Proposition: The complete complement cannot be deleted by changing returned names

Suppose the terminal residual operation is realized completely by an isometry $W_B:B\to B^{\prime }$, including every unreturned record store and measurement environment. Then

Its accessible channel is the same $\mathcal{N}$.

Proof. Tracing $B^{\prime }$ after the isometry is tracing $B$, so the accessible channel is unchanged.

For any comparison state ${\tau }_B$, the isometry maps it to a comparison state on $B^{\prime }$ and preserves trace norm with references. This gives ${\delta }_{\mathrm{full}}({\mathcal{C}}^{\prime })\le {\delta }_{\mathrm{full}}(\mathcal{C})$. A CPTP left inverse of the isometric channel exists by Proposition prop:global. Applying it to any replacement comparison on $B^{\prime }$ gives the reverse inequality.