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QP-PQ, quantum probability and white noise analysis

Quantum Probability and Related Topics

Luigi Accardi, Franco Fagnola

Published Jan 1, 2013• 27 citations
Publisher

Abstract

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.

DOI
10.1142/8680

Related papers

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Existence of the Fock Representation for Current Algebras of the Galilei Algebra (L Accardi et al.) Modular Structures and Landau Levels (F Bagarello) On Spectral Approach to Pascal White Noise Functionals (A Barhoumi) Spectral Analysis for Twisted Waveguides (P Briet) Classification of Invarient State of Generic Quantum Markov Semigroups: The Gaussian Gauge (S Hachiha) On Difficulties Appearing in the Study of Stochastic Volterra Equations (A Karczewska) Entanglement Protection and Generation in Two Atom System (M Orszag) Hilbert Molecules - Square Roots of Positive Maps (M Skeide) Multiparameter Quantum Stochastic Processes (J Spring) and other papers.

WORLD SCIENTIFIC eBooks•Jan 1, 2011•0 citations

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WORLD SCIENTIFIC eBooks•Jan 1, 2010•0 citations

White Noise Analysis and Quantum Information

Luigi Accardi, Louis H. Y. Chen, Takeyuki Hida +3 more

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.

Lecture notes series, Institute For Mathematical Sciences•Apr 18, 2017•1 citation

Quantum Stochastics

Mou-Hsiung Chang

The classical probability theory initiated by Kolmogorov and its quantum counterpart, pioneered by von Neumann, were created at about the same time in the 1930s, but development of the quantum theory has trailed far behind. Although highly appealing, the quantum theory has a steep learning curve, requiring tools from both probability and analysis and a facility for combining the two viewpoints. This book is a systematic, self-contained account of the core of quantum probability and quantum stochastic processes for graduate students and researchers. The only assumed background is knowledge of the basic theory of Hilbert spaces, bounded linear operators, and classical Markov processes. From there, the book introduces additional tools from analysis, and then builds the quantum probability framework needed to support applications to quantum control and quantum information and communication. These include quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups and processes, and large-time asymptotic behavior of quantum Markov semigroups.

Cambridge University Press eBooks•Feb 5, 2015•8 citations

Introduction and Summary

Mou-Hsiung Chang

It is widely known that the commutative probability theory based on Kolmogorov's measure theoretic axioms and the quantum probability theory based on von Neumann's postulates were both created at about the same time in the 1930s (see, e.g., Kolmogorov [Kol50] and von Neumann [vNeu55] for the origins of these 2 very different but parallel lines of theory). Following Kolmogorov's original work, theory of the classical Markov processes has been the subject of intensive research for the last few decades. The classical theory, its applications, and their connections to other areas of research have been systematically developed by many prominent probabilists as illuminated by their widely referenced monographs such as those of Dynkin [Dyn62], Blumenthal and Getoor [BG68], Ethier and Kurtz [EK85], Renuv and Yor [RY99], and others. On the other hand, development of a complete theory of quantum stochastics such as quantum Markov processes has been progressing at a much slower pace in comparison with that of its classical counterpart. This is perhaps because development of a theory of quantum stochastics and quantum Markov processes requires an unusually large number of tools from operator theory and perhaps also because the probabilistic and analytical tools for understanding sample path behaviors of quantum stochastic processes have yet to be developed. The main goal of this monograph is to give a systematic exploration of relevant topics in quantum Markov processes in hopes of stimulating further research along this line and of stimulating interest in the classical stochastics community for research in its quantum counterpart. This monograph is written largely based on the current account of relevant research results by widely surveying relevant results contributed by many prominent researchers in quantum probability, quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups, strong quantum Markov processes, and large time asymptotic behaviors of quantum Markov semigroups through a systematic and self-contained introduction/presentation of these very interesting topics in an attempt to illuminate the rigor and beauty of quantum stochastics and quantum Markov processes.

Cambridge University Press eBooks•Feb 9, 2015•0 citations

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Luigi Accardi, Wolfgang Freudenberg, Michael Schürmann

A Combinatorial Identity and Its Application to Gaussian Measure (L Accardi, H-H Kuo & A I Stan) Integral Representation of Positive Operator on Infinite Dimensional Space of Entire Functions (W Ayed & H Ouerdiane) Comparison of Some Methods of Quantum State Estimation (D Petz et al.) Entropic Bounds and Continual Measurements (A Barchielli & G Lupieri) Covariant Quantum Stochastic Flows and Their Dilation (V Belavkin & L Gregory) Alicki-Fannes and Hudson-Parthasarathy Evolution Equations (A C R Belton) Functional Integrals Over Smolyanov Surface Measures for Evolution Equations in a Riemannian Manifold (Ya A Butko) Quantum Probabilistic Model for the Financial Market (O Choustova) Note on the Time Operator (T Hida) An Analytic Double Product Integral (R L Hudson) To Quantum Mechanics Through Gaussian Integration and the Taylor Expansion of Functionals of Classical Fields (A Yu Khrennikov) A Theorem on Liftings of Statistical Operators (J Kupsch) Thermodynamical Formalism for Quasi-Local C*-Systems and Fermion Grading Symmetry (H Moriya) Micro-Macro Duality and an Attempt Towards Measurement Scheme of Quantum Fields (I Ojima) The Levy Laplacian Acting on Some Class of Levy Functionals (K Saito & A H Tsoi) On the General Form of the Integral-Sigma Lemma in Symmetric Fock Space (K Schubert) Spatial E0-Semigroups are Restrictions of Inner Automorphism Groups (M Skeide) A Characterization of Poisson Noise (Si Si) On Two Conjectures in Segal-Bargmann Analysis (S B Sontz) Note on Quantum Mutual Type Entropies and Capacity (N Watanabe) and other papers.

