Recovery map stability for the Data Processing Inequality

The Data Processing Inequality (DPI) says that the Umegaki relative entropy $S(\rho \mathrm{\mid }\mathrm{\mid }\sigma ):=\mathrm{T}\mathrm{r}[\rho (\log \rho -\log \sigma )]$ 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 $$S(\rho \mathrm{\mid }\mathrm{\mid }\sigma )-S({\rho }_{\mathcal{N}}\mathrm{\mid }\mathrm{\mid }{\sigma }_{\mathcal{N}})\ge {\left(\frac{1}{8\pi }\right)}^4\mathrm{\parallel }{\mathrm{\Delta }}_{\sigma ,\rho }{\mathrm{\parallel }}^{-2}\mathrm{\parallel }{\mathcal{R}}_{{\rho }_{\mathcal{N}}}-\sigma {\mathrm{\parallel }}_1^4$$ where ${\mathrm{\Delta }}_{\sigma ,\rho }$ 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.