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 \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.