Theorem: Normal outgoing-packet channel and Choi state

The selected packet channel is the normal CPTP map

Its normalized reference-output state is

Equations (eq:fourchars) determine its selected Weyl characteristic functions exactly.

The diagonal outputs are equal-weight mixtures of the two displaced baseline packet states corresponding to $-z{a}_1-a_2$ and $-z{a}_1+a_2$. The off-diagonal outputs are interference operators and are not positive preparation states.

Proof. Both source endpoint factors act only on $S\otimes {\mathcal{F}}_{\mathrm{s}}(\mathfrak{k})$. Apply them to $\rho \otimes {\rho }_{\mathfrak{k}}$, trace the source and unselected participating modes, and use (eq:branches). This proves the trace-class formula and complete positivity with trace preservation on every source operator and reference extension.

For a diagonal source input there are two orthogonal final source $X$-branches, each of weight $1/2$. Each full-field Weyl conjugation displaces the selected Gaussian baseline. For off-diagonals the two branch coefficients have opposite signs; their trace is zero by (eq:fourchars) at $l=0$. There is no measure-and-prepare interpretation for those terms.