Theorem: Finite communication representation

Let $E\subset L^2(B)$ have dimension $J$. Prepare arbitrary inputs in $E\oplus E$ in the two incoming channels, with orthogonal incoming modes in vacuum. The restriction of multiplication by $S_{\ell }(\omega )$ is an isometry into $E_D^{\prime }\oplus E_F^{\prime }$, where each output space is spanned by the relevant output components of the $2J$ incoming packet functions and has dimension at most $2J$.

On total input occupation at most $N$, its bosonic lift is an exact finite isometry into at most $4J$ output modes with total occupation at most $N$. Tracing one output gives a finite CPTP communication channel.

Proof. Restriction of a unitary one-particle operator is isometric. Its output components lie in the specified spans. On the $n$-particle sector, the symmetric tensor power is isometric and preserves total number. Take the direct sum for $0\le n\le N$, then partial trace.