  In this paper we return to the problem of reduced-state dynamics in the
presence of an interacting environment. The question we investigate is how to
appropriately model a particular system evolution given some knowledge of the
system-environment interaction. When the experimenter takes into account
certain known features of the interaction such as its invariant subspaces or
its non-local content, it may not be possible to consistently model the system
evolution over a certain time interval using a standard Stinespring dilation,
which assumes the system and environment to be initially uncorrelated. Simple
examples demonstrating how restrictions can emerge are presented below. When
the system and environment are qubits, we completely characterize the set of
unitaries that always generate reduced dynamics capable of being modeled using
a consistent Stinespring dilation. Finally, we show how any initial
correlations between the system and environment can be certified by observing
the system transformation alone during certain joint evolutions.
