Theorem: Normal scalar-output channel

Define

The conditional finite packet states are

The source-to-packet channel is the normal CPTP map

This includes all source matrix units and all references.

Proof. Conjugating a selected Weyl operator by $W(r_s)$ multiplies its baseline characteristic function by the displacement phase determined by (eq:alpha). This is the characteristic function of (eq:packetstates). The latter is a normal density by Lemma lem:gaussian, so characteristic-function uniqueness proves normality of the restricted output. The channel formula follows from Theorem thm:fieldchannel and is CPTP by its measure-and-prepare form.