Lemma: Optimal replacement recovery

Every replacement accessible channel has

Also,

Consequently every accessible channel satisfies ${\epsilon }_{\mathrm{rec}}(\mathcal{N})\le a_d$.

Proof. A decoder following a replacement channel prepares a fixed state ${\omega }_C$. Its output on ${\Phi }_d$ is $I_Q/d\otimes {\omega }_C$, with Bell overlap $1/d^2$. Testing the projector ${\Phi }_d$ gives trace distance at least $a_d$.

For the matching upper bound, let $W_{ab}$, $0\le a,b<d$, be the generalized Weyl unitaries, with $W_{00}=I$. Their uniform twirl is

Writing the twirl as its identity term plus its remaining CPTP average gives

for a channel $\mathcal{M}$. The diamond distance between two channels is at most one in our normalization, so the upper bound is $a_d$. The Bell test gives equality. Ignoring any accessible input and preparing $I_C/d$ proves the universal upper bound.