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

arXiv:2008.07355

Continuous time random walks modeling of quantum measurement and fractional equations of quantum stochastic filtering and control

Vassili Nikitich Kolokoltsov

Published Aug 17, 2020• 1 citation
arXivHTMLPDF

Abstract

Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.

DOI
10.48550/arxiv.2008.07355

Related papers

CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control

Vassili Nikitich Kolokoltsov

Abstract Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.

Fractional Calculus and Applied Analysis•Feb 1, 2022•3 citations

Eventum Mechanics of Quantum Trajectories: Continual Measurements, Quantum Predictions and Feedback Control

V. P. Belavkin

Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.

arXiv:math-ph/0702079•Feb 23, 2007•10 citations

Quantum jump trajectories, hybrid systems, non-Hermitian evolutions, quantum/classical walks

Alberto Barchielli

Quantum stochastic master equations of jump type are formulated in a general way and connections with quantum/classical hybrid systems and quantum filtering theory are discussed. By introducing the notion of ``typical trajectory", we show how to recursively construct the solution of the non-linear stochastic master equation (the conditional state). Moreover, by the notion of ``exclusive probability densities" we can describe all the probabilities related to the jumps, in particular, the waiting times of the jumps and their probability distributions. This general formulation and the idea of hybrid system allow to unify and generalize different fields: evolutions under non-Hermitian Hamiltonians, unitary dynamics interspersed by quantum channels at random times, quantum renewal processes, continuous time open quantum walks, Lindblad rate equation, ...

quant-pharXiv:2605.01411•Journal of Physics A Mathematical and Theoretical•May 5, 2026•0 citations

Quantum filtering and propagation of chaos for open quantum systems, with applications to quantum feedback control and quantum mean-field games

Vassili Nikitich Kolokoltsov

The theory of quantum filtering (of quantum continuous measurements) was developed by V.P. Belavkin about 40 years ago. Since then it attracted attention of numerous investigators including mathematicians, theoretical and experimental physicists. However, the rigorous mathematical theory of the filtering equations for mixed states in basic infinite-dimensional quantum systems remained an open problem, which was resolved by the author recently. This survey paper presents in full the mathematical theory of quantum filtering equations, their rigorous derivation from basic principles, the corresponding law of large number limits (propagation of chaos) and related topics. Applications to feedback control, quantum dynamic and mean-field games are discussed.

arXiv:2607.08507•Jul 9, 2026•0 citations

Quantum trajectories in random environment: the statistical model for a heat bath

Ion Nechita, Clément Pellegrini

In this paper, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath has a clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.

arXiv:0908.2754•Confluentes Mathematici•May 30, 2009•5 citations

Quantum Trajectories in Random Environment: the Statistical Model for a Heat Bath

Ion Nechita, Clément Pellegrini

In this article, we derive the stochastic master equations corresponding to the sta-tistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.

•Jan 1, 2009•4 citations
Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.
Abstract Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.
Quantum stochastic master equations of jump type are formulated in a general way and connections with quantum/classical hybrid systems and quantum filtering theory are discussed. By introducing the notion of ``typical trajectory", we show how to recursively construct the solution of the non-linear stochastic master equation (the conditional state). Moreover, by the notion of ``exclusive probability densities" we can describe all the probabilities related to the jumps, in particular, the waiting times of the jumps and their probability distributions. This general formulation and the idea of hybrid system allow to unify and generalize different fields: evolutions under non-Hermitian Hamiltonians, unitary dynamics interspersed by quantum channels at random times, quantum renewal processes, continuous time open quantum walks, Lindblad rate equation, ...
The theory of quantum filtering (of quantum continuous measurements) was developed by V.P. Belavkin about 40 years ago. Since then it attracted attention of numerous investigators including mathematicians, theoretical and experimental physicists. However, the rigorous mathematical theory of the filtering equations for mixed states in basic infinite-dimensional quantum systems remained an open problem, which was resolved by the author recently. This survey paper presents in full the mathematical theory of quantum filtering equations, their rigorous derivation from basic principles, the corresponding law of large number limits (propagation of chaos) and related topics. Applications to feedback control, quantum dynamic and mean-field games are discussed.
In this paper, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath has a clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.
In this article, we derive the stochastic master equations corresponding to the sta-tistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.