Manuscript Hawk02-D@1 · Hawk02-D.tex · section · e6e4b72f

Channels, complete residuals, and reference-stable estimates

Channels, complete residuals, and reference-stable estimates · lines 217–266

Hawk02-D@1#eq:recovery-definition, Hawk02-D@1#eq:leakage-definition

All logarithms used for parameter selection are natural. For normal states and channels, write

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

The diamond norm retains arbitrary external references. The input code $C$ has finite dimension $d$ .

For an isometry

V : C ⟶ R \otimes B,

define

N_{V} (X) = Tr_{B} (V X V^{†}), C_{V} (X) = Tr_{R} (V X V^{†}) .

Here $B$ is the entire inaccessible output, including preparation purifiers and apparatus outputs whenever they are present.

The replacement channel associated with a state $σ$ is

T_{σ} (X) = Tr (X) σ .

Define

ε_{rec} (N) δ_{full} (C) = D CPTP in f d_{⋄} (DN, id_{C}), = σ in f d_{⋄} (C, T_{σ}) .

For the declared pure preparations in the benchmark, $C$ is the physical complete complement of the same evolution used to calculate $N$ .

Standard channel conventions and finite-reference reduction are as in Watrous. The estimates needed below are stated explicitly.

Displayed formulas in this passage

Hawk02-D@1#eq:recovery-definition · line 249
$ε_{rec} (N) δ_{full} (C) = D CPTP in f d_{⋄} (DN, id_{C}), = σ in f d_{⋄} (C, T_{σ}) .$
Hawk02-D@1#eq:leakage-definition · line 249
$ε_{rec} (N) δ_{full} (C) = D CPTP in f d_{⋄} (DN, id_{C}), = σ in f d_{⋄} (C, T_{σ}) .$
unlabeled · line 221
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
unlabeled · line 232
$V : C ⟶ R \otimes B,$
unlabeled · line 236
$N_{V} (X) = Tr_{B} (V X V^{†}), C_{V} (X) = Tr_{R} (V X V^{†}) .$
unlabeled · line 245
$T_{σ} (X) = Tr (X) σ .$