In my research this summer, I have become interested in lower bounds on the standard "Umegaki quantum relative entropy".
For two non-negative matrices $X$ and $Y$, the Umegaki quantum relative entropy is defined as
$$\;$$ $$S(X,Y) = \mathrm{Tr}[X(\ln(X)-\ln(Y))]\;\;\;\;\; (*)$$ $$\;$$
Since I am interested in the particular physics setting of Gibbs state density matrices, I can also take $X$ and $Y$ positive definite with $\mathrm{Tr}[X] = \mathrm{Tr}[Y] = 1$ and the eigenvalues of both $X$ and $Y$ are bounded above by 1.
The lower bound on $(*)$ reported by Hiai and Petz (1993) is
$$\;$$ $$S(X,Y) \geq \frac{1}{p} \mathrm{Tr}[X ln(Y^{-p/2} X^p Y^{-p/2})]\;\;\;\;\;\;\;\;(**) $$ $$\;$$
where $p > 0$. Given that for Gibbs state density matrices, I can write
$$\;$$ $$X = \frac{e^{-\beta H_X}}{Z_X}$$ $$Y = \frac{e^{-\beta H_Y}}{Z_Y}$$ $$\;$$
where $\beta \in \mathbb{R}_{\geq 0}$, $Z_X \equiv \mathrm{Tr}[\exp(-\beta H_X)]$, $Z_Y \equiv \mathrm{Tr}[\exp(-\beta H_Y)]$ and $H_X$ and $H_Y$ are hermitian matrices, is it possible to simplify the right hand side of $(**)$? I know it smells like Baker-Campbell-Hausdorff but I haven't been able to obtain anything nice using that approach.