Lemma: Optimal recovery from a replacement channel

Set

If $\mathcal{N}$ is a replacement channel, then

Proof. Every decoder of a replacement channel is a replacement channel ${\mathcal{T}}_{\tau }$. Its Bell-input output is $I_Q/d\otimes \tau$, with Bell overlap $d^{-2}$. Thus every decoder has error at least $1-d^{-2}$.

For the matching upper bound, the decoder can prepare $I_C/d$. The completely depolarizing channel is the uniform average of the $d^2$ Weyl unitary channels. Separating the identity term and bounding the remaining channel differences by two gives

The Bell lower bound makes this an equality.