Quantum Probability and Infinite Dimensional Analysis: Proceedings of the 26th Conference

L. Accardi, W. Freudenberg, M. Schurmann9789812708519, 9812708510

This volume contains the latest results in the fields of quantum probability and infinite dimensional analysis. The contributions range from classical probability, ‘pure’ functional analysis and foundations of quantum mechanics to applications in mathematical physics, quantum information theory and modern mathematical finance. This diversity illustrates that research in quantum probability and infinite dimensional analysis is very active and strongly involved in modern mathematical developments and applications.

---

Table of contents :
CONTENTS……Page 8
Foreword……Page 6
A Combinatorial Identity and Its Application to Gaussian Measures L. Accardi, H.-H. Kuo and A. I. Stan……Page 12
1. Introduction……Page 13
2. A Combinatorial Identity……Page 15
3. Standard Gaussian Probability Measure……Page 17
4. Final Comments……Page 22
References……Page 23
1. Introduction……Page 24
2. Preliminaries and notations……Page 26
3. Laplace operators……Page 27
4. Feynman formulas……Page 28
5. Additional remarks……Page 34
References……Page 36
1. Introduction……Page 37
2. The Boson-Fock Case……Page 38
3. The q-Deformed Fock Case……Page 41
References……Page 43
1. Introduction: The Square of the Delta Function……Page 44
2. The Square of the Delta Function Revisited……Page 46
3. The Cube of the Delta Function……Page 49
4. The General Case……Page 51
References……Page 55
1. Introduction……Page 56
2. Preliminaries……Page 57
3. The Dispersion Function……Page 59
4. Convergence of the Rescaled Densities……Page 60
5. The Drift……Page 61
References……Page 63
1. Introduction……Page 64
2. Positive generalized function in two infinite dimensional variables……Page 68
3. Positive operator in L(F (N‘), F (M’)*)……Page 72
References……Page 73
1. Introduction……Page 75
2. Preliminaries about two level systems……Page 77
4. Quality of the estimates……Page 78
5. Bayesian state estimation……Page 79
6. Least squares state estimation……Page 81
8.1. Number of measurements……Page 82
8.2. The length of the Bloch vector……Page 85
9. Conclusion……Page 87
References……Page 88
1.1. Notations and preliminaries……Page 90
1.2.1. Relative and mutual entropies……Page 91
1.3. Continual measurements……Page 92
2.2. The letter states……Page 94
2.3. Probabilities and states derived from 0……Page 95
2.4. The general setup……Page 96
3.2. The state s and the main bound……Page 97
3.3. Quantum information gain……Page 98
References……Page 99
1. Introduction……Page 101
2. q-Symmetric Tensor Product……Page 102
3. Generalized q-Fock Spaces……Page 106
4. Duality Theorems……Page 109
References……Page 111
1. Introduction……Page 113
2.1. Preliminaries……Page 115
2.2. The Quantum Ito Algebm……Page 117
2.3. The Calculus of Quantum Stochastic Flows……Page 121
3.1. Classification of Generators……Page 124
3.2. Dilation……Page 128
4.1. Covariant Quantum Stochastic Flows and Dynamical Expectations……Page 133
4.2. Covariant Flow Generators……Page 134
4.3. Covariant Dilations……Page 136
References……Page 137
1. Introduction……Page 139
2. Results……Page 140
References……Page 144
1. Boolean quantum stochastic differential equation……Page 145
2. Second quantization……Page 150
3. Hudson-Parthasarathy quantum stochastic differential equation……Page 153
4. An example……Page 154
References……Page 155
1. Introduction……Page 156
2. Preliminaries……Page 157
3. Functional integrals corresponding to the Cauchy-Dirichlet problem for the heat equation……Page 160
4. Functional integrals representing solution of the Cauchy Problem for the Schoedinger equation……Page 162
References……Page 164
1. Introduction……Page 167
2. Financial phase-space……Page 169
3. Financial Pilot Wave……Page 171
References……Page 173
1. Introduction……Page 174
2. Interacting Fock spaces……Page 175
3. Central limit theorem for symmetric measures……Page 178
References……Page 182
1. Introduction……Page 184
2. Statement of the problem……Page 185
3. Reducing the problem to an operator norm……Page 186
4. Solution for the qubit case……Page 187
References……Page 191
1. Introduction……Page 192
2.1. Interacting Fock Space……Page 193
2.2. Interaction of Cavity and the External Field……Page 194
2.3. Quantum Stochastic Process……Page 195
3. Modelling of Single QED System……Page 196
3.2. Input-Output Relation of the Open loop System……Page 197
3.3. Transfer Function of the Open-Loop Quantum System……Page 198
