Interpolation problems by completely positive maps

Given commuting families of Hermitian matrices {A1, …, Ak} and {B1, …, Bk}, conditions for the existence of a completely positive map L, such that L(Aj) = Bj for j = 1, …,k, are studied. Additional properties such as unital or / and trace preserving on the map ? are also considered. Connections of the study to dilation theory, matrix inequalities, unitary orbits, and quantum information science are mentioned.