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

The channel construction

Hybrid states, controllers, and transition instruments › The channel construction · lines 349–369

VII-G@1#eq:instrument-complete, VII-G@1#eq:hybrid-map

For each $g, h \in G$ , let

Ψ_{h ∣ g} : L (H) ⟶ L (H)

be completely positive. Require

h \sum Ψ_{h ∣ g}^{*} (I) = I (g \in G) .

Nonzero branches are permitted only on certified edges.

Define

(F ρ)_{h} = g \sum Ψ_{h ∣ g} (ρ_{g}) .

Here and below the adjoint is characterized by $Tr (B Ψ (X)) = Tr (Ψ^{*} (B) X)$ .

Displayed formulas in this passage

VII-G@1#eq:instrument-complete · line 356
$h \sum Ψ_{h ∣ g}^{*} (I) = I (g \in G) .$
VII-G@1#eq:hybrid-map · line 363
$(F ρ)_{h} = g \sum Ψ_{h ∣ g} (ρ_{g}) .$
unlabeled · line 352
$Ψ_{h ∣ g} : L (H) ⟶ L (H)$