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

Proposition: Admissible rerouting

Balance laws and admissibility without postselection › Complete fallback design · lines 1401–1425 · proof included or linked

VII-G@1#prop:fallback, VII-G@1#eq:routing

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Suppose ${Φ_{a ∣ g}}_{a}$ is a complete instrument and $a \mapsto Γ_{a} (g)$ is a fixed routing table whose outputs all have valid certificates, including any declared no-update output. Then

Ψ_{h ∣ g} = a : Γ_{a} (g) = h \sum Φ_{a ∣ g}

is a complete hybrid transition.

Observation compatibility holds if each routed map satisfies

Φ_{a ∣ g}^{*} (O_{Γ_{a} (g)}) \subseteq O_{g} .

Proof. Grouping completely positive maps preserves complete positivity, and summing all groups gives the original trace-preserving instrument. Summing the indicated adjoint inclusions gives the branch inclusions.

Displayed formulas in this passage

VII-G@1#eq:routing · line 1407
$Ψ_{h ∣ g} = a : Γ_{a} (g) = h \sum Φ_{a ∣ g}$
unlabeled · line 1414
$Φ_{a ∣ g}^{*} (O_{Γ_{a} (g)}) \subseteq O_{g} .$

Proposition: Admissible rerouting

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