WORLD SCIENTIFIC eBooks•Jan 1, 2007•1 citation
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.
Existence of the Fock Representation for Current Algebras of the Galilei Algebra (L Accardi et al.) Modular Structures and Landau Levels (F Bagarello) On Spectral Approach to Pascal White Noise Functionals (A Barhoumi) Spectral Analysis for Twisted Waveguides (P Briet) Classification of Invarient State of Generic Quantum Markov Semigroups: The Gaussian Gauge (S Hachiha) On Difficulties Appearing in the Study of Stochastic Volterra Equations (A Karczewska) Entanglement Protection and Generation in Two Atom System (M Orszag) Hilbert Molecules - Square Roots of Positive Maps (M Skeide) Multiparameter Quantum Stochastic Processes (J Spring) and other papers.
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.
The classical probability theory initiated by Kolmogorov and its quantum counterpart, pioneered by von Neumann, were created at about the same time in the 1930s, but development of the quantum theory has trailed far behind. Although highly appealing, the quantum theory has a steep learning curve, requiring tools from both probability and analysis and a facility for combining the two viewpoints. This book is a systematic, self-contained account of the core of quantum probability and quantum stochastic processes for graduate students and researchers. The only assumed background is knowledge of the basic theory of Hilbert spaces, bounded linear operators, and classical Markov processes. From there, the book introduces additional tools from analysis, and then builds the quantum probability framework needed to support applications to quantum control and quantum information and communication. These include quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups and processes, and large-time asymptotic behavior of quantum Markov semigroups.
It is widely known that the commutative probability theory based on Kolmogorov's measure theoretic axioms and the quantum probability theory based on von Neumann's postulates were both created at about the same time in the 1930s (see, e.g., Kolmogorov [Kol50] and von Neumann [vNeu55] for the origins of these 2 very different but parallel lines of theory). Following Kolmogorov's original work, theory of the classical Markov processes has been the subject of intensive research for the last few decades. The classical theory, its applications, and their connections to other areas of research have been systematically developed by many prominent probabilists as illuminated by their widely referenced monographs such as those of Dynkin [Dyn62], Blumenthal and Getoor [BG68], Ethier and Kurtz [EK85], Renuv and Yor [RY99], and others. On the other hand, development of a complete theory of quantum stochastics such as quantum Markov processes has been progressing at a much slower pace in comparison with that of its classical counterpart. This is perhaps because development of a theory of quantum stochastics and quantum Markov processes requires an unusually large number of tools from operator theory and perhaps also because the probabilistic and analytical tools for understanding sample path behaviors of quantum stochastic processes have yet to be developed. The main goal of this monograph is to give a systematic exploration of relevant topics in quantum Markov processes in hopes of stimulating further research along this line and of stimulating interest in the classical stochastics community for research in its quantum counterpart. This monograph is written largely based on the current account of relevant research results by widely surveying relevant results contributed by many prominent researchers in quantum probability, quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups, strong quantum Markov processes, and large time asymptotic behaviors of quantum Markov semigroups through a systematic and self-contained introduction/presentation of these very interesting topics in an attempt to illuminate the rigor and beauty of quantum stochastics and quantum Markov processes.
A Combinatorial Identity and Its Application to Gaussian Measure (L Accardi, H-H Kuo & A I Stan) Integral Representation of Positive Operator on Infinite Dimensional Space of Entire Functions (W Ayed & H Ouerdiane) Comparison of Some Methods of Quantum State Estimation (D Petz et al.) Entropic Bounds and Continual Measurements (A Barchielli & G Lupieri) Covariant Quantum Stochastic Flows and Their Dilation (V Belavkin & L Gregory) Alicki-Fannes and Hudson-Parthasarathy Evolution Equations (A C R Belton) Functional Integrals Over Smolyanov Surface Measures for Evolution Equations in a Riemannian Manifold (Ya A Butko) Quantum Probabilistic Model for the Financial Market (O Choustova) Note on the Time Operator (T Hida) An Analytic Double Product Integral (R L Hudson) To Quantum Mechanics Through Gaussian Integration and the Taylor Expansion of Functionals of Classical Fields (A Yu Khrennikov) A Theorem on Liftings of Statistical Operators (J Kupsch) Thermodynamical Formalism for Quasi-Local C*-Systems and Fermion Grading Symmetry (H Moriya) Micro-Macro Duality and an Attempt Towards Measurement Scheme of Quantum Fields (I Ojima) The Levy Laplacian Acting on Some Class of Levy Functionals (K Saito & A H Tsoi) On the General Form of the Integral-Sigma Lemma in Symmetric Fock Space (K Schubert) Spatial E0-Semigroups are Restrictions of Inner Automorphism Groups (M Skeide) A Characterization of Poisson Noise (Si Si) On Two Conjectures in Segal-Bargmann Analysis (S B Sontz) Note on Quantum Mutual Type Entropies and Capacity (N Watanabe) and other papers.