the quantum filter for the single photon field may be obtained from this equation.

The continuously monitored field observable that corresponds to the conditional expectation (\[eq:matrix-c-exp-def\]) is $I \otimes Y(t)$, and from (\[eq:Y-out\]) we have the corresponding output equation for the extended system: $$d(I \otimes Y(t)) = I \otimes (L(t)+L^\ast(t)) dt + I \otimes (S(t) dB(t) +
S^\ast(t) dB^\ast(t) ) .$$ In what follows we will make use of the following lemma concerning expectations of the process $$V(t) = \int_0^t ( S(s) dB(s) + S^\ast(s) dB^\ast(s) )$$ with respect to the single photon state.

\[lemma:Z-compensated-mtg\] For any $K \in \mathscr{Y}_s$, we have $$\begin{aligned}
\mathbb{E}_{11}[ (V(t) - V(s)) K ] &=& \mathbb{E}_{10}[ \int_s^t S(r)
\xi(r)dr K] + \mathbb{E}_{01}[ \int_s^t S^\ast(r) \xi^\ast(r)dr K] .
\label{eq:V-compensated} \end{aligned}$$

Equation (\[eq:V-compensated\]) is obtained using the additive decomposition (\[eq:factor-additive-1\]) and Lemma \[lemma:expectation-basic\]. [$\Box$]{}

Note that an operator $K$ in the unital commutative algebra $I \otimes \mathscr{Y}_t$ has the form $K=I \otimes \tilde K$, where $\tilde K \in \mathscr{Y}_t$. By the spectral theorem, [@BHJ07 Theorem 3.3], we may identify $K$ and $\tilde K$, both of which are equivalent to a classical stochastic process $K_t(s)$, $0 \leq s
\leq t$. In the remainder of this paper, we use 