= Convex roof extension
{title2=$F(\rho)=\inf_{\rho=\sum_jp_j\pi_j}\sum_jp_j\mu(\pi_j)$}
= Convex roof
{synonym}
A convex roof extension extends a function $\mu$ on <pure states> to arbitrary <density operators> by minimizing its ensemble average over all <pure state> decompositions. It agrees with $\mu$ on <pure states>: every positive-weight state in a decomposition of a rank-one <density operator> has the same one-dimensional support. Concatenating nearly minimizing ensembles proves that $F$ is a <convex function>. Moreover, every <convex function> that is no larger than $\mu$ on <pure states> is no larger than $F$, by applying convexity to each decomposition. Infima need not be attained; an arbitrarily small additive error suffices when $F$ is finite.
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