Quantum Probability and Related Topics: Proceedings of the 30th Conference, Santiago, Chile, 23-28 November 2009 (Qp-Pq: Quantum Probability and White Noise Analysis)

This volume contains current work at the frontiers of research in quantum probability, infinite dimensional stochastic analysis, quantum information and statistics. It presents a carefully chosen collection of articles by experts to highlight the latest developments in those fields. Included in this volume are expository papers which will help increase communication between researchers working in these areas. The tools and techniques presented here will be of great value to research mathematicians, graduate students and applied mathematicians. Title ......Page 3 Copyright ......Page 4 Foreword ......Page 5 Contents ......Page 7 1. Introduction and statement of the problem ......Page 9 2. Random variables in CeHeisc(l) ......Page 12 3. Infinite divisibility ......Page 14 4.1. Positive definite matrices ......Page 15 5. Positive definite kernels ......Page 20 5.1. Schur's lemma ......Page 22 6. Functions of positive definite matrices and kernels ......Page 23 7. Conditionally positive definite kernels ......Page 25 8. Infinitely divisible kernels ......Page 34 9. The Kolmogorov decomposition theorem for C-valued PD kernels ......Page 35 11. Infinitely divisible kernels and Boson Fock spaces ......Page 36 References ......Page 39 1.1. Summary of the mathematical theory ......Page 42 1.1.1. The role of the Hilbert-Schmidt operators ......Page 44 2. Application to Landau levels ......Page 48 2.1. A second representation ......Page 50 3. What for unbounded operators? ......Page 53 References ......Page 58 1. Introduction ......Page 60 2. A non Markovian stochastic Schrodinger equation ......Page 62 3. Random Hamiltonian ......Page 68 4. Projection techniques and closed master equations with memory ......Page 71 References ......Page 74 1. Introduction ......Page 76 2.1. Jacobi matrix and its spectral measure ......Page 78 2.2. The Pascal one-mode type interacting Fock space ......Page 79 3.1. Jacobi fields and spectral theorem ......Page 83 3.2. Pascal Jacobi field and its spectral measure ......Page 85 4. The white noise Pascal process ......Page 90 References ......Page 96 Quantum White Noise Analysis and Quantum Stochastic Equations V. P. Belavkin, J. Heo and U.-C. Ji ......Page 98 1. Introduction ......Page 99 2.1.1. Newton, Wiener and Poisson Ito algebras ......Page 101 2.1.2. Newton-Leibniz differentials as matrices ......Page 102 2.l.3. Matrix implementation of classical Ito algebras ......Page 103 2.1.4. Classical separable Levy-Ito algebras ......Page 104 2.2.1. Quantum matrix-implemented HP-Iti3 algebra ......Page 105 2.2.2. The general affine and Heisenberg Ito algebras ......Page 106 2.2.4. The modular Ito algebras and dual B*-algebras ......Page 108 3.1.1. Quantum adapted processes and martingales ......Page 111 3.1.2. Quantum stochastic integrators and integrands ......Page 113 3.1.3. The associative quantum stochastic covariation ......Page 114 3.2.1. Quantum object-output stochastic processes ......Page 116 3.2.2. Quantum stochastic Schrodinger equation ......Page 117 3.2.3. Quantum stochastic Langevin equation ......Page 118 4. Quantum Hidden Markov Dynamics and Belavkin QS Flows ......Page 119 4.1.1. The generalized Lindblad equation ......Page 120 4.1.2. Unraveling the hidden Markov dynamics ......Page 122 4.1.3. The Belavkin unraveling flow and equation ......Page 123 4.2.1. The standard pairing and marginalization ......Page 124 4.2.2. The dual and reduced QS dynamics ......Page 125 4.2.3. The Belavkin modular master equation ......Page 127 4.2.4. The general QS completely positive dynamics ......Page 129 References ......Page 131 1. Introduction. ......Page 133 2. The constant twisting. ......Page 135 3. The essential spectrum. ......Page 136 4. The discrete spectrum. ......Page 137 References ......Page 138 1. Introduction ......Page 139 2. The decoherence-free subalgebra ......Page 141 3. Characterisation of N(T) ......Page 144 4. A rotation invariant quantum Markov semigroup ......Page 152 References ......Page 154 1. Introduction ......Page 156 2. Main results ......Page 157 3. Applications ......Page 161 3.1. The n-photon absorption emission process ......Page 162 3.2. The asymmetric exclusion QMS ......Page 167 References ......Page 170 1. Introduction ......Page 171 2. The investigated indirect measurement scheme ......Page 173 2.1. Mathematical problem statement ......Page 174 3. Conditional probability density function approaches ......Page 175 3.1. A conditional histogram approach ......Page 176 3.2. Bayesian state filtering ......Page 178 4.1. The martingale generated by the state evolution ......Page 181 4.2. Stopping