Lemma: Optimal error of a replacement channel

For every replacement channel with input dimension $d$,

For every channel with input $C$, ${\epsilon }_{\mathrm{rec}}\le q_d$.

Proof. Any decoder applied to a replacement channel produces another replacement channel ${\mathcal{T}}_{\tau }$ on $C$. Its Bell-input output is $I_Q/d\otimes \tau$, whose Bell overlap is $d^{-2}$. The Bell-projector test therefore gives error at least $q_d$.

For attainment, use the decoder that always prepares ${\tau }_C$. Let $U_{ab}$, $0\le a,b<d$, be the Weyl unitaries. Their uniform twirl is ${\mathcal{T}}_{{\tau }_C}$, so

The triangle inequality and diamond norm one of each channel give half-diamond norm at most $q_d$. The Bell test gives equality. The same constant decoder can be used after any channel, proving the final assertion.