Relative entropy of classically flagged states ($D({⨁}_i{p}_i{\rho }_i\Vert {⨁}_i{p}_i{\sigma }_i)={\sum }_i{p}_{i}D({\rho }_i\Vert {\sigma }_i)$)

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.