= Relative-entropy identity for rank-one dephasing
{title2=$D_{\mathrm{rel}}(\rho\|\Delta(\rho))=S(\Delta(\rho))-S(\rho)$}
Because $\log\Delta(\rho)$ is diagonal, $\operatorname{Tr}(\rho\log\Delta(\rho))=\operatorname{Tr}(\Delta(\rho)\log\Delta(\rho))$. Hence $D_{\mathrm{rel}}(\rho\|\Delta(\rho))=S(\Delta(\rho))-S(\rho)$. <Support inclusion under rank-one dephasing> handles zero probabilities, and <Klein's inequality> makes the difference nonnegative. Equality holds exactly when the state was already diagonal.
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