Theorem: Exact specified-decoder performance

Its Bell fidelity is $(1+3\eta )/4$, and its worst-case target overlap is $\eta$. No optimality is asserted.

Proof. Phase correction makes the off-diagonal coefficient $\eta$. The odd words are replaced by $I_2/2$. The occupied input has decoded $|1⟩$ probability ${\eta }^2+\eta (1-\eta )=\eta$; vacuum is unchanged. Identity and replacement by $|0⟩$ have half-diamond distance one, witnessed by $|1⟩$. The replacement Bell overlap is $1/4$. Every target overlap is at least $\eta$, attained by $|1⟩$.