λEnter Calculus CompanionRegistered manuscripts, kept distinct

Manuscript Hawk02-A@1 · Hawk02-A.tex · definition · 7fbae745

Definition: Distances

Declared objects, references, and cutoffs · lines 254–271

Back to manuscript

For states,

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

For two channels with the same input and output spaces,

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

For channel differences on a $d$ -dimensional input,

∥ Φ - Ψ ∥_{⋄} = ρ_{C Q} sup ∥ ((Φ - Ψ) \otimes id_{Q}) (ρ_{C Q}) ∥_{1},

where a reference of dimension $d$ suffices.

Displayed formulas in this passage

unlabeled · line 256
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1} .$
unlabeled · line 260
$d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
unlabeled · line 264
$∥ Φ - Ψ ∥_{⋄} = ρ_{C Q} sup ∥ ((Φ - Ψ) \otimes id_{Q}) (ρ_{C Q}) ∥_{1},$

Nearby in this section