A phase-matched packet model and an explicit decoder

Fix one normalized compact-band outgoing packet $h$, and put

For $\lambda >0$, choose comparison coefficients

Then

The endpoint unitary $U_0$ acts only on the source and this one outgoing packet. The packet's initial density is the geometric density of mean ${\nu }_h$.

The coefficients in (eq:matched) are not compact bulk tests. They are a comparison model; compact realizations are constructed in Section sec:compact.

Let $|\chi ⟩=(|0⟩+\mathrm{i}|1⟩)/\sqrt{2}$, the $Y=+1$ source state. Define the deterministic packet decoder

Its output source basis is identified with the original message. It uses a fresh independent qubit in $|\chi ⟩$, the inverse two-pulse unitary on that qubit and packet, and packet discard. It is CPTP on every normal packet input.