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

TL;DR

This paper explores the conditions under which positive maps induce preorders on Hermitian matrices when certain constraints are relaxed, leading to new measures of non-positive semidefiniteness with desirable properties.

Contribution

It characterizes the existence of positive maps without unitality or trace-preservation, defining two preorders and associated measures for Hermitian matrices.

Findings

Two new preorders on Hermitian matrices are introduced.

Unique measures quantifying non-positive semidefiniteness are constructed.

The measures have important monotonicity properties.

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.