White Noise Analysis And Quantum Information

Luigi Accardi, Louis H Y Chen;Takeyuki Hida;Masanori Ohya;Si SiNoboru Watanabe

• Publisher: WSPC
• Year: 2017
• eBook ISBN: 9789813225473
• Topic: Mathematics
• Subtopic: Computer Science General
• Index: Mathematics

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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.

--> Contents:

• Extensions of Quantum Theory Canonically Associated to Classical Probability Measures (Luigi Accardi)
• Hida Distribution Construction of Indefinite Metric (ϕ p ) d ( d ≥ 4) Quantum Field Theory (Sergio Albeverio and Minoru W Yoshida)
• A Mathematical Realization of von Neumann's Measurement Scheme (Masanari Asano, Masanori Ohya and Yuta Yamamori)
• On Random White Noise Processes with Memory for Time Series Analysis (Christopher C Bernido and M Victoria Carpio-Bernido)
• Self-Repelling (Fractional) Brownian Motion: Results and Open Questions (Jinky Bornales and Ludwig Streit)
• Normal Approximation for White Noise Functionals by Stein's Method and Hida Calculus (Louis H Y Chen, Yuh-Jia Lee and Hsin-Hung Shih)
• Sensitive Homology Searching Based on MTRAP Alignment (Toshihide Hara and Masanori Ohya)
• Some of the Future Directions of White Noise Theory (Takeyuki Hida)
• Local Statistics for Random Selfadjoint Operators (Peter D Hislop and Maddaly Krishna)
• Multiple Markov Properties of Gaussian Processes and Their Control (Win Win Htay)
• Quantum Stochastic Differential Equations Associated with Square of Annihilation and Creation Processes (Un Cig Ji and Kalyan B Sinha)
• Itô Formula for Generalized Real and Complex White Noise Functionals (Yuh-Jia Lee)
• Quasi Quantum Quadratic Operators of 𝕄 2 (ℂ) (Farrukh Mukhamedov)
• New Noise Depending on the Space Parameter and the Concept of Multiplicity (Si Si)
• A Hysteresis Effect on Optical Illusion and Non-Kolmogorovian Probability Theory (Masanari Asano, Andrei Khrennikov, Masanori Ohya and Yoshiharu Tanaka)
• Note on Entropy-Type Complexity of Communication Processes (Noboru Watanabe)

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--> Readership: Mathematicians, physicists, biologists, and information scientists as well as advanced undergraduates, and graduate students studying in these fields. All researchers interested in the study of Quantum Information and White Noise Theory. -->
Keywords:White Noise Analysis;Quantum Information;Quantum Probability;Bioinformatics;Genes;Adaptive Dynamics;Entanglement;Quantum Entropy;Non-Kolmogorovian Probability;Infinite Dimensional AnalysisReview: Key Features:

• Mainly focused on quantum information theory and white noise analysis in line with the fields of infinite dimensional analysis and quantum probability
• White noise analysis is in a leading position of the analysis on modern stochastic analysis, and this volume contains contributions to the development of these new exciting directions

NORMAL APPROXIMATION FOR WHITE NOISE FUNCTIONALS BY STEIN'S METHOD AND HIDA CALCULUS

Louis H. Y. Chen

Department of Mathematics
National University of Singapore
10 Lower Kent Ridge Road, Singapore 119076
[email protected]

Yuh-Jia Lee

Department of Applied Mathematics
National University of Kaohsiung
Kaohsiung, Taiwan 811
[email protected]

Hsin-Hung Shih

Department of Applied Mathematics
National University of Kaohsiung
Kaohsiung, Taiwan 811
[email protected]

In this chapter, we establish a framework for normal approximation for white noise functionals by Stein’s method and Hida calculus. Our work is inspired by that of Nourdin and Peccati (Probab. Theory Relat. Fields 145 (2009), 75-118), who combined Stein’s method and Malliavin calculus for normal approximation for functionals of Gaussian processes.

1.Introduction

Stein’s method, introduced by C. Stein [39] in his 1972 paper, is a powerful way of determining the accuracy of normal approximation to the distribution of a sum of dependent random variables. It has been extended to approximations by a broad class of other probability distributions such as the Poisson, compound Poisson, and Gamma distributions, and to approximations on finite as well as infinite dimensional spaces. The results of these approximations have ...

Table of contents

• Cover
• Halftitle
• Series Editors
• Title
• Coyright
• Dedication
• Foreword
• Preface
• Contents
• Extensions of Quantum Theory Canonically Associated to Classical Probability Measures
• Hida Distribution Construction of Indefinite Metric (ϕp)d (d ≥ 4) Quantum Field Theory
• A Mathematical Realization of von Neumann’s Measurement Scheme
• On Random White Noise Processes with Memory for Time Series Analysis
• Self-Repelling (Fractional) Brownian Motion Results and Open Questions
• Normal Approximation for White Noise Functionals by Stein’s Method and Hida Calculus
• Sensitive Homology Searching Based on MTRAP Alignment
• Some of the Future Directions of White Noise Theory
• Local Statistics for Random Selfadjoint Operators
• Multiple Markov Properties of Gaussian Processes and Their Control
• Quantum Stochastic Differential Equations Associated with Square of Annihilation and Creation Processes
• Itô Formula for Generalized Real and Complex White Noise Functionals
• Quasi Quantum Quadratic Operators of M2(C)
• New Noise Depending on the Space Parameter and the Concept of Multiplicity
• A Hysteresis Effect on Optical Illusion and Non-Kolmogorovian Probability Theory
• Note on Entropy-Type Complexity of Communication Processes