Recovery map stability for the Data Processing Inequality

Understand

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 $\sigma_{\mathcal N} := {\mathcal E}_\tau \sigma$.