More on a trace inequality in quantum information theory

It is known that for a completely positive and trace preserving (cptp) map ${\cal N}$, $\text{Tr}$ $\exp$$\{ \log σ$ $+$ ${\cal N}^\dagger [\log {\cal N}(ρ)$ $-\log {\cal N}(σ)] \}$ $\leqslant$ $\text{Tr}$ $ρ$ when $ρ$, $σ$, ${\cal N}(ρ)$, and ${\cal N}(σ)$ 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.