More on a trace inequality in quantum information theory

TL;DR

This paper extends a key trace inequality in quantum information theory to cases where the involved matrices are nonnegative rather than strictly positive, and offers an alternative proof for the original case.

Contribution

It generalizes a known trace inequality to nonnegative matrices and provides an alternative proof for the strictly positive case.

Findings

Extended the trace inequality to nonnegative matrices.

Provided an alternative proof for the strictly positive case.

Clarified conditions under which the inequality holds.

Abstract

It is known that for a completely positive and trace preserving (cptp) map ${\cal N}$, $\text{Tr}$ $\exp$$\{ \log \sigma$ $+$ ${\cal N}^\dagger [\log {\cal N}(\rho)$ $-\log {\cal N}(\sigma)] \}$ $\leqslant$ $\text{Tr}$ $\rho$ when $\rho$, $\sigma$, ${\cal N}(\rho)$, and ${\cal N}(\sigma)$ are strictly positive. We state and prove a relevant version of this inequality for the hitherto unaddressed case of these matrices being nonnegative. Our treatment also provides an alternate proof for the strictly positive case.