  Given two sets of quantum states {A_1, ..., A_k} and {B_1, ..., B_k},
represented as sets of density matrices, necessary and sufficient conditions
are obtained for the existence of a physical transformation T, represented as a
trace-preserving completely positive map, such that T(A_i) = B_i for i = 1,
..., k. General completely positive maps without the trace-preserving
requirement, and unital completely positive maps transforming the states are
also considered.
