Manuscript En02-B@1 · En02-B-v1.tex · section · ca28dee4

Encoding, access, and three recovery tasks

Encoding, access, and three recovery tasks · lines 298–322

En02-B@1#eq:diamond

Equation (eq:recovery) is an equality of linear maps. Consequently,

(D_{A} N_{A} \otimes id_{R}) (ρ_{ab R}) = Tr_{b} ρ_{ab R}

for every reference and every input.

For approximate recovery we use

d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄}, d_{⋄} (D_{A} N_{A}, Tr_{b}) \leq ε .

The diamond norm is the completely bounded trace norm. For channel differences, it controls every input entangled with a reference; standard finite-dimensional channel conventions are those of Watrous (Chapters 2–3).

Displayed formulas in this passage

En02-B@1#eq:diamond · line 310
$d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄}, d_{⋄} (D_{A} N_{A}, Tr_{b}) \leq ε .$
unlabeled · line 302
$(D_{A} N_{A} \otimes id_{R}) (ρ_{ab R}) = Tr_{b} ρ_{ab R}$

Nearby in this section

section: Encoding, access, and three recovery tasks
definition: Definition: Recovery tasks
remark: Remark: A selected state is not a channel certificate