= Relative entropy of classically flagged states
{title2=$D(\bigoplus_i p_i\rho_i\|\bigoplus_i p_i\sigma_i)=\sum_ip_iD(\rho_i\|\sigma_i)$}
For common classical weights, the <matrix logarithm> on each positive-weight block is $\log(p_i\rho_i)=(\log p_i)I+\log\rho_i$ on its support. The two logarithms of $p_i$ cancel in the <quantum relative entropy>, giving the displayed identity. Zero-weight blocks are omitted, including if their unweighted relative entropy is infinite. If a positive-weight block violates support inclusion, both sides are infinite. Applying the <data-processing inequality for quantum relative entropy> to the <partial trace> over the flag gives <joint convexity of quantum relative entropy>.
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