Two kinds of maps that describe evolution of states of a subsystem comingfrom dynamics described by a unitary operator for a larger system, maps definedfor fixed mean values and maps defined for fixed correlations, are found to bequite different for the same unitary dynamics in the same situation in thelarger system. An affine form is used for both kinds of maps to find necessaryand sufficient conditions for inverse maps. All the different maps with thesame homogeneous part in their affine forms have inverses if and only if thehomogeneous part does. Some of these maps are completely positive; others arenot, but the homogeneous part is always completely positive. The conditions foran inverse are the same for maps that are not completely positive as for mapsthat are. For maps defined for fixed mean values, the homogeneous part dependsonly on the unitary operator for the dynamics of the larger system, not on anystate or mean values or correlations. Necessary and sufficient conditions foran inverse are stated several different ways: in terms of the maps of matrices,basis matrices, density matrices, or mean values. The inverse maps aregenerally not tied to the dynamics the way the maps forward are. Atrace-preserving completely positive map that is unital can not have an inversethat is obtained from any dynamics described by any unitary operator for anystates of a larger system.
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