Quantum Stochastic Calculus and Quantum Nonlinear Filtering

V. P. BELAVKIN

چکیده

A ?–algebraic inde nite structure of quantum stochastic (QS) calculus is introduced and a continuity property of generalized nonadapted QS integrals is proved under the natural integrability conditions in an in nitely dimensional nuclear space. The class of nondemolition output QS processes in quantum open systems is characterized in terms of the QS calculus, and the problem of QS nonlinear ltering with respect to nondemolition continuous measurments is investigated. The stochastic calculus of a posteriori conditional expectations in quantum observed systems is developed and a general quantum ltering stochastic equation for a QS process is derived. An application to the description of the spontaneous collapse of the quantum spin under continuous observation is given. Introduction The problem of description of continuous observation and ltering in quantum dynamical systems can be e¤ectively solved in the framework of quantum stochastic (QS) calculus of nondemolition input-output processes rst developed for quantum unitary Markovian evolutions in [1]. In contrast to classical probability theory, the conditional expectations de ning à posteriori states of a quantum system with respect to a subalgebra may not exist in general and the existence depends on the algebra of observables and on the à priori state. During the preparation of the measurement of a Quantum System the change necessary to produce the à priori compatible state as a mixture of the à posteriori states is referred to in quantum physics as the demolition of the system. The latter involves a change in the initial state by the reduction of the algebra of the system during such a preparation. The nondemolition principle provides a su¢ cient condition for the algebra of the quantum system to be prepared for the measurement in an initial state. The mathematical formulation of the nondemolition principle for the observability of a class of quantum processes was given in [2] and investigated in subsequent papers [3, 4]. This fundamental principle of quantum measurement theory means that if a QS process Xt is indirectly observable by the measurement of another process Yt then X and Y must satisfy the one sided commutativity condition [Xt; Ys] XtYs YsXt = 0 for all t s but not for t < s. Date : 20 September, 1989. Key words and phrases. Quantum stochastic calculus, Quantum Langevin equations, Quantum nondemolition processes, Quantum conditional expectations, Quantum nonlinear ltering. Published in: Journal of Multivariate Analysis, 42 (2) 171–201 (1992). This paper is in nal form and no version of it will be submitted for publication elsewhere. 1