Theorem: Complete restoration

For the interface (eq:interface), the complete channel $\mathcal{V}$ has the CPTP left inverse

Moreover, for every reference $Q$ and every operator $X$ on $Q\otimes C$,

The left inverse requires access to the joint output $RB$.

Proof. The compression term in (eq:global-decoder) is completely positive. The other term is a positive-effect measurement followed by preparation and is completely positive. Their traces sum to $\mathrm{Tr}Y$, so ${\mathcal{L}}_V$ is trace preserving. On $VX{V}^{†}$, the second term vanishes and the first returns $X$. An isometric embedding preserves all nonzero singular values, including after tensoring with a reference identity. This proves (eq:global-preservation).