  Linear maps of matrices describing evolution of density matrices for a
quantum system initially entangled with another are identified and found to be
not always completely positive. They can even map a positive matrix to a matrix
that is not positive, unless we restrict the domain on which the map acts.
Nevertheless, their form is similar to that of completely positive maps. Only
some minus signs are inserted in the operator-sum representation. Each map is
the difference of two completely positive maps. The maps are first obtained as
maps of mean values and then as maps of basis matrices. These forms also prove
to be useful. An example for two entangled qubits is worked out in detail.
Relation to earlier work is discussed.
