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arXiv:quant-ph/0510028

Quantum Diffusion, Measurement and Filtering

V. P. Belavkin

Published Oct 4, 2005• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0510028

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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.
A brief presentation is given of the basic concepts in quantum probability theory in analogy to the classical ones, as well as the necessary information about noncommutative stochastic integration and its explicit representation in the Fock space. Algebraic differential relations that define quantum diffusion motion with nondemolition observation are derived. In the Markov case we obtain the stochastic equation of quantum diffusion filtering which is analogous to the Zakai equation of classical nonlinear filtering. A quantum linear stochastic model with continuous observation is given for which this equation is reduced to a linear stochastic quantum filtering equation of Kalman-Busi type and to the operator Riccati equation. We also consider an example of observing the coordinate of a free quantum particle, for which the solution of a stationary quantum filtering problem is obtained.
A quantum stochastic model for an open dynamical system (quantum receiver) and output multi-channel of observation with an additive nonvacuum quantum noise is given. A quantum stochastic Master equation for the corresponding instrument is derived and quantum stochastic filtering equations both for the Heisenberg operators and the reduced density matrix of the system under the nondemolition observation are found. Thus the dynamical problem of quantum filtering is generalized for a noncommutative output process, and a quantum stochastic model and optimal filtering equation for the dynamical estimation of an input Markovian process is found. The results are illustrated on an example of optimal estimation of an input Gaussian diffusion signal, an unknown gravitational force say in a quantum optical or Weber's antenna for detection and filtering a gravitational waves.
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure, functional integrals, quantum stochastic differential equations and classical stochastic differential equations (SDE's). Various types of SDE's are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both. In this paper we introduce a class of linear SDE's for trace-class operators, relevant to the theory of continual measurements, and we recall how such SDE's are related to instruments and master equations and, so, to the general formulation of quantum mechanics. We do not present the Hilbert space formulation of such SDE's and we make some mathematical simplifications: no time dependence is introduced into the coefficients and only bounded operators on the Hilbert space of the quantum system are considered. Then we introduce the notion of a posteriori state and the non linear SDE satisfied by such states and we give conditions from which such equation is assured to preserve pure states and to send any mixed state into a pure one for large times. Finally we review the known results about the existence and uniqueness of invariant measures in the purely diffusive case and we give some concrete examples of physical systems.
Abstract. We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous– spectrum and continuous-in-time observations. The “collapsed state–vector” after the “objectification ” is simply treated as a random vector of the a posteriori state given by the quantum filtering, i.e., the conditioning of the a priori induced state on the corresponding reduced algebra. The nonlinear phenomenological equation of “continuous spontaneous localization ” has been derived from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments. 1. The status of quantum measurement theory
Measurements continuous in time have been consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after a measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented for the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.
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.