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 $$S(rho||sigma) - S(rho_{mathcal N} ||sigma_{mathcal N} ) geq left(frac{1}{8pi}right)^{4} |Delta_{sigma,rho}|^{-2} | {mathcal R}_{rho_{mathcal N}} -sigma|_1^4 $$ 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.