Manuscript Em03-G@1 · Em03-G-v1.tex · proposition · 0bb3a8a3

Proposition: Exact coherence-dependent scalar gap

Coherence witnesses, moments, and local observable access › A direct scalar coherence witness · lines 1569–1599 · proof included or linked

Em03-G@1#prop:coherencegap, Em03-G@1#eq:coherencegap

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The sine-effect probability gap for these two inputs is

Δ_{X} (l) = \frac{∣ sin B _{l} ∣}{2} [q (l - 2 a_{1}) + q (l + 2 a_{1})] .

It is nonzero whenever $sin B_{l} \neq = 0$ . For the matched packet comparison, $l = 2 λ v$ gives

Δ_{X} (l) = \frac{1}{2} (1 + e^{- 8 V_{h} λ^{2}}) .

Proof. The input difference is $X = E_{01} + E_{10}$ . Equation (eq:fourchars) gives

N_{G} (X) (W (l)) = - i sin B_{l} [q (l - 2 a_{1}) + q (l + 2 a_{1})] .

Take the imaginary part and divide by two. For the comparison, $B_{l} = - 2 θ_{0} = - π /2$ ; one Gaussian argument is zero and the other is $4 λ v$ .

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Em03-G@1#eq:coherencegap · line 1572
$Δ_{X} (l) = \frac{∣ sin B _{l} ∣}{2} [q (l - 2 a_{1}) + q (l + 2 a_{1})] .$
unlabeled · line 1582
$Δ_{X} (l) = \frac{1}{2} (1 + e^{- 8 V_{h} λ^{2}}) .$
unlabeled · line 1591
$N_{G} (X) (W (l)) = - i sin B_{l} [q (l - 2 a_{1}) + q (l + 2 a_{1})] .$

Proposition: Exact coherence-dependent scalar gap

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