Lemma: Replacement and Choi benchmarks

A replacement accessible channel has recovery error $a_d$; every accessible channel has recovery error at most $a_d$. For

put $J_{\mathrm{n}}(\Phi )=({\mathrm{id}}_Q\otimes \Phi )({\Phi }_d)$. Then

If $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{\tau })\le \eta$, then

Proof. A decoded replacement has Bell output $I_Q/d\otimes {\omega }_C$ and Bell overlap $d^{-2}$, giving the lower bound $a_d$. Preparing $I_C/d$ attains it. Indeed, depolarization is the uniform average of the $d^2$ Weyl conjugations; separating its identity term gives diamond distance $1-d^{-2}$.

The first Choi inequality uses the Bell input. For a pure referenced input, write $|\psi ⟩=(L\otimes I)|{\Phi }_d⟩$, where $\mathrm{Tr}(L^{†}L)=d$. Then ${\Vert L\Vert }_{\infty }^2\le d$. Its output difference is obtained from the normalized Choi difference by multiplication with $L\otimes I$ on both sides, giving the second inequality. Convexity handles mixed inputs. The final assertion follows by postprocessing contraction and the triangle inequality.