Spectral geometric mean and trace characterizations

TL;DR

This paper characterizes positive linear functionals on matrix algebras using spectral geometric mean inequalities, introduces new trace inequalities, and explores their implications in quantum fidelity.

Contribution

It provides novel characterizations of positive linear functionals via spectral geometric mean inequalities and introduces related trace inequalities in quantum information theory.

Findings

Characterization of positive linear functionals as multiples of the trace using spectral geometric mean.

New inequalities involving the spectral geometric mean and the average of matrices.

A trace inequality related to quantum fidelity that does not characterize the trace.

Abstract

We use nearly parallel pure states to characterize positive linear functionals $\phi$ on $\mathbb{M}_n$ as positive multiples of the trace if and only if $\phi(A \natural B) \leq \sqrt{\phi(A) \phi(B)}$ for all positive definite matrices $A$ and $B$. Here $A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $\phi(A \natural B) \leq \phi((A+B)/2)$ for all positive definite matrices $A$ and $B$. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.