Manuscript Hawk04-C@1 · Hawk04-C-v1.tex · definition · 94d46352

Definition: Recovery, replacement, and distances

Finite carriers, complete outputs, and access · lines 276–307

Hawk04-C@1#def:distances, Hawk04-C@1#eq:distances, Hawk04-C@1#eq:recovery-error, Hawk04-C@1#eq:leakage

For $σ \in D (B)$ , define the replacement channel

T_{σ} (X) = Tr (X) σ .

Use half-normalized trace and diamond distances:

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

For differences of channels with input $C$ , the latter distance is the supremum of the referenced output trace distance over input states on $Q \otimes C$ ; a reference of dimension $d$ suffices. Set

ε_{rec} (N) δ_{full} (E) = D : L (R) \to L (C) CPTP in f d_{⋄} (DN, id_{C}), = σ \in D (B) in f d_{⋄} (E, T_{σ}) .

A decoder is deterministic, and its output is quantum. A normalized successful branch alone is not a decoder in (eq:recovery-error).

Displayed formulas in this passage

Hawk04-C@1#eq:distances · line 283
$d_{tr} (ρ, η) = \frac{1}{2} ∥ ρ - η ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
Hawk04-C@1#eq:recovery-error · line 292
$ε_{rec} (N) δ_{full} (E) = D : L (R) \to L (C) CPTP in f d_{⋄} (DN, id_{C}), = σ \in D (B) in f d_{⋄} (E, T_{σ}) .$
Hawk04-C@1#eq:leakage · line 292
$ε_{rec} (N) δ_{full} (E) = D : L (R) \to L (C) CPTP in f d_{⋄} (DN, id_{C}), = σ \in D (B) in f d_{⋄} (E, T_{σ}) .$
unlabeled · line 279
$T_{σ} (X) = Tr (X) σ .$

Definition: Recovery, replacement, and distances

Displayed formulas in this passage

Nearby in this section