Proposition: Approximate cone certificate

Suppose a complete implemented channel $\tilde{\Phi }$ satisfies $d_{\diamond }(\tilde{\Phi },\Phi )\le e$, where $\Phi$ factors through the initial cone snapshot. Two inputs with equal cone snapshots have implemented output reductions at trace distance at most $\min \{1,2e\}$. The estimate is reference stable when the corresponding joint cone–reference reductions agree.

Proof. The two ideal outputs agree. Insert that common output between the implemented outputs and use the triangle inequality. Each implemented–ideal discrepancy is at most $e$. Trace distance is at most one. The diamond allowance includes references.