Table Of Content

This page intentionally left blank Quantum Processes, Systems, and Information Anewandexcitingapproachtothebasicsofquantumtheory,thisundergraduatetextbook containsextensivediscussionsofconceptualpuzzlesandover800exercisesandproblems. Beginning with three elementary “qubit” systems, the book develops the formalismof quantumtheory, addressesquestionsofmeasurementanddistinguishability, andexplores the dynamics of quantum systems. In addition to the standard topics covered in other textbooks,italsocoverscommunicationandmeasurement,quantumentanglement,entropy andthermodynamics,andquantuminformationprocessing. Thistextbookgivesabroadviewofquantumtheorybyemphasizingdynamicalevolution, and exploring conceptual and foundational issues. It focuses on contemporary topics, includingmeasurement,timeevolution,opensystems,quantumentanglement,andtherole ofinformation. Benjamin Schumacher is Professor of Physics at Kenyon College. He coined the term “qubit” and invented quantum data compression, among other contributions to quantum informationtheory. MichaelD.WestmorelandisProfessorofMathematicsatDenisonUniversity.Trainedas analgebraist,formanyyearshehasresearchednonstandardlogics,modelsofcomputation, andquantuminformationtheory. The authors are long-time research collaborators and have made numerous joint contri- butionstoquantumchannelcapacitytheoremsandotheraspectsofquantuminformation science. Quantum Processes, Systems, and Information BENJAMIN SCHUMACHER KenyonCollege MICHAEL D. WESTMORELAND DenisonUniversity CAMBRIDGEUNIVERSITYPRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521875349 © B. Schumacher and M. Westmoreland 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13 978-0-511-68003-8 eBook (EBL) ISBN-13 978-0-521-87534-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page ix 1 Bitsandquanta 1 1.1 Informationandbits 1 1.2 Wave–particleduality 5 Problems 12 2 Qubits 15 2.1Thephotonintheinterferometer15 2.2 Spin1/2 28 2.3 Two-levelatoms 36 2.4 Qubitsandisomorphism 43 Problems 45 3 Statesandobservables 47 3.1 Hilbertspace 47 3.2 Operators 54 3.3 Observables 60 3.4 Adjoints 64 3.5 Eigenvaluesandeigenvectors 68 Problems 77 4 Distinguishabilityandinformation 79 4.1 Quantumcommunication 79 4.2 Distinguishability 83 4.3 Theprojectionruleanditslimitations 85 4.4 Quantumcryptography 88 4.5 Theuncertaintyrelation 92 Problems 96 5 Quantumdynamics 98 5.1 Unitaryevolution 98 5.2 TheSchrödingerequation 102 5.3 Quantumclock-making 105 5.4 Operatorsandsymmetries 107 Problems 114 vi Contents 6 Entanglement 117 6.1 Compositesystems 117 6.2 Interactionandentanglement 121 6.3 A4π world 123 6.4 Conditionalstates 126 6.5 EPR 131 6.6 Bell’stheorem 133 6.7 GHZ 136 Problems 137 7 Informationandebits 140 7.1 Decodinganddistinguishability 140 7.2 Theno-cloningtheorem 142 7.3 Ebits 146 7.4 Usingentanglement 148 7.5 Whatisquantuminformation? 151 Problems 155 8 Densityoperators 158 8.1 Beyondstatevectors 158 8.2 Matrixelementsandeigenvalues 163 8.3 Distinguishingmixedstates 166 8.4 TheBlochsphere 168 8.5 Timeevolution 171 8.6 Uniformdensityoperators 173 8.7 Thecanonicalensemble 175 Problems 178 9 Opensystems 182 9.1 Opensystemdynamics 182 9.2 Informationallyisolatedsystems 185 9.3 TheLindbladequation 188 9.4 Heatandwork 191 9.5 Measurementsonopensystems 194 9.6 Informationandopensystems 196 Problems 198 10 Aparticleinspace 202 10.1 Continuousdegreesoffreedom 202 10.2 Continuousobservables 207 10.3 Wavepackets 213 10.4 Reﬂectionandrecoil 216 10.5 Moredimensionsofspace 218 10.6 Hownottothinkaboutψ 220 Problems 221 vii Contents 11 Dynamicsofafreeparticle 224 11.1 Dynamicsin1-D 224 11.2 Freeparticlesin1-D 229 11.3 Particleonacircle 233 11.4 Particleinabox 235 11.5 Quantumbilliards 239 Problems 243 12 Spinandrotation 247 12.1 Spin-ssystems 247 12.2 Orbitalangularmomentum 254 12.3 Rotation 257 12.4 Addingspins 260 12.5 Isospin 264 Problems 266 13 Laddersystems 268 13.1 Raisingandloweringoperators 268 13.2 Oscillators 270 13.3 Coherentstates 275 13.4 Thermalstatesofaladdersystem 278 Problems 280 14 Manyparticles 282 14.1 Two-particlewavefunctions 282 14.2 Centerofmassandrelativecoordinates 284 14.3 Identicalparticles 288 14.4 Energylevels 293 14.5 Exchangeeffects 295 14.6 Occupationnumbers 298 Problems 304 15 Stationarystatesin1-D 306 15.1 Wavefunctionsandpotentials 306 15.2 Reﬂecting,scattering,andboundstates 311 15.3 Apotentialstep 315 15.4 Scatteringfromasquarebarrier 318 15.5 Boundstatesinasquarewell 322 15.6 Thevariationalmethod 326 15.7 Parametersandscaling 329 Problems 332 16 Boundstatesin3-D 335 16.1 Centralpotentials 335 16.2 Theisotropicoscillator 338 viii Contents 16.3 Hydrogen 341 16.4 Someexpectations 345 Problems 347 17 Perturbationtheory 349 17.1 Shiftingtheenergylevels 349 17.2 Dealingwithdegeneracy 352 17.3 Perturbingthedynamics 353 17.4 Cross-sections 359 Problems 364 18 Quantuminformationprocessing 366 18.1 Quantumcircuits 366 18.2 Computersandalgorithms 371 18.3 Nuclearspins 375 18.4 NMRintherealworld 381 18.5 Pulsesequences 384 Problems 387 19 Classicalandquantumentropy 390 19.1 Classicalentropy 390 19.2 Classicaldatacompression 395 19.3 Quantumentropy 398 19.4 Quantumdatacompression 403 19.5 Entropyandthermodynamics 407 19.6 Bitsandwork 411 Problems 416 20 Errorcorrection 419 20.1 Undoingerrors 419 20.2 Classicalcommunicationanderrorcorrection 420 20.3 Quantumcommunicationanderrorcorrection 423 20.4 Error-correctingcodes 427 20.5 Informationandisolation 432 Problems 435 Appendix A Probability 437 Appendix B Fourierfacts 444 Appendix C Gaussianfunctions 451 Appendix D Generalizedevolution 453 Index 463