Theorem: Complete isometric preservation

For every isometry $V:C\to AE$, the complete channel ${\mathrm{Ad}}_V(X)=VX{V}^{†}$ has a deterministic CPTP left inverse. Moreover, for every operator $X$ on $QC$,

Proof. For any fixed code state ${\omega }_C$, define

Both terms are completely positive, their traces sum to $\mathrm{Tr}Y$, and $\mathcal{L}(VX{V}^{†})=X$. An isometric embedding preserves all nonzero singular values. Apply this observation also to $I_Q\otimes V$.