Theorem: Finite-chain channel estimate

For every code with maximal particle number $K$,

The same bound holds for the complete output channels on a common enlarged Fock space, and therefore for either reduced output of any fixed common partition.

Proof. Remove the scalar onsite phase. The powers of the finite and infinite adjacency operators applied to $e_0$ agree through degree $L$. Each adjacency operator has norm at most two. The difference between evolved one-particle source vectors therefore has norm at most

The same estimate holds for the label isometries $J_{t,L}$ and $J_{t,\infty }$.

For contractions $J_1,J_2$, tensor telescoping gives

Restriction to the antisymmetric subspace cannot increase the norm. On the direct sum of sectors $k\le K$, the second-quantized difference is therefore at most $K\Vert J_1-J_2\Vert$. Composition with $W$ preserves this upper bound. Apply Lemma lem:isometry and then partial trace.