Quantum Probability and Infinite Dimensional Analysis - Proceedings of the 26th Conference (Qp-Pq: Quantum Probability and White Noise Analysis)

CONTENTS Foreword A Combinatorial Identity and Its Application to Gaussian Measures L. Accardi, H.-H. Kuo and A. I. Stan 1. Introduction 2. A Combinatorial Identity 3. Standard Gaussian Probability Measure 4. Final Comments References Feynman Formulas for Evolution Equations with Levy Laplacians on Manifolds L. Accardi and 0. G. Smolyanov 1. Introduction 2. Preliminaries and notations 3. Laplace operators 4. Feynman formulas 5. Additional remarks References On the Fock Representation of the Renormalized Powers of Quantum White Noise L. Accardi and A. Boukas 1. Introduction 2. The Boson-Fock Case 3. The q-Deformed Fock Case References Powers of the Delta Function L. Accardi and A . Boukas 1. Introduction: The Square of the Delta Function 2. The Square of the Delta Function Revisited 3. The Cube of the Delta Function 4. The General Case References Dispersion Relations in the Stochastic Limit of Quantum Theory L. Accardi and F. G. Cubillo 1. Introduction 2. Preliminaries 3. The Dispersion Function 4. Convergence of the Rescaled Densities 5. The Drift Acknowledgements References Integral Representation of Positive Operator on Infinite Dimensional Space of Entire Functions W. Ayed and H. Ouerdiane 1. Introduction 2. Positive generalized function in two infinite dimensional variables 3. Positive operator in L(F (N‘), F (M’)*) References Comparison of Some Methods of Quantum State Estimation Th. Baier, D. Petz, K. M. Hangos and A. Magyar 1. Introduction 2. Preliminaries about two level systems 3. Measurements on qubits 4. Quality of the estimates 5. Bayesian state estimation 6. Least squares state estimation 7. Simulation experiments 8. Results of the experiments 8.1. Number of measurements 8.2. The length of the Bloch vector 9. Conclusion References Entropic Bounds and Continual Measurements A. Barchielli and G. Lupieri 1. Introduction 1.1. Notations and preliminaries 1.1.1. A quantum/classical algebra 1.2. Quantum channels and entropies 1.2.1. Relative and mutual entropies 1.2.2. Channels 1.3. Continual measurements 2. The initial state and the measurement 2.1. Ensembles 2.2. The letter states 2.3. Probabilities and states derived from 0 2.4. The general setup 3. Mutual entropies and informational bounds 3.1. The state qt and the classical information 3.2. The state s and the main bound 3.3. Quantum information gain Acknowledgments References Generalized q-Fock Spaces and Duality Theorems A. Barhoumi and H. Ouerdiane 1. Introduction 2. q-Symmetric Tensor Product 3. Generalized q-Fock Spaces 4. Duality Theorems References Covariant Quantum Stochastic Flows and Their Dilation V. Belavlcin and L. Gregory 1. Introduction 2. Quantum Stochastic Flows 2.1. Preliminaries 2.2. The Quantum Ito Algebm 2.3. The Calculus of Quantum Stochastic Flows 3. Dilation of Quantum Stochastic Flows 3.1. Classification of Generators 3.2. Dilation 4. Covariance 4.1. Covariant Quantum Stochastic Flows and Dynamical Expectations 4.2. Covariant Flow Generators 4.3. Covariant Dilations References Alicki-Fannes and Hudson-Parthasarathy Evolution Equations A . C. R. Belton 1. Introduction 1.1. Conventions 2. Results Acknowledgements References Boson Cocycle as the Second Quantization of the Boolean Cocycle A . Ben Ghorbal and F. Fagnola 1. Boolean quantum stochastic differential equation 2. Second quantization 3. Hudson-Parthasarathy quantum stochastic differential equation 4. An example References Functional Integrals over Smolyanov Surface Measures for Evolutionary Equations on a Riemannian Manifold Ya. A . Butlco 1. Introduction 2. Preliminaries 3. Functional integrals corresponding to the Cauchy-Dirichlet problem for the heat equation 4. Functional integrals representing solution of the Cauchy Problem for the Schoedinger equation Acknowledgments References Quantum Probabilistic Model for the Financial Market 0. Choustova 1. Introduction 2. Financial phase-space 3. Financial Pilot Wave References A New Proof of a Quantum Central Limit Theorem for Symmetric Measures V. Crismale and Y. G. Lu 1. Introduction 2. Interacting Fock spaces 3. Central limit theorem for symmetric measures References On the Most Efficient Unitary Transformation for Programming Quantum Channels G. M. D’Ariano and P. Perinotti 1. Introduction 2. Statement of the problem 3. Reducing the problem to an operator norm 4. Solution for the qubit case References Stability Analysis of Quantum Mechanical Feedback Control System P. K. Das and B. C. Roy 1. Introduction 2. Preliminaries and Notations 2.1. Interacting Fock Space 2.2. Interaction of Cavity and the External Field 2.3. Quantum Stochastic Process 3. Modelling of Single QED System 3.1. State Equation of the Cavity 3.2. Input-Output Relation of the Open loop System 3.3. Transfer Function of the Open-Loop Quantum System 4. Mathematical Model of the Feedback Control of the Cavity QED Using Beam Splitter 5. Nyquist Stability Analysis of the Quantum Feedback Control System 6. Conclusion References Markov States on Quasi-Local Algebras F. Fidaleo 1. Preliminaries 2. Markov states on linearly ordered sets 3. Diagonalizable Markov states 4. Non diagonalizable examples of Markov states References Some Open Problems in Information Geometry P. Gibilisco and T. Isola 1. Orlicz geometry and statistical manifolds 2. Quantum Fisher information and uncertainty principle 3. Schur-convexity of curvature for statistical models References Introduction to Determinantal Point Processes from a Quantum Probability Viewpoint A . D. Gottlieb 1. Introduction 2. Determinantal probability measures on finite sets 3. Determinantal finite point processes 4. Determinantal processes of infinitely many points Acknowledgments References Note on the Time Operator T. Hida 1. Introduction 2. Time operator 3. Second order process. 