*quantum filter* for the quantum conditional expectation (see, e.g. [@BHJ07 Definition 3.13]) $$\hat X(t) = \mathbb{E}_{\eta\Psi}[ X(t) \, \vert \, \mathscr{Y}_t ] .
\label{eq:cond-exp}$$ This conditional expectation is well defined, since $X(t)$ commutes with the subspace $\mathscr{Y}_t $ (non-demolition condition). The conditional estimate $\hat X(t)$ is affiliated to $\mathscr{Y}_t$ (written in abbreviated fashion as $\hat X(t) \in \mathscr{Y}_t$) and is characterized by the requirement that $$\mathbb{E}_{\eta\Psi} [ \hat X(t) K ] = \mathbb{E}_{\eta\Psi} [ X(t) K ]
\label{eq:c-exp-def}$$ for all $K \in \mathscr{Y}_t$.

Single Photon Input Fields {#sec:photon}
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Single Photon Fields States {#sec:photon-state}
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In this section we consider the continuous-mode single photon state $\vert \Psi \rangle = \vert 1_\xi \rangle$ defined by [@RL00 sec. 6.3], [@GM08 eq. (9)] $$\vert 1_\xi \rangle = B^\ast(\xi) \vert 0  \rangle,  
\label{eq:xi-create}$$ where $\xi$ is a complex valued function such that $\int_0^\infty \vert
\xi(s) \vert^2 ds =1$, and $\vert 0 \rangle$ is the vacuum state of the field. Expression (\[eq:xi-create\]) says that the single photon wavepacket with temporal shape $\xi$ is created from the vacuum using the field operator $B^\ast(\xi)$.

The Hilbert space for the composite system is $$\mathfrak{H} = \mathfrak{H}_S \otimes \mathfrak{F}= \mathfrak{H}_S \otimes \mathfrak{F}_{t]} \otimes \mathfrak{F}_{(t},$$ where here we have exhibited 