Manuscript Oai069-A@1 · Oai069-A.tex · section · a4913aab

Distances and complete comparisons

Phase 1: fix the information task › Distances and complete comparisons · lines 314–330

For states, write

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

For finite-dimensional channels, use

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

Equivalently, for channels this is the supremum trace distance of their outputs on arbitrary reference-entangled input states. The finite-dimensional stabilization and channel conventions are standard Watrous. Our bounds below are proved in the reference-state formulation.

Displayed formulas in this passage

unlabeled · line 317
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1} .$
unlabeled · line 322
$d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$

Nearby in this section

definition: Definition: Claim specification