Quantum Measurement (eBook)

Preface 6
Contents 8
1 Introduction 14
1.1 Background and Content 14
1.2 Statistical Duality---an Outline 18
References 22
Part I Mathematics 24
2 Rudiments of Hilbert Space Theory 25
2.1 Basic Notions and the Projection Theorem 25
2.2 The Fréchet--Riesz Theorem and Bounded Linear Operators 29
2.3 Strong, Weak, and Monotone Convergence of Nets of Operators 32
2.4 The Projection Lattice mathcalP(mathcalH) 34
2.5 The Square Root of a Positive Operator 36
2.6 The Polar Decomposition of a Bounded Operator 38
2.7 Orthonormal Sets 39
2.8 Direct Sums of Hilbert Spaces 43
2.9 Tensor Products of Hilbert Spaces 44
2.10 Exercises 46
Reference 48
3 Classes of Compact Operators 49
3.1 Compact and Finite Rank Operators 49
3.2 The Spectral Representation of Compact Selfadjoint Operators 52
3.3 The Hilbert--Schmidt Operator Class mathcalHS(mathcalH) 57
3.4 The Trace Class mathcalT(mathcalH) 59
3.5 Connection of the Ideals mathcalT(mathcalH) and mathcalHS(mathcalH) with the Sequence Spaces ell1 and ell2 61
3.6 The Dualities mathcalC(mathcalH)ast = mathcalT(mathcalH) and mathcalT(mathcalH)ast = mathcalL(mathcalH) 64
3.7 Linear Operators on Hilbert Tensor Products and the Partial Trace 67
3.8 The Schmidt Decomposition of an Element of mathcalH1otimesmathcalH2 71
3.9 Exercises 73
4 Operator Integrals and Spectral Representations: The Bounded Case 75
4.1 Classes of Sets and Positive Measures 75
4.2 Measurable Functions 77
4.3 Integration with Respect to a Positive Measure 77
4.4 The Hilbert Space L2(?,mathcalA,?) 80
4.5 Complex Measures and Integration 81
4.6 Positive Operator Measures 83
4.7 Positive Operator Bimeasures 87
4.8 Integration of Bounded Functions with Respect ƒ 92
4.9 The Connection Between (Semi)Spectral Measures and (Semi)Spectral Functions 95
4.10 A Riesz--Markov--Kakutani Type Representation Theorem for Positive Operator Measures 97
4.11 The Spectral Representation of Bounded Selfadjoint Operators 98
4.12 The Spectrum of a Bounded Operator 105
4.13 The Spectral Representations of Unitary and Other Normal Operators 106
4.14 Exercises 108
References 111
5 Operator Integrals and Spectral Representations: The Unbounded Case 113
5.1 Elementary Notes on Unbounded Operators 113
5.2 Integration of Unbounded Functions with Respect to Positive Operator Measures 116
5.3 Integration of Unbounded Functions with Respect to Projection Valued Measures 119
5.4 The Cayley Transform 122
5.5 The Spectral Representation of an Unbounded Selfadjoint Operator 125
5.6 The Support of the Spectral Measure of a Selfadjoint Operator 127
5.7 Applying a Borel Function to a Selfadjoint Operator 129
5.8 One-Parameter Unitary Groups and Stone's Theorem 131
5.9 Taking Stock: Hilbert Space Theory and Its Use in Quantum Mechanics 135
5.10 Exercises 137
References 138
6 Miscellaneous Algebraic and Functional Analytic Techniques 139
6.1 Normal and Positive Linear Maps on mathcalL(mathcalH) 139
6.2 Basic Notions of the Theory of C*-algebras and Their Representations 142
6.3 Algebraic Tensor Products of Vector Spaces 146
6.4 Completions 147
6.5 Exercises 147
References 148
7 Dilation Theory 149
7.1 Completely Positive Linear Maps 149
7.2 A Bilinear Dilation Theorem 151
7.3 The Stinespring and Naimark Dilation Theorems 156
7.4 Normal Completely Positive Operators from mathcalL(mathcalH) into mathcalL(mathcalK) 159
7.5 Naimark Projections of Operator Integrals 164
7.6 Operations and Instruments 165
7.7 Measurement Dilation 171
7.8 Exercises 173
References 174
8 Positive Operator Measures: Examples 175
8.1 The Canonical Spectral Measure and Its Fourier-Plancherel Transform 175
8.2 Restrictions of Spectral Measures 178
8.3 Smearings and Convolutions 180
8.4 Phase Space Operator Measures 184
8.5 Moment Operators and Spectral Measures 187
8.6 Semispectral Measures and Direct Integral Hilbert Spaces 190
8.7 A Dirac Type Formalism: An Elementary Approach 194
8.8 Exercises 197
References 198
Part II Elements 200
9 States, Effects and Observables 201
9.1 States 202
9.2 Effects 206
