Characterization of preorders induced by positive maps in the set of Hermitian matrices

Abstract: Uhlmann showed that there exists a positive, unital and trace-preserving map transforming a Hermitian matrix $A$ into another $B$ if and only if the vector of eigenvalues of $A$ majorizes that of $B$. In this work I characterize the existence of such a transformation when one of the conditions of unitality or trace preservation is dropped. This induces two possible preorders in the set of Hermitian matrices and I argue how this can be used to construct measures quantifying the lack of positive semidefiniteness of any given Hermitian matrix with relevant monotonicity properties. It turns out that the measures in each of the two formalisms are essentially unique.