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arXiv:quant-ph/0509205 · Springer eBooks

Continuous non-demolition observation quantum filtering and optimal estimation

V. P. Belavkin

Published Apr 7, 2008• 13 citations
arXivHTMLPDF
DOI
10.1007/3-540-53862-3_176

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A quantum stochastic model for an open dynamical system (quantum receiver) and output multi-channel of observation with an additive nonvacuum quantum noise is given. A quantum stochastic Master equation for the corresponding instrument is derived and quantum stochastic filtering equations both for the Heisenberg operators and the reduced density matrix of the system under the nondemolition observation are found. Thus the dynamical problem of quantum filtering is generalized for a noncommutative output process, and a quantum stochastic model and optimal filtering equation for the dynamical estimation of an input Markovian process is found. The results are illustrated on an example of optimal estimation of an input Gaussian diffusion signal, an unknown gravitational force say in a quantum optical or Weber's antenna for detection and filtering a gravitational waves.
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.