Manuscript Hawk01-B@1 · Hawk01-B.tex · section · cfa976f1

Preservation criteria and the complete complement

Preservation criteria and the complete complement · lines 185–209

Hawk01-B@1#eq:normalization, Hawk01-B@1#eq:Phi

All Hilbert spaces in the channel theorems are finite-dimensional. Logarithms and entropies use base two. For states and channels, respectively, define

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

The diamond norm is the stabilized trace norm in the Schrödinger picture. Its dual is the completely bounded operator norm used in the Heisenberg-picture formulation of Kretschmann–Schlingemann–Werner KSW2006.

Let $A$ be a code of dimension $d \geq 2$ , contained in one declared energy and charge sector. Let $S ≅ A$ be an external reference, and write

∣ Φ_{d} ⟩^{S A} = \frac{1}{d} i = 0 \sum^{d - 1} ∣ i ⟩_{S} ∣ i ⟩_{A} .

Displayed formulas in this passage

Hawk01-B@1#eq:normalization · line 190
$D_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, D_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
Hawk01-B@1#eq:Phi · line 204
$∣ Φ_{d} ⟩^{S A} = \frac{1}{d} i = 0 \sum^{d - 1} ∣ i ⟩_{S} ∣ i ⟩_{A} .$

Preservation criteria and the complete complement

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