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Lecture notes series, Institute For Mathematical Sciences

White Noise Analysis and Quantum Information

Luigi Accardi, Louis H. Y. Chen, Takeyuki Hida, Masanori Ohya, Si Si, Noboru Watanabe

Published Apr 18, 2017• 1 citation
Publisher

Abstract

This volume is to pique the interest of many researchers in the fields of infinite dimensional analysis and quantum probability. These fields have undergone increasingly significant developments and have found many new applications, in particular, to classical probability and to different branches of physics. These fields are rather wide and are of a strongly interdisciplinary nature. For such a purpose, we strove to bridge among these interdisciplinary fields in our Workshop on IDAQP and their Applications that was held at the Institute for Mathematical Sciences, National University of Singapore from 3-7 March 2014. Readers will find that this volume contains all the exciting contributions by well-known researchers in search of new directions in these fields.

DOI
10.1142/10582

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# Part 1: # White Noise Functional Approach to Polymer Entanglements (C C Bernido & M V Carpio-Bernido) # White Noise Analysis, Quantum Field Theory, and Topology (A Hahn) # A Topic on Noncanonical Representations of Gaussian Processes (Y Hibino) # Integral Representation of Hilbert-Schimdt Operators on Boson Fock Space (U C Ji) # The Dawn of White Noise Analysis (I Kubo) # White Noise Stochastic Integration (H-H Kuo) # Connes-Hida Calculus and Bismut-Quillen Superconnections (R Leandre & H Ouerdiane) # A Quantum Decomposition of Levy Processes (Y-J Lee & H-H Shih) # Generalized Entanglement and Its Classification (T Matsuoka) # A White Noise Approach to Fractional Brownian Motion (D Nualart) # Adaptive Dynamics in Quantum Information and Chaos (M Ohya) # Micro-Macro Duality in Quantum Physics (I Ojima) # White Noise Measures Associated to the Solutions of Stochastic Differential Equations (H Ouerdiane) # A Remark on Sets of Infinite Dimensional Spaces with Full or Zero Capacity (J Ren & M Rockner) # An Infinite Dimensional Laplacian in White Noise Theory (K Saito) # Invariance of Poisson Noise (Si Si et al.) # Nonequilibrium Steady States with Bose-Einstein Condensates (S Tasaki & T Matsui) # Multidimensional Skew Reflected Diffusions (G Trutnau) # On Quantum Mutual Type Entropies and Quantum Capacity (N Watanabe) # Part 2: # White Noise Calculus and Stochastic Calculus (L Accardi & A Boukas)

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This volume is to pique the interest of many researchers in the fields of infinite dimensional analysis and quantum probability. These fields have undergone increasingly significant developments and have found many new applications, in particular, to classical probability and to different branches of physics. These fields are rather wide and are of a strongly interdisciplinary nature. For such a purpose, we strove to bridge among these interdisciplinary fields in our Workshop on IDAQP and their Applications that was held at the Institute for Mathematical Sciences, National University of Singapore from 3-7 March 2014. Readers will find that this volume contains all the exciting contributions by well-known researchers in search of new directions in these fields.
Some thoughts on the infinite dimensional harmonic analysis, T. Hida and Y.-H. Mimachi a quarter century of white noise theory, H.-H. Kuo the Clark formula of generalized Wiener functionals, Y.-J. Lee and H.-H. Shih a treatment of quantum Baker's map by chaos degree, K. Inoue et al inverse S-transform, Wick product and overcompleteness of exponential vectors, N. Obata a Hausdorff-Young inequality for white noise analysis, H.-H. Kuo et al integral transform and Segal-Bargmann representation associated to q-Charlier polynomials, N. Asai topics on complex Gaussian random fields, S. Si and W.W. Htay quantum information in space and time and theory of stochastic processes, I.V. Volovich notions of independence in algebraic probability theory and set partition statistics, Y. Hashimoto Poisson noise analysis based on the Levy Laplacian, A. Ishikawa et al study on the power of potential fluctuation in living cells, M. Yamanoi some properties of measure-valued processes with singular branching rate, I. Doku. (Part contents)
Quantum theory is one of the most important intellectual developments in the early twentieth century.The confluence of mathematics and quantum physics emerged arguably from von Neumann's seminal work on the spectral theory of linear operators.This volume arose from a twomonth workshop held at the Institute for Mathematical Sciences at the National University of Singapore in July-September 2008 on mathematical physics, focusing specifically on operator algebras in quantum theory.This volume is essentially written for graduate students and young researchers so that they can acquire a gentle introduction to the application of operator algebras to quantum information sciences, chaotic and many-body problems.Several lecture notes delivered during the workshop by experts in the field were specially commissioned for this volume.
The present volume contains contributions of the 32nd International Conference on Quantum Probability and Related Topics held in Levico Terme (Italy) in June 2011 as the annual meeting of the Association for Quantum Probability and Innite Dimensional Analysis (AQPIDA). The\nconference was sponsored by CIRM Bruno Kessler Foundation, Centro V. Volterra and MIUR PRIN 2009 project "Quantum Probability and Applications\nto Information Theory".\nThe goal of the conference was to communicate new results in quantum probability and infinite dimensional analysis and related elds. A wide range of topics was discussed: white noise and higher powers of white\nnoise, quantum Markov semigroups and applications to quantum statistical mechanics, quantum stochastic calculus, stochastic differential equations, quantum Levy processes, product systems and interacting Fock spaces, algebraic probability, realizations of classical processes, quantum probability in bio-systems, complexity, entanglement and measurement showing the highly interdisciplinary activity of the field and deep connections with many areas of mathematics.\nSelected contributions highlighting the latest developments, collected after the conference, refereed by one or more experts, are now presented in this volume that will be of great value for researchers, applied mathematicians and graduate students.
# Part 1: # White Noise Functional Approach to Polymer Entanglements (C C Bernido & M V Carpio-Bernido) # White Noise Analysis, Quantum Field Theory, and Topology (A Hahn) # A Topic on Noncanonical Representations of Gaussian Processes (Y Hibino) # Integral Representation of Hilbert-Schimdt Operators on Boson Fock Space (U C Ji) # The Dawn of White Noise Analysis (I Kubo) # White Noise Stochastic Integration (H-H Kuo) # Connes-Hida Calculus and Bismut-Quillen Superconnections (R Leandre & H Ouerdiane) # A Quantum Decomposition of Levy Processes (Y-J Lee & H-H Shih) # Generalized Entanglement and Its Classification (T Matsuoka) # A White Noise Approach to Fractional Brownian Motion (D Nualart) # Adaptive Dynamics in Quantum Information and Chaos (M Ohya) # Micro-Macro Duality in Quantum Physics (I Ojima) # White Noise Measures Associated to the Solutions of Stochastic Differential Equations (H Ouerdiane) # A Remark on Sets of Infinite Dimensional Spaces with Full or Zero Capacity (J Ren & M Rockner) # An Infinite Dimensional Laplacian in White Noise Theory (K Saito) # Invariance of Poisson Noise (Si Si et al.) # Nonequilibrium Steady States with Bose-Einstein Condensates (S Tasaki & T Matsui) # Multidimensional Skew Reflected Diffusions (G Trutnau) # On Quantum Mutual Type Entropies and Quantum Capacity (N Watanabe) # Part 2: # White Noise Calculus and Stochastic Calculus (L Accardi & A Boukas)