Simulation of Quantum filter for estimating the spin of an electron

The Belavkin filter is a real-time quantum filter for dynamically estimating atomic observables and states based on non-demolition measurements, i.e. measurements that commute at different times and with future values of the observables. This filter is a generalisation of the classical Kushner filter based on quantum mechanics, and we propose to simulate it by using the H-P stochastic equation and the Belavkin equations without randomness. Randomness enters only when we consider the distribution of observables in specific states of the system and bath. We apply the Belavkin filter to estimate the spin of the electron.