Characterizing Schwarz maps by tracial inequalities

TL;DR

This paper explores tracial inequalities for Schwarz maps, extending known inequalities for 2-positive maps to a broader class of positive maps, with implications for mathematical physics.

Contribution

It establishes a new tracial inequality for Schwarz maps, broadening the understanding of positivity conditions and their applications in physics.

Findings

The tracial inequality holds for a wider class of positive maps.

Connections to monotonicity principles in mathematical physics are discussed.

New results derived from the inequality enhance understanding of Schwarz maps.

Abstract

Let $\phi$ be a linear map from the $n\times n$ matrices ${\mathcal M}_n$ to the $m\times m$ matrices ${\mathcal M}_m$. It is known that $\phi$ is $2$-positive if and only if for all $K\in {\mathcal M}_n$ and all strictly positive $X\in {\mathcal M}_n$, $\phi(K^*X^{-1}K) \geq \phi(K)^*\phi(X)^{-1}\phi(K)$. This inequality is not generally true if $\phi$ is merely a Schwarz map. We show that the corresponding tracial inequality ${\rm Tr}[\phi(K^*X^{-1}K)] \geq {\rm Tr}[\phi(K)^*\phi(X)^{-1}\phi(K)]$ holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.