Proposition: A finite complementary Choi certificate

Let $\sigma \in \mathsf{D}(B)$ and suppose

Then

If a justified model approximation satisfies $d_{\diamond }({\mathcal{E}}_{\mathrm{phys}},\mathcal{E})\le \lambda$ on the same complete residual interface, then

A physical recovery conclusion additionally requires ${\mathcal{E}}_{\mathrm{phys}}$ to be the complete complement of the declared physical accessible channel.

Proof. Use Lemma lem:choi with $J_{{\mathcal{T}}_{\sigma }}=I_Q/d\otimes \sigma$, then apply Theorem thm:approx. For the physical comparison, use the triangle inequality. The final sentence retains the hypothesis needed to apply that theorem to the physical channel.