Proposition: Approximate scheduled no-signalling

Suppose an actual complete channel $\tilde{\mathcal{C}}$ and a scheduled ideal channel $\mathcal{C}$ satisfy

For two input states with equal reductions on ${\mathsf{Sch}}^{-}(Y)$, their actual output reductions on $Y$ differ in trace distance by at most $\min \{1,2e\}$. This estimate is reference stable.

Proof. Their ideal output reductions are equal. Insert that common reduction between the two actual outputs and use the triangle inequality. Each actual–ideal discrepancy is at most $e$. The half-diamond allowance includes arbitrary references Watrous.