  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.
