Theorem: Complete unitary preservation

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

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

For every trace-class $X$, including operators on a code–reference space,

For the Bell input, the complete output satisfies

Proof. The composition gives $V^{†}V=I_C$. The two terms in (eq:restore) are completely positive. Their traces sum to $\mathrm{Tr}Y$, and $\mathcal{L}(VX{V}^{†})=X$.

An isometric embedding preserves nonzero singular values, proving the trace-norm identity. For the Bell input, the complete output is pure with $d$ equal Schmidt coefficients. Its reduced entropies are both ${\log }_{2}d$, and the joint entropy is zero.