  Two kinds of maps that describe evolution of states of a subsystem coming
from dynamics described by a unitary operator for a larger system, maps defined
for fixed mean values and maps defined for fixed correlations, are found to be
quite different for the same unitary dynamics in the same situation in the
larger system. An affine form is used for both kinds of maps to find necessary
and sufficient conditions for inverse maps. All the different maps with the
same homogeneous part in their affine forms have inverses if and only if the
homogeneous part does. Some of these maps are completely positive; others are
not, but the homogeneous part is always completely positive. The conditions for
an inverse are the same for maps that are not completely positive as for maps
that are. For maps defined for fixed mean values, the homogeneous part depends
only on the unitary operator for the dynamics of the larger system, not on any
state or mean values or correlations. Necessary and sufficient conditions for
an inverse are stated several different ways: in terms of the maps of matrices,
basis matrices, density matrices, or mean values. The inverse maps are
generally not tied to the dynamics the way the maps forward are. A
trace-preserving completely positive map that is unital can not have an inverse
that is obtained from any dynamics described by any unitary operator for any
states of a larger system.
