Theorem: Hamiltonian-derived complete emission

The prepared emission isometry is

It satisfies

With the entire field accessible and memory residual,

Their exact errors are

When $p(t)>0$, an optimal decoder compresses $J_t=p(t{)}^{-1/2}{L}_t$ and replaces the complement by $I_C/d$.

Proof. The label decomposition of Theorem thm:C2 gives (eq:WSF). Unitarity gives $|A{|}^2+{\Vert f\Vert }^2=1$, and label orthogonality gives the operator normalization.

The ground/excited memory sectors and vacuum/one-particle field sectors are orthogonal. The two partial traces therefore give the displayed channels on every operator and reference extension.

Absorb the phase of $A$ into the residual embedding when its weight is nonzero. Apply Lemma lem:erasure with $r=q(t)$. Its endpoint rules avoid division by zero.