= Monotonicity of a convex roof under a quantum instrument
{title2=$F(\rho)\geq\sum_kP_kF(\rho_k)$}
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 $P_k\rho_k=\sum_jp_jq_{k|j}\sigma_{jk}$. Convexity gives $\sum_kP_kF(\rho_k)\leq\sum_{j,k}p_jq_{k|j}F(\sigma_{jk})\leq\sum_jp_j\mu(\pi_j)$. Taking the <infimum> over input ensembles proves the result, even when conditional outputs of pure inputs are mixed, provided the averages are well defined.
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