4. Gaussian case 5. Nonlinear case 6. Diffusion process References On the Dynamical Symmetric Algebra of Ageing: Lie Structure, Representations and Appell Systems M. Henkel, R. Schott, S. Stoimenov and J. Unterberger 1. Introduction 2. A brief perspective on the algebra alt 2.1. The abstract Lie algebra alt 2.2. Central extensions: an introduction 2.3. Infinite-dimensional eztension of alt 2.4. Some results on representations of W 3. Appell systems References An Analytic Double Product Integral R. L. Hudson 1. Introduction 2. The double time orthogonal dilation [3] 3. Stochastic differential equations 4. Epilogue References On Generalized Quantum Turing Machine and Its Application S. Iriyama and M. Ohya 1. Introduction 2. Generalized Quantum Turing Machine 2.1. Computational class for GQTM 3. SAT Problem 4. SAT algorithm in GQTM 4.1. Computational complexity of the SAT algorithm References Dynamics with Infinite Number of Derivatives for Level Truncated Noncommutative Interaction L. Joukovskaya 1. Introduction 2. Level-truncated action and equations of motion 3. Stress tensor, energy conservation, pressure 4. Interacting open-closed SFT model 5. Arbitrary number of interacting fields Acknowledgments References A Logarithmic Sobolev Inequality for an Interacting Spin System Under a Geometric Reference Measure A . Joulin and N. Privault 1. Logarithmic Sobolev inequality for the geometric distribution 2. Logarithmic Sobolev inequality for an interacting spin system 3. Proof of Theorem 2.2 References To Quantum Mechanics through Gaussian Integration and the Taylor Expansion of Functionals of Classical Fields A . Yu. Khrennikov 1. Introduction 2. Quantum mechanics as a projection of a classical model with the infinite-dimensional state space 3. Pure quantum states as Gaussian statistical mixtures 4. Pure states as one-dimension projections of spatial white-noise References Hyperbolic Quantization A . Khrennikov and G. Segre 1. Introduction 2. Hyperbolic Hilbert space and hyperbolic Fock space 3. Hyperbolic Quantum Mechanics 4. Harmonic oscillator in Hyperbolic Quantum Mechanics 5. Hyperbolic-quantization of the electromagnetic field References Discrete Energy Spectrum in Discrete Time Dynamics A . Khrennikov and Ya. Volovich 1. Introduction 2. Motion in central field with constant radius 3. Energy spectrum for various models 3.1. Physical energy spectra and corresponding potentials 3.2. Energy spectrum for given potentials 4. Discussion and conclusions Acknowledgments References Convolution Associated with the Free cosh-Law A . D. Krystek and L. J. Wojakowski 1. Definitions 2. Central limit theorem 3. Poisson limit theorem References A Theorem on Liftings of Statistical Operators J. Kupsch 1. Introduction 2. The problem of lifting 3. Restriction mapping for observables 4. Remarks about Zwanzig projectors References Positive Maps between M2(C) and Mn(C). On Decomposability of Positive Maps between M2(C) and Mn(C) W. A . Majewski and M. Marciniak 1. Introduction 2. Main results References Thermodynamical Formalism for Quasi-Local C*-Systems and Fermion Grading Symmetry H. Moriya References Micro-Macro Duality and an Attempt Towards Measurement Scheme of Quantum Fields I. Ojirna 1. Micro-quantum systems vs. macro-classical systems 2. Sectors and order parameters as q-c correspondence 3. Intrasectorial structure & MASA as q-c correspondence References The Levy Laplacian Acting on Some Class of Levy Functionals K. Saito and A . H. Tsoi Introduction 1. Preliminaries 2. The Levy Laplacian acting on the Levy functionals 3. Extensions of the Levy Laplacian 4. Associated infinite dimensional stochastic processes Acknowledgments References On the General Form of the Integral-sigma Lemma in Symmetric Fock Space K. Schubert 1. Introduction 2. The Symmetric Fock Space 3. Generalized Binomial Coefficients 4. The Integral-Sigma Lemma in Symmetric Fock Space References Spatial Eo-Semigroups are Restrictions of Inner Automorphism Groups M. Skeide 1. Units and inductive limits 2. Proof of the main theorem and its supplement 3. An open problem References A Characterization of Poisson Noise 1. Poisson noise and its probability distributions 2. Characterization of Poisson noise as compared with Gaussian noise 3. Fractional power distribution in terms of Poisson noise 4. Concluding remark References On Two Conjectures in Segal-Bargmann Analysis S. B. Sontz 1. Notations and Definitions 2. The Two Conjectures 3. Concluding Remarks References Note on Quantum Mutual Type Entropies and Capacity N. Watanabe 1. Introduction 2. Quantum Channels 2 .1. Attenuation channel 3. Ohya Mutual Entropy and Capacity 3.1. Quantum capacity 3.2. Semi-classical mutual entropy 4. Quantum Mutual Type Entropies References