  In most studies of the capacity of quantum channels, it is assumed that the
errors in each use of the channel are independent. However, recent work has
begun to investigate the effects of memory or correlations in the error, and
has led to suggestions that there can be interesting non-analytic behaviour in
the capacity of such channels. In a previous paper we pursued this issue by
connecting the study of channel capacities under correlated error to the study
of critical behaviour in many-body physics. This connection enables the use of
techniques from many-body physics to either completely solve or understand
qualitatively a number of interesting models of correlated error with analogous
behaviour to associated many-body systems. However, in order for this approach
to work rigorously, there are a number of technical properties that need to be
established for the lattice systems being considered. In this article we
discuss these properties in detail, and establish them for some classes of
many-body system.
