Convex roof extension ($F(\rho )={\inf }_{\rho ={\sum }_j{p}_j{\pi }_j}{\sum }_j{p}_j\mu ({\pi }_j)$)

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.

If a convex roof extension is non-increasing on average for every pure input to a quantum instrument, then it is non-increasing on average for mixed inputs too. For an input ensemble $\{p_j,{\pi }_j\}$, let $q_{k|j}$ and ${\sigma }_{jk}$ be its conditional outcome probabilities and states. Linearity of each unnormalized outcome map gives $P_k{\rho }_k={\sum }_j{p}_j{q}_{k|j}{\sigma }_{jk}$. Convexity gives ${\sum }_k{P}_{k}F({\rho }_k)\le {\sum }_{j,k}{p}_j{q}_{k|j}F({\sigma }_{jk})\le {\sum }_j{p}_j\mu ({\pi }_j)$. Taking the infimum over input ensembles proves the result, even when conditional outputs of pure inputs are mixed, provided the averages are well defined.