Manuscript Oai265-A@2 · Oai265-A.tex · section · c0a34a45

Norms and tasks

Stage 5: complete channels and the entropy separation › Norms and tasks · lines 1233–1268

For states and channels, use

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

The diamond norm includes arbitrary untouched reference systems Watrous.

Let $C ≃ C^{d}$ , $d \geq 2$ , and let $V : C \to R \otimes B$ be an isometry. Its accessible and complete complementary channels are

N (X) = Tr_{B} (V X V^{†}), E (X) = Tr_{R} (V X V^{†}) .

Define information-theoretic recovery error and complete leakage by

ε_{rec} (N) = D CPTP in f d_{⋄} (DN, id_{C}),

δ_{leak} (E) = σ_{B} in f d_{⋄} (E, R_{σ_{B}}), R_{σ} (X) = Tr (X) σ .

A resource-constrained decoder is a further, separate optimization.

Displayed formulas in this passage

unlabeled · line 1236
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥Φ - Ψ ∥_{⋄} .$
unlabeled · line 1247
$N (X) = Tr_{B} (V X V^{†}), E (X) = Tr_{R} (V X V^{†}) .$
unlabeled · line 1253
$ε_{rec} (N) = D CPTP in f d_{⋄} (DN, id_{C}),$
unlabeled · line 1259
$δ_{leak} (E) = σ_{B} in f d_{⋄} (E, R_{σ_{B}}), R_{σ} (X) = Tr (X) σ .$