[@VWSRVSKW06]. For instance, the light may interact with a system, say an atom, quantum dot, or cavity, and this system may be used as a quantum memory, [@MHNWGBRH97], or to control the pulse shape of the single photon state [@GM08]. When light interacts with a quantum system, information about the system is contained in the scattered light. This information may be useful for monitoring the behavior of the system, or for controlling it. The topic of this paper concerns the extraction of information from the scattered light when the incoming light is placed in a single photon state $\vert \Psi \rangle = \vert 1_\xi \rangle$, or a superposition of coherent states $\vert \Psi \rangle = \sum_j \alpha_j \vert f_j \rangle$, as illustrated in Figure \[fig:filter-one-1\].

![A system initialized in a state $\vert \eta \rangle$ coupled to a field in a state $\vert \Psi  \rangle$ (single photon or superposition of coherent states). The output field is continuously monitored by homodyne detection (assumed perfect) to produce a classical measurement signal $Y(t)$. The output $Y(t)$ is filtered to produce estimates $\hat X(t)=\pi_t(X)$ of system operators $X$ at time $t$. []{data-label="fig:filter-one-1"}](images/system-filter-qip)

The problem of extracting information from continuous measurement of the scattered light is 