Manuscript VII-J@1 · VII-J.tex · section · d8c01b04

Finite channels and operational influence

Finite channels and operational influence · lines 303–330

Let the input factors be indexed by a finite set $X$ :

H_{X} = x \in X ⨂ H_{x} .

For $S \subseteq X$ , define

A_{X} (S) = L (x \in S ⨂ H_{x}) \otimes I_{X ∖ S} .

Output factors may have a different index set $Y$ .

A channel is completely positive and trace preserving. Its adjoint is defined by

Tr (B Φ (Z)) = Tr (Φ^{*} (B) Z) .

Adjoints are unital. We use

d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .

The diamond norm, without the factor $1/2$ , is denoted explicitly.

Displayed formulas in this passage

unlabeled · line 307
$H_{X} = x \in X ⨂ H_{x} .$
unlabeled · line 311
$A_{X} (S) = L (x \in S ⨂ H_{x}) \otimes I_{X ∖ S} .$
unlabeled · line 320
$Tr (B Φ (Z)) = Tr (Φ^{*} (B) Z) .$
unlabeled · line 324
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$