On measurements of the parameters of a quantum random process

The quantum analog of the Ornstein–Uhlenbeck process is considered. It describes a quantum oscillator with damping in thermodynamic equilibrium with a reservoir and subject to an external disturbance. The problem of estimating the unknown amplitude of the disturbance using the results of continuous (in time) quantum measurements on the process with allowance for reduction of the state is solved. Two procedures are considered: Continuous measurement of the complex amplitude and continuous measurement of the coordinate of the oscillator in the Hiesenberg representation. Exact and asymptotic expressions are found for the dispersions of the corresponding estimates, and from them a criterion follows for comparing the two procedures. These expressions are compared with the “reductionless” bound which is derived from the quantum theory of detection and estimation.