measurement of an observable $Y(t)$ of the output field $B_{out}(t)$. The filtering problem of interest in this paper is to determine the conditional state from which estimates $\hat X(t)$ of system operators $X$ may be determined at time $t$ based on knowledge of the observables $\{ Y(s)$, $0 \leq s \leq t \}$.

In what follows the system $S$ is assumed to be defined on a Hilbert space $\mathfrak{H}_S$, with an initial state denoted $\vert  \eta \rangle   \in \mathfrak{H}_S$. The input field $B_{in}$ is described in terms of annihilation $B(\xi)$ and creation $B^\ast(\xi)$ operators defined on a Fock space $\mathfrak{F}$, [@KRP92 Chapter II], [@BHJ07 Section 4]. Quantum expectation will be denoted by the symbol $\mathbb{E}$, and when we wish to display the underlying state, we employ subscripts; for example, $\mathbb{E}_{\eta\Psi}$ denotes quantum expectation with respect to the state $\vert \eta \rangle \otimes \vert \Psi \rangle$.

The dynamics of the system will be described using the quantum stochastic calculus, [@HP84], [@GC85], [@KRP92], [@GZ00], [@BHJ07]. Quantum stochastic integrals are defined in terms of fundamental field operators $B(t)$, $B^\ast(t)$ and $\Lambda(t)$, [@KRP92 Chapter II], [@BHJ07 Section 4].[^5] The non-zero Ito products for the field operators are $$dB(t) dB^\ast(t) = dt, \ \ dB(t) 