Manuscript Hawk05-B@1 · Hawk05-B.tex · section · f2e5d4f5

Finite interfaces, reference preservation, and recovery

Finite interfaces, reference preservation, and recovery · lines 244–274

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 retains arbitrary reference extensions Watrous (Chapter 3).

Define replacement, recovery error, and complete leakage by

T_{τ} (X) ε_{rec} (N) δ_{full} (C) = Tr (X) τ, = D : R \to C CPTP in f d_{⋄} (DN, id_{C}), = τ_{B} \in S (B) in f d_{⋄} (C, T_{τ_{B}}) .

We abbreviate

a_{d} = 1 - d^{- 2}, b_{d} = 1 - d^{- 1} .

Compactness ensures that these finite-dimensional infima are attained. Entropies are in bits:

S (ρ) = - Tr (ρ lo g_{2} ρ) .

Displayed formulas in this passage

unlabeled · line 256
$T_{τ} (X) ε_{rec} (N) δ_{full} (C) = Tr (X) τ, = D : R \to C CPTP in f d_{⋄} (DN, id_{C}), = τ_{B} \in S (B) in f d_{⋄} (C, T_{τ_{B}}) .$
unlabeled · line 247
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
unlabeled · line 266
$a_{d} = 1 - d^{- 2}, b_{d} = 1 - d^{- 1} .$
unlabeled · line 271
$S (ρ) = - Tr (ρ lo g_{2} ρ) .$