Manuscript VII-G@1 · VII-G.tex · proposition · 54f561e8

Proposition: Composition

Hybrid states, controllers, and transition instruments › The channel construction · lines 433–456 · proof included or linked

VII-G@1#prop:composition-channel, VII-G@1#eq:branch-composition

If a second transition has branches $Λ_{k ∣ h}$ , the composed channel has branches

Ω_{k ∣ g} = h \sum Λ_{k ∣ h} \circ Ψ_{h ∣ g} .

Its certificates are concatenations of the declared edge certificates.

Proof. Substitution into the two block formulas gives (eq:branch-composition). Sums and compositions preserve complete positivity. Moreover,

k \sum Ω_{k ∣ g}^{*} (I) = h \sum Ψ_{h ∣ g}^{*} (k \sum Λ_{k ∣ h}^{*} (I)) = h \sum Ψ_{h ∣ g}^{*} (I) = I .

The certificate assertion means concatenation of already declared certificates; it introduces no new geometric existence claim.

Displayed formulas in this passage

VII-G@1#eq:branch-composition · line 437
$Ω_{k ∣ g} = h \sum Λ_{k ∣ h} \circ Ψ_{h ∣ g} .$
unlabeled · line 448
$k \sum Ω_{k ∣ g}^{*} (I) = h \sum Ψ_{h ∣ g}^{*} (k \sum Λ_{k ∣ h}^{*} (I)) = h \sum Ψ_{h ∣ g}^{*} (I) = I .$