9.3 Observables 210
9.4 State Changes 218
9.5 Compound Systems 223
9.6 Exercises 231
References 233
10 Measurement 235
10.1 Measurement Schemes 236
10.2 Instruments 240
10.3 Sequential, Joint and Mixed Measurements 242
10.4 Examples of Measurement Schemes 246
10.5 Repeatable Measurements 257
10.6 Ideal Measurements 258
10.7 Correlations, Disturbance and Entanglement 262
10.8 Appendix 265
10.9 Exercises 268
References 269
11 Joint Measurability 271
11.1 Definitions and Basic Results 271
11.2 Alternative Definitions 275
11.3 Regular Observables 276
11.4 Sharp Observables 279
11.5 Compatibility, Convexity, and Coarse-Graining 281
11.6 Exercises 283
References 284
12 Preparation Uncertainty 285
12.1 Indeterminate Values of Observables 286
12.2 Measures of Uncertainty 286
12.3 Examples of Preparation Uncertainty Relations 290
12.4 Exercises 294
References 295
13 Measurement Uncertainty 296
13.1 Conceptualising Error and Disturbance 297
13.2 Comparing Distributions 299
13.3 Error Bar Width 303
13.4 Value Comparison Error 308
13.5 Connections 311
13.6 Unsharpness 312
13.7 Finite Outcome Observables 316
13.8 Appendix 321
13.9 Exercises 323
References 323
Part III Realisations 325
14 Qubits 326
14.1 Qubit States and Observables 326
14.2 Preparation Uncertainty Relations for Qubits 329
14.3 Compatibility of a Pair of Qubit Effects 331
14.4 Excursion: Compatibility of Three Qubit Effects 336
14.5 Approximate Joint Measurements of Qubit Observables 338
14.6 Appendix 347
14.7 Exercises 349
References 350
15 Position and Momentum 351
15.1 The Weyl Pairs 351
15.2 Preparation Uncertainty Relations for Q and P 356
15.3 Approximate Joint Measurements of Q and P 359
15.4 Measuring Q and P with a Single Measurement Scheme 361
15.5 Appendix 364
15.6 Exercises 369
References 370
16 Number and Phase 372
16.1 Covariant Observables 373
16.2 Canonical Phase 377
16.3 Phase Space Phase Observables 384
16.4 Number-Phase Complementarity 386
16.5 Other Phase Theories 389
16.6 Exercises 391
References 392
17 Time and Energy 394
17.1 The Concept of Time in Quantum Mechanics 394
17.2 Time in Nonrelativistic Classical Mechanics 396
17.3 Covariant Time Observables in Nonrelativistic Quantum Mechanics 399
17.4 Exercises 407
References 407
18 State Reconstruction 409
18.1 Informational Completeness 410
18.2 The Pauli Problem 412
18.3 State Reconstruction 417
18.4 Exercises 425
References 427
19 Measurement Implementations 429
19.1 Arthurs--Kelly Model 430
19.2 Photon Detection, Phase Shifters and Beam Splitters 437
19.3 Balanced Homodyne Detection and Quadrature Observables 443
19.4 Eight-Port Homodyne Detection and Phase Space Observables 447
19.5 Eight-Port Homodyne Detection and Phase Observables 451
19.6 Mach--Zehnder Interferometer 456
19.7 Exercises 465
References 465
Part IV Foundations 467
20 Bell Inequalities and Incompatibility 468
20.1 Bell Inequalities and Compatibility: General Observations 468
20.2 Bell Inequalities and Joint Probabilities 470
20.3 Bell Inequality Violation and Nonlocality 473
20.4 Bell Inequality Violation and Incompatibility 474
20.5 Exercises 478
References 478
21 Measurement Limitations Due to Conservation Laws 480
21.1 Measurement of Spin Versus Angular Momentum Conservation 480
21.2 The Yanase Condition 482
21.3 The Wigner--Araki--Yanase Theorem 482
21.4 A Quantitative Version of the WAY Theorem 484
21.5 Position Measurements Obeying Momentum Conservation 485
21.6 A Measurement-Theoretic Interpretation of Superselection Rules 488
21.7 Exercises 490
References 490
22 Measurement Problem 492
22.1 Preliminaries 492
22.2 Reading of Pointer Values 494
22.3 The Problem of Objectification 496
22.4 Exercises 499
References 499
23 Axioms for Quantum Mechanics 501
23.1 Statistical Duality and Its Representation 502
23.2 Quantum Logic 509
23.3 Filters and the Projection Postulate 516
23.4 Hilbert Space Coordinatisation 520
23.5 The Role of Symmetries in the Representation Theorem 526
23.6 The Case of the Complex Field 532
23.7 Exercises 534
References 536