Theorem: Complete outgoing endpoint

For the model of Section sec:model, the outgoing wave operator

restricts to a unitary

Let $J_S:{\mathbb{C}}^n\to \mathfrak{h}$ include the source modes and define

Then $F_{\mathrm{out}}$ is an isometry.

In the outgoing representation, the complete prepared evolution converges in operator norm on the finite code to

Its complete complementary channel is constant:

Consequently,

No terminal release operation is appended to the dynamics.

Proof. Proposition prop:spectral proves that $h$ is purely absolutely continuous. The identical sine-transform argument proves that $h_B$ is purely absolutely continuous. The decoupled source contributes only a finite-dimensional pure-point subspace. Hence

The perturbation consists of the $n$ source–boundary bonds and has rank at most $2n$, so it is trace class.

The external scattering theorem therefore gives an isometric wave operator from ${\mathfrak{h}}_B$ onto all of $\mathfrak{h}$. It is unitary between these spaces, proving that $F_{\mathrm{out}}$ is an isometry.

Completeness also gives the inverse asymptotic relation

for every $f\in \mathfrak{h}$, with the limiting vector embedded in the reservoir summand. One can see this directly by writing $f={\Omega }_{+}g$: the norm of the difference is the norm of

which tends to zero by the definition of ${\Omega }_{+}$.

Apply this to the finitely many source label vectors. Strong convergence on their finite-dimensional span is operator-norm convergence. Tensor telescoping, as in Theorem thm:finite-chain, upgrades it to the sectors with at most $K$ particles. Composition with the finite code $W$ gives operator-norm convergence of the complete prepared isometries.

In the limit there is no source excitation. The carrier was already fixed vacuum by Proposition prop:loading. All reservoir output modes are included in the declared outgoing access space. Taking the complementary partial trace therefore gives the constant channel above.

The outgoing code map $\Gamma (F_{\mathrm{out}})W$ is an isometry. Its inverse on its range, completed by state preparation on the orthogonal complement as in (eq:complete-decoder), gives exact recovery.