Joint concavity of quantum fidelity ($F({\sum }_x{p}_x{\rho }_x,{\sum }_x{p}_x{\sigma }_x)\ge {\sum }_x{p}_{x}F({\rho }_x,{\sigma }_x)$)

In the unsquared convention, $F({\sum }_x{p}_x{\rho }_x,{\sum }_x{p}_x{\sigma }_x)\ge {\sum }_x{p}_{x}F({\rho }_x,{\sigma }_x)$. Use Uhlmann's theorem to choose component purifications with nonnegative maximizing overlaps, and then construct matching flagged purifications of a quantum ensemble. Their overlap is the weighted sum on the right; the fidelity of the mixtures maximizes over all such overlaps.