Proposition: Complete restoration does not identify a residual

A finite isometry has a CPTP left inverse on its complete output. This does not imply recovery from a chosen component or constancy of its complement.

Proof. For a state ${\tau }_C$, define

It is CP and trace preserving, and $\mathcal{L}(V\rho V^{†})=\rho$. For $V|\psi ⟩=|0{⟩}_R\otimes |\psi {⟩}_B$, the $R$-channel is constant and the $B$-channel is the identity.