Manuscript Oai283-A@1 · Oai283-A.tex · proposition · 6fb75895

Proposition: Bell overlap is not a dimension-free worst-case bound

Operational certification of the logical channel › What Bell tests can establish · lines 1075–1092 · proof included or linked

Oai283-A@1#prop:Bell

For $d \geq 2$ , let $V = I - 2∣0 ⟩ ⟨ 0∣$ . Then

d_{⋄} (Ad_{V}, id) = 1, ⟨ Ω_{d} ∣ J_{n} (Ad_{V}) ∣ Ω_{d} ⟩ = (1 - \frac{2}{d})^{2} .

Thus Bell overlap can tend to one while worst-case channel error remains maximal.

Proof. The inputs $(∣0 ⟩ + ∣1 ⟩) / 2$ under identity and $V$ become orthogonal. The Bell overlap is $∣ Tr V / d ∣^{2}$ .

Displayed formulas in this passage

unlabeled · line 1078
$d_{⋄} (Ad_{V}, id) = 1, ⟨ Ω_{d} ∣ J_{n} (Ad_{V}) ∣ Ω_{d} ⟩ = (1 - \frac{2}{d})^{2} .$