Manuscript Oai290-A@1 · Oai290-A.tex · section · 7c6e104a

Metrics

Objects and the three ledgers › Metrics · lines 298–325

Oai290-A@1#eq:errors

For states and channels, set

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

For a returned logical channel,

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

The first quantity concerns a specified decoder. The second permits all deterministic CPTP decoders unless a smaller class is explicitly declared.

Every referenced input state obeys

d_{tr} ((id_{Q} \otimes E) (ω_{QC}), ω_{QC}) \leq e (E) .

Consequently every joint measurement has total-variation error at most $e (E)$ .

Displayed formulas in this passage

Oai290-A@1#eq:errors · line 307
$e (E) = d_{⋄} (E, id_{C}), ε_{rec} (N) = D CPTP in f d_{⋄} (D N, id_{C}) .$
unlabeled · line 301
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
unlabeled · line 319
$d_{tr} ((id_{Q} \otimes E) (ω_{QC}), ω_{QC}) \leq e (E) .$