Theorem: Hawk02-A conditional certification

Suppose Assumption ass:physical holds. Let ${\eta }_j$ be the stepwise allowances and let $e_{\mathrm{prep}},e_{\mathrm{cut}},e_{\mathrm{end}}$ be the additional CPTP comparison allowances actually used. Set

If the complete comparison complement satisfies ${\delta }_{\mathrm{leak}}(\hat{\mathcal{C}})\le \lambda$, then:

1. the actual and comparison complete channels are within $e_{\mathrm{tot}}$ in half-diamond distance;

2. the actual residual leakage obeys

   $${\delta }_{\mathrm{leak}}(\mathcal{C})\le \min \{1,\lambda +e_{\mathrm{tot}}\};$$

3. arbitrary-code, reference-preserving recovery satisfies

   $${\epsilon }_{\mathrm{rec}}(\mathcal{N})\le \min \left\{1,e_{\mathrm{tot}}+\sqrt{2\lambda },\sqrt{2(\lambda +e_{\mathrm{tot}})}\right\};$$

4. if $\lambda =e_{\mathrm{tot}}=0$, exact recovery exists.

These conclusions certify the supplied transfer interface. They do not derive that interface from the emission spectrum.

Proof. Theorem thm:sequential, applied also to the additional comparison maps, proves (i). Theorem thm:stability gives (ii) and (iii). For (iv), the actual complement is constant and Theorem thm:constant applies. The transfer interface is an explicit hypothesis in (P3), whereas Corollary cor:spectrum disproves its determination by the listed thermal diagnostics.