4. Mathematical Model of the Feedback Control of the Cavity QED Using Beam Splitter……Page 199
5. Nyquist Stability Analysis of the Quantum Feedback Control System……Page 201
6. Conclusion……Page 203
References……Page 204
1. Preliminaries……Page 207
2. Markov states on linearly ordered sets……Page 209
3. Diagonalizable Markov states……Page 211
4. Non diagonalizable examples of Markov states……Page 214
References……Page 215
1. Orlicz geometry and statistical manifolds……Page 216
2. Quantum Fisher information and uncertainty principle……Page 217
3. Schur-convexity of curvature for statistical models……Page 220
References……Page 221
1. Introduction……Page 223
2. Determinantal probability measures on finite sets……Page 225
3. Determinantal finite point processes……Page 229
References……Page 232
1. Introduction……Page 235
3. Second order process…….Page 236
4. Gaussian case……Page 238
5. Nonlinear case……Page 239
6. Diffusion process……Page 242
References……Page 243
1. Introduction……Page 244
2.1. The abstract Lie algebra alt……Page 245
2.3. Infinite-dimensional eztension of alt……Page 246
3. Appell systems……Page 248
References……Page 251
1. Introduction……Page 252
2. The double time orthogonal dilation [3]……Page 253
3. Stochastic differential equations……Page 254
References……Page 260
2. Generalized Quantum Turing Machine……Page 262
2.1. Computational class for GQTM……Page 265
3. SAT Problem……Page 266
4. SAT algorithm in GQTM……Page 267
References……Page 268
1. Introduction……Page 269
2. Level-truncated action and equations of motion……Page 270
3. Stress tensor, energy conservation, pressure……Page 271
4. Interacting open-closed SFT model……Page 273
5. Arbitrary number of interacting fields……Page 275
References……Page 276
1. Logarithmic Sobolev inequality for the geometric distribution……Page 278
2. Logarithmic Sobolev inequality for an interacting spin system……Page 280
3. Proof of Theorem 2.2……Page 283
References……Page 284
1. Introduction……Page 285
2. Quantum mechanics as a projection of a classical model with the infinite-dimensional state space……Page 286
3. Pure quantum states as Gaussian statistical mixtures……Page 289
4. Pure states as one-dimension projections of spatial white-noise……Page 290
References……Page 291
2. Hyperbolic Hilbert space and hyperbolic Fock space……Page 293
4. Harmonic oscillator in Hyperbolic Quantum Mechanics……Page 294
5. Hyperbolic-quantization of the electromagnetic field……Page 296
References……Page 298
1. Introduction……Page 299
2. Motion in central field with constant radius……Page 300
3.1. Physical energy spectra and corresponding potentials……Page 302
3.2. Energy spectrum for given potentials……Page 303
Acknowledgments……Page 304
References……Page 305
1. Definitions……Page 306
2. Central limit theorem……Page 307
3. Poisson limit theorem……Page 309
References……Page 312
2. The problem of lifting……Page 313
3. Restriction mapping for observables……Page 316
References……Page 317
1. Introduction……Page 319
2. Main results……Page 321
References……Page 328
Thermodynamical Formalism for Quasi-Local C*-Systems and Fermion Grading Symmetry H. Moriya……Page 330
References……Page 333
1. Micro-quantum systems vs. macro-classical systems……Page 334
2. Sectors and order parameters as q-c correspondence……Page 336
3. Intrasectorial structure & MASA as q-c correspondence……Page 337
References……Page 340
Introduction……Page 341
1. Preliminaries……Page 342
2. The Levy Laplacian acting on the Levy functionals……Page 343
3. Extensions of the Levy Laplacian……Page 344
4. Associated infinite dimensional stochastic processes……Page 345
References……Page 347
1. Introduction……Page 349
2. The Symmetric Fock Space……Page 350
3. Generalized Binomial Coefficients……Page 352
4. The Integral-Sigma Lemma in Symmetric Fock Space……Page 353
References……Page 357
Spatial Eo-Semigroups are Restrictions of Inner Automorphism Groups M. Skeide……Page 359
2. Proof of the main theorem and its supplement……Page 362
3. An open problem……Page 364
References……Page 365
1. Poisson noise and its probability distributions……Page 367
2. Characterization of Poisson noise as compared with Gaussian noise……Page 370
3. Fractional power distribution in terms of Poisson noise……Page 373
References……Page 374
1. Notations and Definitions……Page 376
2. The Two Conjectures……Page 377
References……Page 383
1. Introduction……Page 384
2 .1. Attenuation channel……Page 385
3. Ohya Mutual Entropy and Capacity……Page 386
3.1. Quantum capacity……Page 387
4. Quantum Mutual Type Entropies……Page 388
References……Page 389