+j_{t}([L^{\ast },X]S) dB( t) +j_{t}(\mathcal{L}(X)) dt ,
\label{eq:X-dyn}\end{aligned}$$ where $$\mathcal{L}(X)=\frac{1}{2}L^{\ast }[X,L]+\frac{1}{2}[L^{\ast },X]L-i[ X,H] .$$ The map $X\mapsto \mathcal{L} (X)$ is known as the *Lindblad generator*, while the quartet of maps $X \mapsto \mathcal{L} (X), S^\ast XS -X, \,
S^\ast [X,L ], \, [L^\ast , X] S$ are known as *Evans-Hudson maps*.

The output field is defined by $B_{out}(t) = U(t)^\ast B(t) U(t)$.[^6] In this paper we consider the output field observable $Y(t)$ defined by $$Y(t) = U(t)^\ast Z(t) U(t) , 
 \label{eq:Y-out}$$ where $$Z(t)= B(t)+ B^\ast(t), 
 \label{eq:Z-def}$$ is a quadrature observable of the input field (the counting case $Z(t)=\Lambda(t)$ is discussed briefly in Section \[sec:conclusion\]). Note that both $Z(t)$ and $Y(t)$ are self-adjoint and self-commutative: $[Z(t), Z(s)]=0$ and $[Y(t), Y(s)]=0$. We write $\mathscr{Z}_t$ and $\mathscr{Y}_t$ for the subspaces of commuting operators generated by the observables $Z(s)$, $Y(s)$, $0\leq s \leq t$, respectively.[^7] They are related by the unitary rotation $\mathscr{Y}_t = U(t)^\ast \mathscr{Z}_t U(t)$. Physically, $Y(t)$ may represent the integrated photocurrent arising in an idealized (perfect) homodyne photodetection scheme, as in Figure \[fig:filter-one-1\]. For further information on homodyne detection, we refer the reader to the literature; for example, [@BR04], [@AB03], [@WM10].

The primary goal of this paper is to determine the 