Example: A constant selected residual is insufficient

Let $X|j⟩=|j+1\mathrm{mod}d⟩$ and $Z|j⟩=e^{2\pi ij/d}|j⟩$. Put $W_{ab}=X^a{Z}^b$, for $0\le a,b<d$, and define

This is an isometry. The declared accessible factor is $R$; the complete complement is $K\otimes G$, not $K$.

Proof. Orthogonality of the key basis and unitarity of each $W_{ab}$ give ${V_{\mathrm{key}}}^{†}{V}_{\mathrm{key}}=I_C$. Tracing both key registers gives

The last identity follows by applying the sum to each matrix unit: the phase sum kills off-diagonal units and the shift sum averages diagonal units.

Tracing $R\otimes G$ gives the selected key marginal

Thus that selected residual is constant even though $R$ does not recover the code. The actual complement has matrix elements

It retains input dependence in the joint key coherences. Indeed, because $R$ is constant, the complete $KG$ output exactly recovers the code by Corollary cor:nohiding.