Proposition: Approximate cone certificate

If a complete channel $\tilde{\Phi }$ is within half-diamond distance $e$ of a channel that factors through an ancestor input region, then two inputs with equal reductions on that region have output trace distance at most $\min (1,2e)$. The referenced conclusion requires equal joint ancestor–reference reductions.

Proof. The two ideal reduced outputs agree. Insert their common value between the implemented outputs and use the triangle inequality. Each discrepancy is at most $e$. Tensoring the comparison with the reference gives the joint version.