Lemma: Replacement and classical-output benchmarks

A replacement accessible channel has optimal recovery error $a_d$. A measure-and-prepare channel, followed by any decoder to $C$, has error at least $1-d^{-1}$. Computational readout and basis-state preparation attain the latter bound.

Proof. A decoded replacement has Bell output $I_Q/d\otimes {\tau }_C$ and Bell overlap $d^{-2}$. The Bell-projector test gives error at least $a_d$. Preparing $I_C/d$ attains it: the completely depolarizing channel is the uniform average of the $d^2$ Weyl unitary channels, and separating the identity term gives the matching upper bound.

A measure-and-prepare channel has separable Bell output. A normalized product vector has squared overlap with ${\Phi }_d$ at most $1/d$, by Cauchy–Schwarz; convexity extends this to separable states. The Bell test gives $1-1/d$. Computational dephasing is the uniform average of the $d$ phase unitaries ${Z_d}^k$. Its Bell outputs are orthogonal, giving diamond distance $1-1/d$ from the identity.