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

Interpolation problems by completely positive maps