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

Accessible output algebras

Record compatibility, residuals, and composition › Accessible output algebras · lines 1262–1285

VII-E@1#eq:record-algebra, VII-E@1#eq:observable-distance

For a unital residual algebra $B \subseteq L (H_{c})$ , define

O_{B} = {w \sum ∣ w ⟩ ⟨ w ∣_{R} \otimes B_{w} : B_{w} \in B} .

The record register is classical. Thus $O_{B}$ is the named record–residual observation algebra.

For channels $Θ, Ξ$ with a common output space and a unital output algebra $O$ , define

d_{O} (Θ, Ξ) = \frac{1}{2} ρ sup A = A^{†} \in O ∥ A ∥_{\infty} \leq 1 sup ∣ Tr (A [Θ (ρ) - Ξ (ρ)])∣ .

Displayed formulas in this passage

VII-E@1#eq:record-algebra · line 1266
$O_{B} = {w \sum ∣ w ⟩ ⟨ w ∣_{R} \otimes B_{w} : B_{w} \in B} .$
VII-E@1#eq:observable-distance · line 1278
$d_{O} (Θ, Ξ) = \frac{1}{2} ρ sup A = A^{†} \in O ∥ A ∥_{\infty} \leq 1 sup ∣ Tr (A [Θ (ρ) - Ξ (ρ)])∣ .$