Recovery map stability for the Data Processing Inequality

The Data Processing Inequality (DPI) says that the Umegaki relative entropy $S(\rho||\sigma) := {\rm Tr}[\rho(\log \rho - \log \sigma)]$ is non-increasing under the action of completely positive trace preserving (CPTP) maps. Let ${\mathcal M}$ be a finite dimensional von Neumann algebra and ${\mathcal N}$ a von Neumann subalgebra if it. Let ${\mathcal E}_\tau$ be the tracial conditional expectation from ${\mathcal M}$ onto ${\mathcal N}$. For density matrices $\rho$ and $\sigma$ in ${\mathcal N}$, let $\rho_{\mathcal N} := {\mathcal E}_\tau \rho$ and $\sigma_{\mathcal N} := {\mathcal E}_\tau \sigma$. Since ${\mathcal E}_\tau$ is CPTP, the DPI says that $S(\rho||\sigma) \geq S(\rho_{\mathcal N}||\sigma_{\mathcal N})$, and the general case is readily deduced from this. A theorem of Petz says that there is equality if and only if $\sigma = {\mathcal R}_\rho(\sigma_{\mathcal N} )$, where ${\mathcal R}_\rho$ is the Petz recovery map, which is dual to the Accardi-Cecchini coarse graining operator ${\mathcal A}_\rho$ from ${\mathcal M}$ to ${\mathcal N}$. In it simplest form, our bound is $$S(\rho||\sigma) - S(\rho_{\mathcal N} ||\sigma_{\mathcal N} ) \geq \left(\frac{1}{8\pi}\right)^{4} \|\Delta_{\sigma,\rho}\|^{-2} \| {\mathcal R}_{\rho_{\mathcal N}} -\sigma\|_1^4$$ where $\Delta_{\sigma,\rho}$ is the relative modular operator. We also prove related results for various quasi-relative entropies. Explicitly describing the solutions set of the Petz equation $\sigma = {\mathcal R}_\rho(\sigma_{\mathcal N} )$ amounts to determining the set of fixed points of the Accardi-Cecchini coarse graining map. Building on previous work, we provide a throughly detailed description of the set of solutions of the Petz equation, and obtain all of our results in a simple self, contained manner.