Proposition: No exact intermediate recovery and endpoint optima

For $0\le \eta <1$,

For all $\eta$,

At the endpoints,

In particular, the specified decoder's error one at $\eta =0$ is not the optimal error.

Proof. Let ${\rho }_i={\mathcal{N}}_{\eta }(|i⟩⟨i|)$. These states are diagonal and

For any decoder with error $\epsilon$, its decoded basis inputs are within $\epsilon$ of their respective orthogonal targets. The triangle inequality and contraction imply

Thus $\epsilon \ge t^2/2$.

The specified decoder gives the upper bound $1-\eta$. Another decoder ignores its input and returns $I_2/2$. For Pauli matrices $X,Y,Z$,

The triangle inequality gives its distance to the identity at most $3/4$.

At $\eta =0$, every decoded channel is a replacement ${\mathcal{R}}_{\tau }$. Its Bell output is $I_Q/2\otimes \tau$, whose overlap with $|\Phi ⟩$ is $1/4$, independently of $\tau$. The Bell-projector effect and Lemma lem:contraction give distance at least $3/4$. The maximally mixed replacement attains the preceding upper bound. At $\eta =1$, ${\mathcal{N}}_1={\mathrm{Ad}}_W$, and the specified decoder is its left inverse.