  We consider the problem of transmitting classical information over a
time-invariant channel with memory. A popular class of time-invariant channels
with memory are finite-state-machine channels, where a \emph{classical} state
evolves over time and governs the relationship between the classical input and
the classical output of the channel. For such channels, various techniques have
been developed for estimating and bounding the information rate. In this paper
we consider a class of time-invariant channels where a \emph{quantum} state
evolves over time and governs the relationship between the classical input and
the classical output of the channel. We propose algorithms for estimating and
bounding the information rate of such channels. In particular, we discuss
suitable graphical models for doing the relevant computations.
