Theorem: Certified response and certified quantum return

Suppose a finite prepared proxy is embedded access-preservingly in a common output space and satisfies

Then

If a validated proxy pair witness gives

then

For a specified proxy replacement state, put

Then

and a finite proxy decoder has a full extension satisfying

The last bound means that one may choose the better of the extended proxy decoder and the constant decoder.

Proof. Partial trace gives allowance $b$ for both reduced channels. A local isometric embedding has a CPTP retraction

Embedding and retracting decoders and replacement states preserve the tasks. Lemma lem:distances therefore gives stability.

The pair-witness lower bound follows from Lemma lem:witness and leakage stability. The Choi estimate gives proxy leakage at most $de$. Equation (eq:KSW) supplies a finite decoder of error at most $\sqrt{2de}$. Extend it off the proxy support by fixed-state preparation. Contraction adds $b$. Lemma lem:constant supplies the cap.