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Information and Entropy in Quantum Theory

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Information and Entropy in Quantum Theory Information and Entropy in Quantum Theory O J E Maroney Ph.D. Thesis Birkbeck College University of London Malet Street London WC1E 7HX arXiv:quant-ph/0411172v1 23 Nov 2004 Abstract Recent developments in quantum computing have revived interest in the notion of information as a foundational principle in physics. It has been suggested that information provides a means of interpreting quantum theory and a means of understanding the role of entropy in thermodynam- ics. The thesis presents a critical examination of these ideas, and contrasts the use of Shannon information with the concept of ’active information’ introduced by Bohm and Hiley. We look at certain thought experiments based upon the ’delayed choice’ and ’quantum eraser’ interference experiments, which present a complementarity between information gathered from a quantum measurement and interference effects. It has been argued that these experiments show the Bohm interpretation of quantum theory is untenable. We demonstrate that these experiments depend critically upon the assumption that a quantum optics device can operate as a measuring device, and show that, in the context of these experiments, it cannot be consistently understood in this way. By contrast, we then show how the notion of ’active information’ in the Bohm interpretation provides a coherent explanation of the phenomena shown in these experiments. We then examine the relationship between information and entropy. The thought experiment connecting these two quantities is the Szilard Engine version of Maxwell’s Demon, and it has been suggested that quantum measurement plays a key role in this. We provide the first complete description of the operation of the Szilard Engine as a quantum system. This enables us to demonstrate that the role of quantum measurement suggested is incorrect, and further, that the use of information theory to resolve Szilard’s paradox is both unnecessary and insufficient. Finally we show that, if the concept of ’active information’ is extended to cover thermal density matrices, then many of the conceptual problems raised by this paradox appear to be resolved. 1 Contents 1 Introduction 10 2 Information and Measurement 13 2.1 ShannonInformation................................ ... 13 2.1.1 Communication.................................. 15 2.1.2 Measurements .................................. 16 2.2 QuantumInformation ................................ .. 17 2.2.1 Quantum Communication Capacity . 17 2.2.2 InformationGain................................. 18 2.2.3 Quantum Information Quantities . 20 2.2.4 Measurement ................................... 22 2.3 QuantumMeasurement ............................... .. 24 2.4 Summary ......................................... 29 3 Active Information and Interference 30 3.1 The Quantum Potential as an Information Potential . ........ 31 3.1.1 Non-locality.................................... 31 3.1.2 Formdependence................................. 32 3.1.3 Active, Passive and Inactive Information . .... 33 3.2 Informationandinterference. ...... 34 3.2.1 Thebasicinterferometer. 35 3.2.2 Whichwayinformation ............................. 37 3.2.3 Welcher-wegdevices ............................... 39 3.2.4 Surrealistic trajectories . 43 3.2.5 Conclusion .................................... 48 3.3 Informationandwhichpathmeasurements . ...... 48 3.3.1 Which path information . 49 3.3.2 Welcher-weg information . 52 3.3.3 Locality and teleportation . 55 3.3.4 Conclusion .................................... 59 2 3.4 Conclusion ........................................ 62 4 Entropy and Szilard’s Engine 64 4.1 StatisticalEntropy ................................. ... 65 4.2 Maxwell’sDemon..................................... 67 4.2.1 Information Acquisition . 68 4.2.2 InformationErasure ............................... 69 4.2.3 ”Demonless” Szilard Engine . 72 4.3 Conclusion ........................................ 76 5 The Quantum Mechanics of Szilard’s Engine 79 5.1 Particleinabox ..................................... 81 5.2 BoxwithCentralBarrier .............................. .. 82 5.2.1 Asymptotic solutions for the HBA, V E .................. 86 ≫ 5.3 MoveablePartition................................... 87 5.3.1 FreePiston .................................... 88 5.3.2 PistonandGasononeside ........................... 90 5.3.3 Piston with Gas on both sides . 93 5.4 Lifting a weight against gravity . ... 98 5.5 ResettingtheEngine................................. 102 5.5.1 InsertingShelves ................................. 103 5.5.2 RemovingthePiston............................... 106 5.5.3 ResettingthePiston ............................... 106 5.6 Conclusions........................................ 