acting on the full Hilbert space $\mathfrak{H}$ that is adapted, i.e. $K(t)$ acts trivially on $\mathfrak{F}_{(t}$, the field in the future. Then the expectation with respect to the single photon field may be expressed in terms of the vacuum state as follows: $$\begin{aligned}
\mathbb{E}_{11} [ K(t) ] &=& \mathbb{E}_{00} [ B^-_t(\xi) K(t)
B^{-\ast}_t(\xi) + r(t) K(t) ]  
\label{eq:xi-phi}\end{aligned}$$ where $r(t) = \int_t^\infty \vert \xi(s) \vert^2 ds$.

Master Equation {#sec:master}
---------------

Before deriving the quantum filter, we work out dynamical equations for the unconditioned single photon expectation, [@GEPZ98]. To assist us in evaluating this expectation, we make use of the following lemma.

\[lemma:expectation-basic\] Let $K(t)$ be a bounded quantum stochastic process defined by $$K(t) = \int_0^t M_0(s) ds + \int_0^t M_- (s) dB(s) + \int_0^t M_+(s)
dB^\ast(s) + \int_0^t M_1(s) d\Lambda(s),  
\label{eq:Kt-def}$$ where $M_0$, $M_\pm$ and $M_1$ are bounded and adapted. Then we have $$\begin{aligned}
\mathbb{E}_{11}[ K(t) ] &=& \mathbb{E}_{11}[ \int_0^t M_0(s) ds ] + \mathbb{E}_{10}[ \int_0^t M_- (s) \xi(s) ds ]  \notag \\
&& + \mathbb{E}_{01}[ \int_0^t M_+(s) \xi^\ast(s) ds ] + \mathbb{E}_{00}[
\int_0^t M_1 (s) \vert \xi(s) \vert^2 ds ] ,  \label{eq:Kt-11} \\
\mathbb{E}_{10}[ K(t) ] &=& \mathbb{E}_{10}[ \int_0^t M_0(s) ds ] + \mathbb{E}_{00}[ \int_0^t M_+ (s) \xi^\ast(s) ds 