Corollary: Strict loss of exact full-code correctability

If $\mathcal{C}$ is nonconstant, then ${\epsilon }_{\mathrm{rec}}(\mathcal{N})>0$. Also, if

exact full-code recovery is impossible.

Proof. In finite dimensions the decoder Choi matrices form a compact set: they are positive, satisfy a fixed partial-trace identity, and have fixed trace. Channel distance is continuous, so the infimum in (eq:errors) is attained. If the minimum were zero, Theorem thm:exact would force a constant complement.

For the rank assertion, exact recovery would identify ${\tau }_B=\mathcal{C}(I_C/d)$, and then (eq:rankcapacity) would hold.