Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

WORLD SCIENTIFIC eBooks

Boson Fock Representations of Stochastic Processes

L. Accardi, Takeyuki Hida, WIN WIN HTAY

Published Apr 1, 2001• 0 citations
Publisher

Abstract

A, classification theory of quantum stationary processes similar to the corresponding theory for classical stationary processes is presented, Our main result is the classification of those pairs of classical stationary processes that admit a joint boson Fock canonical representation.
DOI
10.1142/9789812794611_0021

Related papers

CANONICAL REPRESENTATION OF STATIONARY QUANTUM GAUSSIAN PROCESSES

Luigi Accardi, Yuji Hibino

Stimulated by the quantum generalization of the canonical representation theory for Gaussian processes in Ref. 1, we first give the representations (not necessarily canonical) of two stationary Gaussian processes X and Y by means of white noises qt and pt with no assumptions on their commutator. We then assume that qt + ipt annihilates the vacuum state and prove that the representations are the joint Boson–Fock ones if and only if X and Y have a scalar commutator.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Sep 1, 2002•0 citations

Stochastic Processes: From Classical to Quantum

Soon Hoe Lim

The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson and Parthasarathy. We introduce quantum stochastic processes on a boson Fock space and their calculus. Whenever possible, we make connections with the relevant concepts in classical probability theory. As an application of the theory, we introduce the theory of open quantum systems, with emphasis on the physics and modeling aspects of these systems.

arXiv:2407.04005•Jul 4, 2024•0 citations

Boson fock representations of stochastic processes

Luigi Accardi, T. Hida, Win Win Htay

Mathematical Notes•Jan 1, 2000•3 citations

Quantum stochastic integral representations of Fock space operators

Un Cig Ji, Nobuaki Obata

An (unbounded) operator Ξ on Boson Fock space over L 2(R +) is called regular if it is an admissible white noise operator such that the conditional expectations give rise to a regular quantum martingale. We prove that an admissible white noise operator is regular if and only if it admits a quantum stochastic integral representation.

Stochastics•Jun 1, 2009•18 citations

Quantum probability communications : QP-PQ

R. L. Hudson, J. Martin Lindsay, Philippe Biane +1 more

Extensions of Quantum Stochastic Calculus (S Attal) Quantum Ito Algebras: Axioms, Representations, Decompositions (V Belavkin) Free Probability for Probabilists (P Biane) Conditional Expectations on von Neumann Algebras (C Cecchini) Classical Probability Theory: An Outline of Stochastic Integrals and Diffusions (M Emery) Quantum Stochastic Differential Equations (F Fagnola) Canonical Commutation and Anticommutation Relations (M Fannes) Quantum and Classical Stochastic Calculus (A Holevo) An Introduction to Quantum Stochastic Calculus and Some of Its Applications (R Hudson) Stationary Processes in Quantum Probability (B Kummerer)

WORLD SCIENTIFIC eBooks•Jan 1, 1998•1 citation

STATIONARY QUANTUM STOCHASTIC PROCESSES FROM THE COHOMOLOGICAL POINT OF VIEW

Grigori Gennadievich Amosov

Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra $\tilde K=\oplus_{i=1}^{\infty}K_i$ into a ring of (abelian) cohomologies of the one-parameter group $α$ consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from $K_i$ is translated into an additive $i-α$-cocycle j(x). It is shown that (noncommutative) multiplicative markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The $E_0$-semigroup $\tilde β$ on the von Neumann algebra $\cal N$ associated with the markovian perturbation of K-flow j posseses the restriction $\tilde β|_{{\cal N}_0}, {\cal N}_0\subset {\cal N}$, which is conjugate to the flow of Powers shifts $β$ associated with j. It yields for $\tilde β$ an analogue of the Wold decomposition for classical stochastic process on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space $L^2(\mathbb R)$ all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.

arXiv:math/0202192•Quantum Probability and Infinite Dimensional Analysis•Jan 1, 2003•2 citations
The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson and Parthasarathy. We introduce quantum stochastic processes on a boson Fock space and their calculus. Whenever possible, we make connections with the relevant concepts in classical probability theory. As an application of the theory, we introduce the theory of open quantum systems, with emphasis on the physics and modeling aspects of these systems.
Extensions of Quantum Stochastic Calculus (S Attal) Quantum Ito Algebras: Axioms, Representations, Decompositions (V Belavkin) Free Probability for Probabilists (P Biane) Conditional Expectations on von Neumann Algebras (C Cecchini) Classical Probability Theory: An Outline of Stochastic Integrals and Diffusions (M Emery) Quantum Stochastic Differential Equations (F Fagnola) Canonical Commutation and Anticommutation Relations (M Fannes) Quantum and Classical Stochastic Calculus (A Holevo) An Introduction to Quantum Stochastic Calculus and Some of Its Applications (R Hudson) Stationary Processes in Quantum Probability (B Kummerer)
Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra K~=⊕i=1∞Ki\tilde K=\oplus_{i=1}^{\infty}K_iK~=⊕i=1∞​Ki​ into a ring of (abelian) cohomologies of the one-parameter group ααα consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from KiK_iKi​ is translated into an additive i−αi-αi−α-cocycle j(x). It is shown that (noncommutative) multiplicative markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The E0E_0E0​-semigroup β~\tilde ββ~​ on the von Neumann algebra N\cal NN associated with the markovian perturbation of K-flow j posseses the restriction β~∣N0,N0⊂N\tilde β|_{{\cal N}_0}, {\cal N}_0\subset {\cal N}β~​∣N0​​,N0​⊂N, which is conjugate to the flow of Powers shifts βββ associated with j. It yields for β~\tilde ββ~​ an analogue of the Wold decomposition for classical stochastic process on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space L2(R)L^2(\mathbb R)L2(R) all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.