Theorem: Endpoint channel certificate

After the declared identifications, suppose actual and proxy complete channels satisfy

If the proxy complement obeys ${\delta }_{\mathrm{full}}(\hat{\mathcal{C}})\le \zeta$, then

A fixed proxy-optimal decoder has actual error at most

If a feasible state ${\tau }_B$ instead supplies the normalized Choi discrepancy

one may take $\zeta =\min \{1,de\}$.

Proof. Partial traces contract the complete-channel comparison. The triangle inequality proves leakage. Apply a fixed proxy-optimal decoder to the actual accessible channel, use contraction, and then use (eq:ksw). The constant decoder gives the independent $a_d$ cap. Equation (eq:choi) gives the Choi conversion.