111 5.6.1 RaisingCycle................................... 112 5.6.2 LoweringCycle.................................. 113 5.6.3 Summary ..................................... 114 6 The Statistical Mechanics of Szilard’s Engine 116 6.1 StatisticalMechanics................................ .. 117 6.2 Thermalstateofgas ................................ .. 120 6.2.1 Nopartition.................................... 121 6.2.2 Partition raised . 121 6.2.3 ConfinedGas................................... 122 6.2.4 Moving partition . 123 6.3 ThermalStateofWeights ............................. .. 128 6.3.1 Raising and Lowering Weight . 128 6.3.2 InsertingShelf .................................. 131 6.3.3 MeanEnergyofProjectedWeights . 132 3 6.4 Gearing Ratio of Piston to Pulley . 134 6.4.1 Location of Unraised Weight . 135 6.5 TheRaisingCycle .................................... 135 6.6 TheLoweringCycle ................................... 141 6.7 Energy Flow in Popper-Szilard Engine . .. 144 6.8 Conclusion ........................................ 150 7 The Thermodynamics of Szilard’s Engine 152 7.1 FreeEnergyandEntropy ............................. .. 153 7.1.1 OneAtomGas .................................. 154 7.1.2 Weight above height h .............................. 156 7.1.3 Correlations and Mixing . 157 7.2 Raisingcycle ....................................... 160 7.3 LoweringCycle ...................................... 165 7.4 Conclusion ........................................ 168 8 Resolution of the Szilard Paradox 169 8.1 TheRoleoftheDemon ................................. 170 8.1.1 TheRoleofthePiston.............................. 170 8.1.2 Maxwell’sDemons ................................ 172 8.1.3 The Significance of Mixing . 173 8.1.4 GeneralisedDemon................................ 179 8.1.5 Conclusion .................................... 183 8.2 Restoring the Auxiliary . 184 8.2.1 Fluctuation Probability Relationship . 184 8.2.2 ImperfectResetting ............................... 187 8.2.3 TheCarnotCycleandtheEntropyEngine . 199 8.2.4 Conclusion .................................... 201 8.3 Alternative resolutions . .. 202 8.3.1 Information Acquisition . 202 8.3.2 InformationErasure ............................... 204 8.3.3 ’Free will’ and Computation . 207 8.3.4 Quantumsuperposition ............................. 210 8.4 CommentsandConclusions ............................. 215 8.4.1 Criticisms of the Resolution . 215 8.4.2 Summary ..................................... 217 9 Information and Computation 219 9.1 Reversible and tidy computations . .... 219 4 9.1.1 LandauerErasure ................................ 220 9.1.2 Tidy classical computations . 223 9.1.3 Tidyquantumcomputations .. .. .. .. .. .. .. .. .. .. .. 224 9.1.4 Conclusion .................................... 227 9.2 Thermodynamic and logical reversibility . .... 227 9.2.1 Thermodynamically irreversible computation . 228 9.2.2 Logically irreversible operations . 228 9.3 Conclusion ........................................ 230 10 Active Information and Entropy 232 10.1 TheStatisticalEnsemble. .... 232 10.2TheDensityMatrix .................................. 235 10.2.1 SzilardBox .................................... 236 10.2.2 CorrelationsandMeasurement . 237 10.3ActiveInformation ................................. .. 238 10.3.1 TheAlgebraicApproach. 239 10.3.2 CorrelationsandMeasurement . 242 10.4Conclusion ........................................ 248 A Quantum State Teleportation 250 A.1 Introduction...................................... 250 A.2 QuantumTeleportation ............................... 251 A.3 Quantum State Teleportation and Active Information . ........ 252 A.4 Conclusion ........................................ 255 B Consistent histories and the Bohm approach 257 B.1 Introduction...................................... 257 B.2 Historiesandtrajectories . .. .. .. .. .. .. .. .. .. .. .. .... 258 B.3 Theinterferenceexperiment . ..... 260 B.4 Conclusion ........................................ 264 C Unitary Evolution Operators 265 D Potential Barrier Solutions 268 D.1 Oddsymmetry ...................................... 270 D.1.1 E > V ....................................... 270 D.1.2 E = V ....................................... 271 D.1.3 E < V ....................................... 272 D.1.4 Summary ..................................... 273 D.2 Evensymmetry..................................... 273 5 D.2.1 E > V ...................................... 273 D.2.2 E = V ....................................... 274 D.2.3 E < V ....................................... 275 D.2.4 Summary ..................................... 276 D.3 Numerical Solutions to Energy Eigenvalues . ..... 276 E Energy of Perturbed Airy Functions 279 F Energy Fluctuations 282 G Free Energy and Temperature 285 H Free Energy and Non-Equilibrium
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