Manuscript En08-B@1 · En08-B.tex · section · 23cac52a

Reference-stable distances

The frozen model › Reference-stable distances · lines 320–346

En08-B@1#eq:distances

For finite-dimensional compatible channels $Φ, Ψ$ , use

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

One definition of the diamond norm is

∥ Θ ∥_{⋄} = r \geq 1 sup ∥ Z ∥_{1} = 1 sup ∥ (Θ \otimes id_{r}) (Z) ∥_{1} .

In finite input dimension the reference size can be bounded by the input dimension. Channels have diamond norm one, and compatible pre- and post-composition by channels are contractions Watrous2018 (Chapter 3).

The use of a complete channel, rather than selected input tests, is compatible with the finite-channel conventions in the supplied Cheung manuscripts CheungEn05C,CheungEn02C. The corresponding finite telescoping estimate is proved below.

Displayed formulas in this passage

En08-B@1#eq:distances · line 323
$d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄}, d_{tr} (ρ, ω) = \frac{1}{2} ∥ ρ - ω ∥_{1} .$
unlabeled · line 330
$∥ Θ ∥_{⋄} = r \geq 1 sup ∥ Z ∥_{1} = 1 sup ∥ (Θ \otimes id_{r}) (Z) ∥_{1} .$