Manuscript VII-E@1 · VII-E.tex · section · b713cf3c

Effect pullback

Record compatibility, residuals, and composition › Effect pullback · lines 1103–1124

VII-E@1#eq:aggregated, VII-E@1#eq:record-laws

Let

T : L (H_{f}) ⟶ L (H_{c})

be a channel. It may include prescribed evolution followed by a coarse-graining channel. Let ${F_{z} : z \in Z}$ be a fine POVM, ${E_{w} : w \in W}$ a coarse POVM, and $q : Z \to W$ a label map. Define

Π_{w} = z : q (z) = w \sum F_{z} .

Empty sums are zero. The two record laws are

p_{T} (w ∣ ρ) = Tr (E_{w} T (ρ)), p_{f} (w ∣ ρ) = Tr (Π_{w} ρ) .

Displayed formulas in this passage

VII-E@1#eq:aggregated · line 1113
$Π_{w} = z : q (z) = w \sum F_{z} .$
VII-E@1#eq:record-laws · line 1118
$p_{T} (w ∣ ρ) = Tr (E_{w} T (ρ)), p_{f} (w ∣ ρ) = Tr (Π_{w} ρ) .$
unlabeled · line 1106
$T : L (H_{f}) ⟶ L (H_{c})$

Nearby in this section

theorem: Theorem: Exact record compatibility
section: Effect pullback