Continuous Non-Demolition Observation, Quantum Filtering and Optimal Estimation

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.Comment: A revised version of the paper published in the Proceedings of the 1st QCMC conference, Paris 199

Filtering equation for a measurement of a coherent channel

A stochastic model for a continuous photon counting and heterodyne measurement of a coherent source is proposed. A nonlinear filtering equation for the posterior state of a single-mode field in a cavity is derived by using the methods of quantum stochastic calculus. The posterior dynamics is found for the observation of a Bose field being initially in a coherent state. The filtering equations for counting and diffusion processes are given.Comment: 14 pages, 1 figure. This paper was published in JOSA B. The final version is available on OSA websit

Nondemolition Principle of Quantum Measurement Theory

We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous-spectrum and continuous-in-time observations. The "collapsed state-vector" after the "objectification" is simply treated as a random vector of the a posteriori state given by the quantum filtering, i.e., the conditioning of the a priori induced state on the corresponding reduced algebra. The nonlinear phenomenological equation of "continuous spontaneous localization" has been derived from the Schroedinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments.Comment: 23 pages. See also related papers at http://www.maths.nott.ac.uk/personal/vpb/research/mes_fou.html and http://www.maths.nott.ac.uk/personal/vpb/research/cau_idy.htm

Quantum Stochastic Positive Evolutions: Characterization, Construction, Dilation

A characterization of the unbounded stochastic generators of quantum completely positive flows is given. This suggests the general form of quantum stochastic adapted evolutions with respect to the Wiener (diffusion), Poisson (jumps), or general Quantum Noise. The corresponding irreversible Heisenberg evolution in terms of stochastic completely positive (CP) maps is constructed. The general form and the dilation of the stochastic completely dissipative (CD) equation over the algebra L(H) is discovered, as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators. A unitary quantum stochastic cocycle, dilating the subfiltering CP flows over L(H), is reconstructed.Comment: 33 page