Joint convexity of quantum relative entropy

Quantum relative entropy obeys $D({\sum }_i{p}_i{\rho }_i\Vert {\sum }_i{p}_i{\sigma }_i)\le {\sum }_i{p}_{i}D({\rho }_i\Vert {\sigma }_i)$. It follows by applying data processing to flagged block-diagonal states and discarding the flag.