Quantum filtering of Markov signals with white quantum noise

Abstract. Time-continuous non-anticipating quantum processes of nonde-molition measurements are introduced as the dynamical realizations of the causal quasi-measurements, which are described in this paper by the adapted operator-valued probability measures on the trajectory spaces of the gener-alized temporal observations in quantum open systems. In particular, the notion of physically realizable quantum \u85lter is de\u85ned and the problem of its optimization to obtain the best a posteriori quantum state is considered. It is proved that the optimal \u85ltering of a quantum Markovian Gaussian signal with the Gaussian white quantum noise is described as a coherent Markovian linear lter generalizing the classical Kalman \u85lter. As an example, the prob-lem of optimal measurement of complex amplitude for a quantum Markovian open oscillator, loaded to a quantum wave communication line, is considered and solved. 1