.3.1 Greenberger-Horne--Zeilinger states ................. 197
6.3.2 The Conflict with Local Realism ................... 198
6.3.3 A Source for Three-Photon GHZ Entanglement ...... 200
6.3.4 Experimental Proof of GHZ Entanglement ........... 204
6.3.5 Experimental Test of Local Realism
Versus Quantum Mechanics ........................ 206
6.4 Entanglement Quantification ............................. 210
6.4.1 Schmidt Decomposition and von Neumann Entropy ... 2lO
6.4.2 Purification Procedures ........................... 212
6.4.3 Conditions for Entanglement Measures .............. 214
6.4.4 Two Measures of Distance Between Density Matrices . 216
6.4.5 Numerics for Two Spin 1/2 Particles ................ 217
6.4.6 Statistical Basis of Entanglement Measure ........... 219
7. Decoherence and Quantum Error Correction . ............ 221
7.1 Introduction ........................................... 221
7.2 Decoherence ........................................... 222
7.2.1 Decoherence: Entanglement Between
Qubits and Environment .......................... 222
7.2.2 Collective Interaction and Scaling .................. 224
7.2.3 Subspace Decoupled From Environment ............. 225
7.2.4 Other Find of Couplings .......................... 225
7.3 Limits to Quantum Computation Due to Decoherence ....... 227
7.4 Error Correction and Fault-Tolerant Computation .......... 232
7.4.1 Symmetrisation Procedures ........................ 232
7.4.2 Classical Error Correction ......................... 234
7.4.3 General Aspects of Quantum Error Correcting Codes. 236
7.4.4 The Three Qubit Code ............................ 237