Theorem: Complete preservation and global restore

Let $J:C\to H_{\mathrm{in}}$ be an isometry, let $|\Omega ⟩$ be a fixed normalized ancilla, and let $U$ be unitary on the complete input space. Then

is an isometry. Its complete channel has the CPTP left inverse

For every code–reference trace-class operator $X$,

Proof. The preparation and evolution compose to an isometry. Both terms of (eq:restore) are completely positive; their traces sum to $\mathrm{Tr}Y$. The second term vanishes on the encoded range, and $\mathcal{L}(VX{V}^{†})=X$.

An isometric embedding preserves nonzero singular values, including after tensoring with a reference identity.