Probabilistic and statistical aspects of quantum theory (Statistics & Probability), 1982

Preface ......Page 3
Table of Contentsr ......Page 8
1. States and measurements ......Page 11
2. Some facts about convex sets ......Page 16
3. Definition of a statistical model ......Page 24
4. The classical statistical model ......Page 25
5. Reduction of statistical model. Classical model with a restricted class of measurements ......Page 30
6. The statistical model of quantum mechanics ......Page 37
7. On the problem of “hidden variables” ......Page 43
8. Comments ......Page 48
1. Operators in a Hilbert space ......Page 53
2. Quantum states and measurements ......Page 60
3. Spectral representation of bounded operators ......Page 62
4. Spectral representation of unbounded operators ......Page 67
5. On realization of measurement ......Page 74
6. Uncertainty relations and compatibility ......Page 78
7. Trace-class operators and Hilbert-Schmidt operators ......Page 83
8. L2 spaces associated with a quantum state ......Page 91
9. Uncertainty relations for measurements with finite second moments ......Page 97
10. Matrix representation of square-summable operators. The commutation operator of a state ......Page 100
11. Comments ......Page 105
1. Statistical model and Galilean relativity ......Page 109
2. One-parameter shift groups and uncertainty relations ......Page 114
3. Kinematics of a quantum particle in one dimension ......Page 119
4. Uniqueness theorem. The Schrodinger and the momentum representations ......Page 124
5. Minimum-uncertainty states. The completeness relation ......Page 127
6. Joint measurements of coordinate and velocity ......Page 130
7. Dynamics of a quantum particle in one dimension ......Page 136
8. Time observable. The \"time-energy\" uncertainty relation ......Page 140
9. Quantum oscillator and phase measurement ......Page 147
10. The coherent-state representation ......Page 153
11. Representations of the rotation group and angular momenta ......Page 158
12. Measuring the angle of rotation ......Page 164
13. Comments ......Page 169
1. Parametric symmetry groups and covariant measurements ......Page 173
2. Structure of covariant measurements ......Page 176
3. The covariant quantum estimation problem ......Page 179
4. Measurements of angular parameters ......Page 184
5. Uncertainty relations for angular quantities ......Page 189
6. Covariant measurements of angular parameter in the case of arbitrary representation of the group T ......Page 193
7. Covariant measurements of a shift parameter ......Page 198
8. The case of irreducible representation ......Page 207
9. Estimation of pure state ......Page 213
10. Measuring parameters of orientation ......Page 218
11. Comments ......Page 222
1. Quasiclassical states of the quantum oscillator ......Page 225
2. The CCR for many degrees of freedom ......Page 229
3. Proof of the Stone-von Neumann uniqueness theorem. The Weyl transform ......Page 233
4. Characteristic function and moments of state ......Page 241
5. Structure of general Gaussian states ......Page 249
6. A characteristic property of Gaussian states ......Page 255
7. Comments ......Page 260
1. Quantum communication channel ......Page 263
2. A lower bound for the variance in one-dimensional case ......Page 266
3. The case of shift parameter ......Page 271
4. Estimation of force by measurements over a trial object ......Page 278
5. A bound for the measurement covariance matrix based on symmetric logarithmic derivatives ......Page 284
6. A bound based on right logarithmic derivatives ......Page 288
7. A general bound for the total mean-square deviation ......Page 295
8. Linear measurements ......Page 302
9. Mean-value estimation for Gaussian states ......Page 308
10. Comments ......Page 312
References ......Page 315
Subject index ......Page 321