  For quantum systems described by finite matrices, linear and affine maps of
matrices are shown to provide equivalent descriptions of evolution of density
matrices for a subsystem caused by unitary Hamiltonian evolution in a larger
system; an affine map can be replaced by a linear map, and a linear map can be
replaced by an affine map. There may be significant advantage in using an
affine map. The linear map is generally not completely positive, but the linear
part of an equivalent affine map can be chosen to be completely positive and
related in the simplest possible way to the unitary Hamiltonian evolution in
the larger system.
