Proposition: Replacement benchmark and lower-bound direction

For input dimension $d$, every replacement channel has optimal recovery error $a_d=1-d^{-2}$. If $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{\omega })\le \eta$, every deterministic decoder has error at least $\max \{0,a_d-\eta \}$.

A qubit pure-output replacement followed by the identity decoder has implemented error one, although its optimal recovery error is $3/4$.

Proof. Every decoded replacement is a replacement. Its Bell output is $I_d/d\otimes \tau$, with Bell overlap $d^{-2}$, giving the lower bound $a_d$.

Preparing $I_d/d$ attains it: the corresponding replacement is the uniform average of $d^2$ Weyl unitary channels. Separating identity gives unhalved diamond norm at most $2a_d$; the Bell witness gives equality. Contraction and the reverse triangle inequality give the approximate lower bound. For a fixed pure output $|0⟩$, input $|1⟩$ gives orthogonal actual and ideal outputs.