  In a partially observed quantum or classical system the information that we
cannot access results in our description of the system becoming mixed even if
we have perfect initial knowledge. That is, if the system is quantum the
conditional state will be given by a state matrix $\rho_r(t)$ and if classical
the conditional state will be given by a probability distribution $P_r(x,t)$
where $r$ is the result of the measurement. Thus to determine the evolution of
this conditional state under continuous-in-time monitoring requires an
expensive numerical calculation. In this paper we demonstrating a numerical
technique based on linear measurement theory that allows us to determine the
conditional state using only pure states. That is, our technique reduces the
problem size by a factor of $N$, the number of basis states for the system.
Furthermore we show that our method can be applied to joint classical and
quantum systems as arises in modeling realistic measurement.
