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

Journal of the London Mathematical Society

Perturbations of Quantum Diffusions

M. Evans, R. L. Hudson

Published Apr 1, 1990• 23 citations
Publisher

Abstract

Quantum diffusions are non-commutative generalisations of classical diffusions on manifolds, in which the manifold i s ‘quantised’ by deforming the algebra of smooth functions thereon to a non-commutative algebra, and in which quantum stochastic calculus replaces the Itô calculus of Brownian motion. Quantum diffusions fall into two categories; inner diffusions, which are in a certain sense cohomologically trivial, and which act by conjugating by a unitary process, and outer diffusions, which do not. A natural perturbation procedure is described for linking two cohomologically equivalent quantum diffusions; it involves the construction of a new type of intertwining unitary process.

DOI
10.1112/jlms/s2-41.2.373

Related papers

Brownian Motion on So(3) Factorises in Fock Space

David Applebaum

Techniques of quantum stochastic calculus are employed to factorise canonical Brownian motion on SO(3) into two non-commutative unitary operator valued processes acting in a suitable Fock space.

Bulletin of the London Mathematical Society•Sep 1, 1991•0 citations

A quantization of diffusion processes on manifolds

Lu, Y.G.

Via the well known chaos decomposition with respect to the multidimensional Brownian motion, we give a possible quantization of a diffusion process (in particular, Brownian motion) on a Riemannian manifold. The quantized diffusion process on Riemannian manifold is a quantum Brownian motion on a Holbert space determined by the momentum algebra.

Note di matematica/Note di matematica - Università degli studi di Lecce•Aug 4, 2009•0 citations

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

Quantum Stochastic Calculus

Kalyanapuram Rangachari Parthasarathy

The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.

Birkhäuser Basel eBooks•Jan 1, 1995•6 citations

From quantum stochastic differential equations to Gisin-Percival state diffusion

Kalyanapuram Rangachari Parthasarathy, A. R. Usha Devi

Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ(L2(R+)⊗(Cn⊕Cn)) and the Hilbert space L2(μ), where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B(t),t≥0}, we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.

arXiv:1705.00520•Journal of Mathematical Physics•Aug 1, 2017•3 citations

Quantum Probability and Applications III: Proceedings of a Conference Held in Oberwolfach, Frg, January 25-31, 1987

Luigi Accardi, Wilhelm von Waldenfels

A note on Meyer's note.- Stochastic integration.- Quantum stochastic parallel transport on non-commutative vector bundles.- Input and output channels in quantum systems and quantum stochastic differential equations.- Some noncommutative Radon-Nikodym Theorems for von Neumann algebras.- Multidimensional quantum diffusions.- An application of de finetti's theorem.- The quantum Langevin equation from the independent-oscillator model.- Quantum poisson processes: Physical motivations and applications.- A noncommutative generalization of conditionally positive definite functions.- Contraction semigroups in L2 over a von neumann algebra.- Survey on a theory of non-commutative stationary markov processes.- Dynamical entropy for quantum systems.- An integral kernel approach to noise.- A note on shifts and cocycles.- Local measures in fock space stochastic calculus and a generalized ito-tanaka formula.- Representation of a class of quantum martingales II.- Conditional expectation in quantum probability.- Mutual quadratic variation and ito's table in quantum stochastic calculus.- On mixing properties of automorphisms of von neumann algebras related to measure space transformations.- First exit time: A theory of stopping times in quantum processes.- A central limit theorem on the free lie group.- Entropy, observability and the generalised second law of thermodynamics.- Remarks on asymptotic properties of groups of the bogoliubov transformations of car c*-algebras.- Linear and non-linear stochastic processes.- Detailed balance and critical slowing down.- Quantum martingales and stochastic integrals.

•May 1, 1988•1 citation
Quantum diffusions are non-commutative generalisations of classical diffusions on manifolds, in which the manifold i s ‘quantised’ by deforming the algebra of smooth functions thereon to a non-commutative algebra, and in which quantum stochastic calculus replaces the Itô calculus of Brownian motion. Quantum diffusions fall into two categories; inner diffusions, which are in a certain sense cohomologically trivial, and which act by conjugating by a unitary process, and outer diffusions, which do not. A natural perturbation procedure is described for linking two cohomologically equivalent quantum diffusions; it involves the construction of a new type of intertwining unitary process.
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.
The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.
Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ(L2(R+)⊗(Cn⊕Cn)) and the Hilbert space L2(μ), where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B(t),t≥0}, we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.
A note on Meyer's note.- Stochastic integration.- Quantum stochastic parallel transport on non-commutative vector bundles.- Input and output channels in quantum systems and quantum stochastic differential equations.- Some noncommutative Radon-Nikodym Theorems for von Neumann algebras.- Multidimensional quantum diffusions.- An application of de finetti's theorem.- The quantum Langevin equation from the independent-oscillator model.- Quantum poisson processes: Physical motivations and applications.- A noncommutative generalization of conditionally positive definite functions.- Contraction semigroups in L2 over a von neumann algebra.- Survey on a theory of non-commutative stationary markov processes.- Dynamical entropy for quantum systems.- An integral kernel approach to noise.- A note on shifts and cocycles.- Local measures in fock space stochastic calculus and a generalized ito-tanaka formula.- Representation of a class of quantum martingales II.- Conditional expectation in quantum probability.- Mutual quadratic variation and ito's table in quantum stochastic calculus.- On mixing properties of automorphisms of von neumann algebras related to measure space transformations.- First exit time: A theory of stopping times in quantum processes.- A central limit theorem on the free lie group.- Entropy, observability and the generalised second law of thermodynamics.- Remarks on asymptotic properties of groups of the bogoliubov transformations of car c*-algebras.- Linear and non-linear stochastic processes.- Detailed balance and critical slowing down.- Quantum martingales and stochastic integrals.