STOCHASTIC DIFFERENTIAL-EQUATIONS AND A POSTERIORI STATES IN QUANTUM-MECHANICS

In recent years a consistent theory describing measurements continuous in time in quantum mechanics has been developed. The result of such a measurement is a ''trajectory'' for one or more quantities observed with continuity in time. Applications are connected especially with detection theory in quantum optics. In such a theory of continuous measurements one can ask what is the state of the system given that a certain trajectory up to time t has been observed. The response to this question is the notion of a posteriori states and a ''filtering'' equation governing the evolution of such states: this turns out to be a nonlinear stochastic differential equation for density matrices or for pure vectors. The driving noise appearing in such an equation is not an external one, but its probability law is determined by tie system itself (it is the probability measure on the trajectory space given by the theory of continuous measurements).