these identifications without further comment. The quantum conditional expectation $\tilde \pi_t( A \otimes X) \in I
\otimes \mathscr{Y}_t$ is well defined because $A \otimes X(t)$ is in the commutant $I \otimes \mathscr{Y}_t^{\prime}$ of the algebra $I \otimes \mathscr{Y}_t$, and is characterized by the requirement that $$\mathbb{E}_{\Sigma} [ \tilde \pi_t(A \otimes X) I \otimes K ] = \mathbb{E}_{\Sigma} [
(A \otimes X(t))( I \otimes K) ]  \label{eq:c-exp-def-matrix}$$ for all $K \in \mathscr{Y}_t$, see, e.g. [@BHJ07 Definition 3.13].

\[thm:matrix-filter\] Assume $\alpha_0 \neq 0$. The conditional expectation $\tilde \pi_t(A \otimes X)$ defined by (\[eq:matrix-c-exp-def\]) for the extended system satisfies $$\begin{aligned}
d\tilde \pi_t(A\otimes X) &=&\tilde  \pi_t( \mathcal{G}_t(A \otimes X) ) dt + \mathcal{H}_t(A\otimes X) dW(t),  
\label{pi_filter}\end{aligned}$$ where $$\begin{aligned}
\mathcal{H}_t(A \otimes X) &=& \tilde  \pi_t( A \otimes (XL+L^\ast X) ) - \tilde \pi_t( A
\otimes X) \pi_t( I \otimes (L+L^\ast) )  \notag \\
&& + \tilde \pi_t( ( A \sigma_+) \otimes XS) \nu \xi(t) + \tilde \pi_t( ( \sigma_-A )
\otimes S^\ast X) \nu^\ast \xi^\ast(t)  \notag \\
&& - \tilde \pi_t( A \otimes X) \tilde \pi_t( ( \sigma_+ \otimes S) \nu \xi(t) +( \sigma_-
\otimes S^\ast ) \nu^\ast \xi^\ast(t) )  \label{eq:matrix-filter}\end{aligned}$$ and $$dW(t) = dY(t) - \tilde \pi_t( I \otimes (L+L^\ast) + ( \sigma_+ \otimes 