Quantum filtering: a reference probability approach

TL;DR

This paper introduces quantum filtering theory using a reference probability approach, covering quantum probability, stochastic calculus, and deriving filtering equations for quantum optical systems.

Contribution

It presents a novel application of reference probability methods to derive quantum filtering equations in both unnormalized and normalized forms.

Findings

Derivation of quantum filtering equations for various quantum optical models

Application of noncommutative probability to quantum filtering

Extension of classical filtering concepts to quantum systems

Abstract

These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. Next we describe the Hudson-Parthasarathy quantum Ito calculus and its use in the modelling of physical systems. Finally, we use a reference probability method to obtain quantum filtering equations, in the Belavkin-Zakai (unnormalized) form, for several system-observation models from quantum optics. The normalized (Belavkin-Kushner-Stratonovich) form is obtained through a noncommutative analogue of the Kallianpur-Striebel formula.