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

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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 Library of Congress Cataloging·in. Publication Data applied for. Die Deutsche Bibliothek· CIP·Einheitsaufnahme The physics of quantum informat ion : quantum cryptography. quantum teleportation. quantum computation I Dirk Bouwmeester ... (ed.). This work is subject to copyright. AII rights are reserved. whether the whole or part of the material is concerned. specifically the rights of translation. reprinting. reuse of iIIustrations. recitation. broadcasting. reproduction on microfilm or in any other way. and storage in data banks. Duplication of this publicat ion or parts thereof is permitted only under the provisions of the German Copyright Law of September 9. 1965. in its current version. and permission for use must always be obtained Crom Springer-Verlag Berlin Heidelberg GmbH . Violations are Iiable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 2000 Originally published by Springer-Verlag Berlin Heidelberg New York in 2000 Softcover reprint ofthe hardcover Ist edition 2000 The use of general descriptive names. registered names, trademarks. etc. in this publication does not imply. even in the absence of a specific statement. that such names are exempt Crom the relevant protective laws and regu1ations and therefore free for general use. Camera-ready by the editors using a Springe T EX macro package. Cover design: Erich Kirchner. Heidelberg Printed on aCld·free paper 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

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