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:0903.4486

Quantum filter processes driven by Markovian white noises have classical versions

Radhakrishnan Balu

Published Mar 25, 2009• 0 citations
arXivHTMLPDF

Abstract

We study quantum filters that are driven by basic quantum noises and construct classical versions. Our approach is based on exploiting the quantum markovian component of the observation and measurement processes of the filters. This approach leads in a natural way the classical versions for a class of quantum filters. We consider quantum white noises derived from Wiener and Poisson processes that drive the signal and measurement processes and derive the recursive filtering equations using classical machinery.

DOI
10.48550/arxiv.0903.4486

Related papers

Quantum Diffusion, Measurement and Filtering

V. P. Belavkin

A brief presentation of the basic concepts in quantum probability theory is given in comparison to the classical one. The notion of quantum white noise, its explicit representation in Fock space, and necessary results of noncommutative stochastic analysis and integration are outlined. Algebraic differential equations that unify the quantum non Markovian diffusion with continuous non demolition observation are derived. A stochastic equation of quantum diffusion filtering generalising the classical Markov filtering equation to the quantum flows over arbitrary *-algebra is obtained. A Gaussian quantum diffusion with one dimensional continuous observation is considered.The a posteriori quantum state difusion in this case is reduced to a linear quantum stochastic filter equation of Kalman-Bucy type and to the operator Riccati equation for quantum correlations. An example of continuous nondemolition observation of the coordinate of a free quantum particle is considered, describing a continuous collase to the stationary solution of the linear quantum filtering problem found in the paper.

arXiv:quant-ph/0510028•Oct 4, 2005•6 citations

Quantum filter for a class of non-Markovian quantum systems

Shibei Xue, M. R. James, Alireza Shabani +2 more

In this paper we present a Markovian representation approach to constructing quantum filters for a class of non-Markovian quantum systems disturbed by Lorentzian noise. An ancillary system is introduced to convert white noise into Lorentzian noise which is injected into a principal system via a direct interaction. The resulting dynamics of the principal system are non-Markovian, which are driven by the Lorentzian noise. By probing the principal system, a quantum filter for the augmented system can be derived from standard theory, where the conditional state of the principal system can be obtained by tracing out the ancillary system.

arXiv:1503.07999•IEEE Conference on Decision and Control (CDC)•Dec 1, 2015•14 citations

A master equation for quantum systems driven by Poisson white noise

J. Łuczka, Mariusz Niemiec

An evolution equation for a reduced statistical operator of quantum systems driven by Poisson white noise is derived. It is applied to a simple system and compared with a counterpart driven by Gaussian white noise.

Journal of Physics A Mathematical and General•Sep 7, 1991•21 citations

Quantum smoother for open quantum systems driven by quantum jump-diffusion processes

Kentaro Ohki

In this paper, we develop a quantum smoothing theory to the class of open quantum systems driven by quantum jump-diffusion processes. A quantum smoothing theory, which is derived in a different senses by different researchers, has been applied for diffusive quantum Markovian systems, however, they only deal with quantum Wiener noises. We show how to derive the quantum smoother for quantum dynamical systems driven by quantum jump-diffusion noises.

Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications•Jul 31, 2019•3 citations

An Introduction to Quantum Filtering

Luc Bouten, Ramon van Handel, M. R. James

This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum Itô calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.

arXiv:math/0601741•SIAM Journal on Control and Optimization•Jan 1, 2007•498 citations

Fundamentals of Classical and Supersymmetric Quantum Stochastic Filtering Theory

Harish Parthasarathy

In this paper, we first derive the discrete time recursive stochastic filtering equations for a classical Markov process when noisy measurements of the process are made with the measurement noise at each discrete sample forms an iid Gaussian or non-Gaussian process. The filter involves deriving general formulae for the conditional mean process given measurements upto that time and also simultaneously recursive formulae for the conditional moments of the estimation error process. In the next section, we build upon the method of John Gough etal, to construct a supersymmetric quantum stochastic filter, ie, the supersymmetric Belavkin filter. The construction of supersymmetric noise in the Hudson Parthasarathy noisy Schrodinger equation is based on constructing Fermionic noise by applying a twist to Bosonic noise, ie, \(dJ=(-1)^{\Lambda}dA, dJ^*=(-1)^{\Lambda}dA^*\). Such a Fermionic noise has memory unlike the Bosonic noise and satisfies the CAR. We prove that when a superposition of Bosonic quantum Brownian motion and Bosonic counting process along with Fermionic counting process is measured through the Hudson-Parthasarathy noisy Schrodinger system with both Bosonic and supersymmetric noise, then it satisfies the non-demolition property and hence conditional expectations can be defined. The construction of the quantum filter here involves taking into consideration all the positive integer powers of the output measurement noise differential and we give an algorithm for computing these powers based on the quantum Ito formula. The method for deriving a countably infinite number of linear equations for the filter coefficients is based on the reference probability method of Gough et.al. The method of constructing supersymmetric noise for driving the HPS equation is based on the work of Timothy Eyre. The main result of this paper is to derive an infinite system of linear equations for the quantum filter coefficients thereby yielding a real time implementable filter for estimating system observables or equivalently, the system mixed state from the most general form of non-demolition measurements comprising Bosonic continuous and counting noise plus Fermionic counting noise.

