Recovery map stability for the Data Processing Inequality

Abstract

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

Version 3 simplifies various parts