relative to the initial state ......Page 183 4.3. Simulation investigations ......Page 184 5. Comparison and discussion ......Page 185 6. Conclusions ......Page 186 References ......Page 187 1. Introduction ......Page 189 2. The Fagnola-Rebolledo Criterion ......Page 195 References ......Page 197 1.1. Independence in algebraic probability space ......Page 198 1.2. Free product of algebras and independence ......Page 200 1.3. The additive convolution associated with a product of states ......Page 202 2. Free, Boolean and monotone products in terms of conditionally free product ......Page 203 3. C-monotone product and c-monotone independence ......Page 206 References ......Page 208 1. Preliminaries ......Page 210 2. Generalized white noise functionals and the rotation group ......Page 211 3. A proposal and some of future directions ......Page 219 References ......Page 221 1. Introduction ......Page 222 2. Volterra equations - resolvent approach ......Page 224 3. Consequences of convolution ......Page 226 4. Probabilistic complications ......Page 228 5. Remarks on resolvent approach and perspectives for stochastic Volterra equations ......Page 232 References ......Page 234 1. Entanglement evolution ......Page 235 1.1. Time evolution of a two-atom system ......Page 236 2.1. Vacuum Reservoir ......Page 238 2.2. Thermal Reservoir ......Page 242 3. Entanglement in a common reservoir ......Page 244 3.1. The model ......Page 246 3.3. General Solution for vacuum reservoir. N 0 ......Page 247 3.4. General Solution for N = 0 ......Page 251 3.5. Discussion ......Page 256 3.6. A curious effect. Creation of delayed entanglement ......Page 259 4. Conclusion ......Page 264 References ......Page 266 Introduction ......Page 269 1. A general quantum setting ......Page 271 1.1. From classical Fisher information via measurement ......Page 272 1.2. The linear mapping ]D ......Page 275 2. Fisher information and covariance ......Page 278 3. Extended monotone metrics ......Page 284 4. Skew information ......Page 286 References ......Page 287 1. Complementary subalgebras ......Page 290 2. Characterization of complementarity ......Page 291 3. Complementary decompositions of Mn(C) ......Page 293 References ......Page 294 1. Introduction and preliminaries ......Page 296 2. Invariant faces and sub-harmonic projections ......Page 297 3. Appendix ......Page 301 References ......Page 302 1. Introduction ......Page 304 2. Kernels with values in a C*-algebra ......Page 306 3. Composing PD-kernels? ......Page 308 4. CPD-kernels ......Page 309 5. Positivity, tensor product, and Schur product ......Page 313 6. More Examples ......Page 319 7. Positivity in *-algebras ......Page 324 References ......Page 328 2. Clifford Construction ......Page 331 2.1. Increments ......Page 332 4. Representation ......Page 334 References ......Page 337 Existence of the Fock representation for current algebras of the Galilei algebra / L. Accardi, A. Boukas and J. Misiewicz -- Modular structures and Landau levels / F. Bagarello -- Stochastic Schrodinger equations and memory / A. Barchielli [und weitere] -- On spectral approach to Pascal white noise functionals / A. Barhoumi, H. Ouerdiane and A. Riahi -- Quantum white noise analysis and quantum stochastic equations / V.P. Belavkin, J. Heo and U.-C. Ji -- Spectral analysis for twisted waveguides / P. Briet -- The decoherence-free subalgebra of a quantum Markov semigroup on [symbol](h) / A. Dhahri, F. Fagnola and R. Rebolledo -- A sufficient condition for all invariant states of a QMS to be diagonal / J.C. Garcia, R. Quezada and L. Pantaleon-Martinez -- State estimation methods using indirect measurements / K.M. Hangos and L. Ruppert -- On the classification of invariant state of generic quantum Markov semigroups : The Gaussian gauge invariant case / S. Hachicha -- Independence generalizing monotone and boolean independences / T. Hasebe -- Roles of white noise in stochastic analysis and some of future directions / T. Hida -- On difficulties appearing in the study of stochastic Volterra equations / A. Karczewska -- Entanglement protection and generation in a two-atom system / M. Orszag -- Introduction to quantum Fisher information / D. Petz and C. Ghinea -- Complementary subalgebras in finite quantum systems / D. Petz and A. Szanto -- On the long-time asymptotics of quantum dynamical semigroups / G.A. Raggio and P.R. Zangara -- Hilbert modules-square roots of positive maps / M. Skeide -- Multiparameter quantum stochastic processes / W.J. Spring Contains work at the frontiers of research in quantum probability, infinite dimensional stochastic analysis, quantum information and statistics. This book includes tools and techniques that is of interest to research mathematicians, graduate students and applied mathematicians.