expectations in the extended system, we need the following lemma, which follows from Lemma \[lemma:expectation-basic\], and makes use of the matrices $$\sigma_+ =  \vert e_1 \rangle \langle e_0 \vert = \left[
\begin{array}{cc}
0 & 1 \\
0 & 0
\end{array}
\right], \ \ \sigma_- =  \vert e_0 \rangle \langle e_1 \vert  = \left[
\begin{array}{cc}
0 & 0 \\
1 & 0
\end{array}
\right].$$

\[lemma:expectation-basic-extended\] Assume $\alpha_0\neq 0$, and let $M(t)$ be bounded and adapted. Then $$\begin{aligned}
\mathbb{E}_\Sigma [ \int_0^t A \otimes M(s) dB(s) ] &=& \nu \mathbb{E}_\Sigma [
\int_0^t ( A \sigma_+)\otimes M(s) \xi(s) ds ], \\
\mathbb{E}_\Sigma [ \int_0^t A \otimes M (s) dB^\ast(s) ] &=& \nu^\ast \mathbb{E}_\Sigma [ \int_0^t (\sigma_-A )\otimes M(s) \xi^\ast(s) ds], \\
\mathbb{E}_\Sigma [ \int_0^t A \otimes M(s) d\Lambda(s) ] &=& \vert \nu
\vert^2 \mathbb{E}_\Sigma [ \int_0^t ( \sigma_- A \sigma_+ )\otimes M(s) \vert
\xi(s) \vert^2 ds ],\end{aligned}$$ where $$\nu = \frac{\alpha_1}{\alpha_0}.$$

In Lemma \[lemma:expectation-basic-extended\], expectations of stochastic integrals with respect to the superposition state $\vert \Sigma \rangle$ are expressed in terms of expectations of non-stochastic integrals again with respect to $\vert \Sigma \rangle$ with the aid of the matrices $\sigma_\pm$ acting on the ancilla system $\mathbb{C}^2$. The action of the field annihilation, creation and 