Qeios•May 17, 2024•1 citation
We study quantum filters that are driven by basic quantum noises and construct classical versions. Our approach is based on exploiting the quantum markovian component of the observation and measurement processes of the filters. This approach leads in a natural way the classical versions for a class of quantum filters. We consider quantum white noises derived from Wiener and Poisson processes that drive the signal and measurement processes and derive the recursive filtering equations using classical machinery.
A brief presentation of the basic concepts in quantum probability theory is given in comparison to the classical one. The notion of quantum white noise, its explicit representation in Fock space, and necessary results of noncommutative stochastic analysis and integration are outlined. Algebraic differential equations that unify the quantum non Markovian diffusion with continuous non demolition observation are derived. A stochastic equation of quantum diffusion filtering generalising the classical Markov filtering equation to the quantum flows over arbitrary *-algebra is obtained. A Gaussian quantum diffusion with one dimensional continuous observation is considered.The a posteriori quantum state difusion in this case is reduced to a linear quantum stochastic filter equation of Kalman-Bucy type and to the operator Riccati equation for quantum correlations. An example of continuous nondemolition observation of the coordinate of a free quantum particle is considered, describing a continuous collase to the stationary solution of the linear quantum filtering problem found in the paper.
In this paper we present a Markovian representation approach to constructing quantum filters for a class of non-Markovian quantum systems disturbed by Lorentzian noise. An ancillary system is introduced to convert white noise into Lorentzian noise which is injected into a principal system via a direct interaction. The resulting dynamics of the principal system are non-Markovian, which are driven by the Lorentzian noise. By probing the principal system, a quantum filter for the augmented system can be derived from standard theory, where the conditional state of the principal system can be obtained by tracing out the ancillary system.
This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum Itô calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.
In this paper, we first derive the discrete time recursive stochastic filtering equations for a classical Markov process when noisy measurements of the process are made with the measurement noise at each discrete sample forms an iid Gaussian or non-Gaussian process. The filter involves deriving general formulae for the conditional mean process given measurements upto that time and also simultaneously recursive formulae for the conditional moments of the estimation error process. In the next section, we build upon the method of John Gough etal, to construct a supersymmetric quantum stochastic filter, ie, the supersymmetric Belavkin filter. The construction of supersymmetric noise in the Hudson Parthasarathy noisy Schrodinger equation is based on constructing Fermionic noise by applying a twist to Bosonic noise, ie, dJ=(−1)ΛdA,dJ∗=(−1)ΛdA∗dJ=(-1)^{\Lambda}dA, dJ^*=(-1)^{\Lambda}dA^*dJ=(−1)ΛdA,dJ∗=(−1)ΛdA∗. Such a Fermionic noise has memory unlike the Bosonic noise and satisfies the CAR. We prove that when a superposition of Bosonic quantum Brownian motion and Bosonic counting process along with Fermionic counting process is measured through the Hudson-Parthasarathy noisy Schrodinger system with both Bosonic and supersymmetric noise, then it satisfies the non-demolition property and hence conditional expectations can be defined. The construction of the quantum filter here involves taking into consideration all the positive integer powers of the output measurement noise differential and we give an algorithm for computing these powers based on the quantum Ito formula. The method for deriving a countably infinite number of linear equations for the filter coefficients is based on the reference probability method of Gough et.al. The method of constructing supersymmetric noise for driving the HPS equation is based on the work of Timothy Eyre. The main result of this paper is to derive an infinite system of linear equations for the quantum filter coefficients thereby yielding a real time implementable filter for estimating system observables or equivalently, the system mixed state from the most general form of non-demolition measurements comprising Bosonic continuous and counting noise plus Fermionic counting noise.