Benchmark energy, transmission, and optimal recovery

Choose weights $w_0,\dots ,w_s$ summing to one and frequencies ${\omega }_x>0$. A raw packet has vacuum and occupied states $|x,a⟩$, $a\in \{0,1\}$, with energies $0,{\omega }_x$. Its declared representation has a label and one rail:

Fix $E_{\mathrm{in}}>3{\max }_x{\omega }_x$. Inject

Branch zero emits vacuum and retains all five qubits. Branch $x>0$ extracts $Q_1,Q_2,Q_3$ in successive packets. After $j$ extractions its core energy is $E_{\mathrm{in}}-j{\omega }_x$; branch zero retains energy $E_{\mathrm{in}}$. These maps are isometries on their declared domains because branches are orthogonal and extraction transfers a tensor factor. Their output energies equal their input energies.

Propagation is the rail-independent isometry

with vacuum mapped to two vacua. Thus $H_C+H_D+H_F=E_{\mathrm{in}}I$ on the evolved input space.

For $S\subseteq \{1,2,3\}$, set

Tracing the branch-labelled core and reflected positions gives orthogonal accessible blocks $(x,S)$ for nonempty $S$. Their rail channels are the corresponding reductions of ${\mathcal{A}}_L$. All empty surviving sets form one constant vacuum block. In particular, all flag weights are message-independent.

Define

and