= Relative entropy of a pinched state
{title2=$D(\rho\|\mathcal P(\rho))=S(\mathcal P(\rho))-S(\rho)$}
The <pinching map> is self-adjoint for the trace pairing, and $\log\mathcal P(\rho)$ is block diagonal. Thus $\operatorname{Tr}\rho\log\mathcal P(\rho)=\operatorname{Tr}\mathcal P(\rho)\log\mathcal P(\rho)$. The kernel of $\mathcal P(\rho)$ is contained in that of $\rho$, so the relative entropy is finite on its support. Expanding the <quantum relative entropy> gives the displayed identity. Its nonnegativity proves entropy increase, with equality exactly when the original state is already block diagonal.
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