= Joint concavity of quantum fidelity
{title2=$F(\sum_xp_x\rho_x,\sum_xp_x\sigma_x)\geq\sum_xp_xF(\rho_x,\sigma_x)$}
In the unsquared convention, $F(\sum_xp_x\rho_x,\sum_xp_x\sigma_x)\geq\sum_xp_xF(\rho_x,\sigma_x)$. 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.
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