Manuscript IV-B@1 · IV-B-v1.tex · lemma · dfb66408

Lemma: Mean separation implies a small-time contrast

Complete binary instruments and quantitative transfer › A general positive-record certificate · lines 454–485 · proof included or linked

IV-B@1#lem:contrast, IV-B@1#eq:positive-record

Let $ψ_{0}, ψ_{1} \in Dom A$ be unit vectors, and put

d = ∣ ⟨ ψ_{1}, A ψ_{1} ⟩ - ⟨ ψ_{0}, A ψ_{0} ⟩ ∣ > 0, S = ∥ A ψ_{0} ∥^{2} + ∥ A ψ_{1} ∥^{2} .

For $θ = π /2$ and $0 < τ \leq d / S$ ,

∣ p_{+} (A; ψ_{1}) - p_{+} (A; ψ_{0}) ∣ \geq \frac{τ d}{4} > 0.

Proof. The spectral theorem and $e^{i x} - 1 - i x \leq x^{2} /2$ give

⟨ ψ, e^{i τ A} ψ ⟩ - 1 - i τ ⟨ ψ, A ψ ⟩ \leq \frac{τ ^{2}}{2} ∥ A ψ ∥^{2} .

At $θ = π /2$ , the record reads the imaginary part. Subtracting the two expectations yields

∣ Δ p_{+} ∣ \geq \frac{τ d}{2} - \frac{τ ^{2} S}{4} \geq \frac{τ d}{4} .

Displayed formulas in this passage

IV-B@1#eq:positive-record · line 463
$∣ p_{+} (A; ψ_{1}) - p_{+} (A; ψ_{0}) ∣ \geq \frac{τ d}{4} > 0.$
unlabeled · line 457
$d = ∣ ⟨ ψ_{1}, A ψ_{1} ⟩ - ⟨ ψ_{0}, A ψ_{0} ⟩ ∣ > 0, S = ∥ A ψ_{0} ∥^{2} + ∥ A ψ_{1} ∥^{2} .$
unlabeled · line 473
$⟨ ψ, e^{i τ A} ψ ⟩ - 1 - i τ ⟨ ψ, A ψ ⟩ \leq \frac{τ ^{2}}{2} ∥ A ψ ∥^{2} .$
unlabeled · line 480
$∣ Δ p_{+} ∣ \geq \frac{τ d}{2} - \frac{τ ^{2} S}{4} \geq \frac{τ d}{4} .$