Manuscript En03-A@1 · En03-A.tex · section · 5398614e

Effect pullback

Scale, records, and residual instruments › Effect pullback · lines 2306–2330

En03-A@1#eq:scale-channel, En03-A@1#eq:aggregated-effects, En03-A@1#eq:record-laws

Let

T = C_{ℓ \to L} \circ Φ_{ℓ} : L (H_{ℓ}) ⟶ L (H_{L})

be a channel. For a fine POVM ${F_{z}}$ , coarse POVM ${E_{w}}$ , and label map $q : Z \to W$ , put

Π_{w} = z : q (z) = w \sum F_{z} .

Empty sums are allowed; ${Π_{w}}$ is a POVM. The compared laws are

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

A fine readout intended at another time must first be pulled back through its evolution.

Displayed formulas in this passage

En03-A@1#eq:scale-channel · line 2309
$T = C_{ℓ \to L} \circ Φ_{ℓ} : L (H_{ℓ}) ⟶ L (H_{L})$
En03-A@1#eq:aggregated-effects · line 2316
$Π_{w} = z : q (z) = w \sum F_{z} .$
En03-A@1#eq:record-laws · line 2322
$p_{T} (w ∣ ρ) = Tr (E_{w} T (ρ)), p_{f} (w ∣ ρ) = Tr (Π_{w} ρ) .$