Proposition: Exact Z-readout with a nonconstant complement

Let $C,A,B$ be qubits and $V|i⟩=|i{⟩}_A|i{⟩}_B$, $i=0,1$. Both $\mathcal{N}$ and $\mathcal{C}$ are computational-basis dephasing. For $F=|0⟩⟨0|$, boundary effect $G=|0⟩⟨0|$ gives ${\eta }_F(G)=0$. Nevertheless exact full-code recovery is impossible, and the identity decoder has error $1/2$.

Proof. Dephasing fixes $F$, so ${\mathcal{N}}^{\ast }(G)=F$. The complement distinguishes the two basis inputs and is nonconstant. For the identity decoder,

The triangle inequality gives error at most $1/2$. On a Bell input the output is an equal mixture of two orthogonal Bell states, giving distance $1/2$ and equality.