Manuscript IV@1 · IV.tex · lemma · 351cb2d7

Lemma: Instrument, calibration, and remainder

Shared conventions and elementary measurement facts › One Ramsey pattern, different generators · lines 539–573 · proof included or linked

IV@1#lem:Ramsey, IV@1#eq:prob, IV@1#eq:Taylor

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The operations in (eq:Kraus) form a normal binary CP instrument, and

p_{+} (Ψ) = \frac{1}{2} + \frac{1}{2} Re (e^{- i θ} ⟨ Ψ, e^{i τ A} Ψ ⟩) .

For normalized $Ψ \in Dom A$ ,

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

For bounded self-adjoint $A, B$ ,

e^{i τ A} - e^{i τ B} \leq ∣ τ ∣ ∥ A - B ∥,

and the change in a single probability at fixed $θ$ is at most half this norm.

Proof. Expansion gives $\sum_{\pm} K_{\pm}^{*} K_{\pm} = I$ , proving normalization. Bounded Kraus maps are normal and CP. The same expansion proves (eq:prob). Integrating the scalar inequality $∣ e^{i x} - 1 - i x ∣ \leq x^{2} /2$ against the spectral measure gives (eq:Taylor). The operator bound follows by differentiating $e^{i (τ - s) A} e^{i s B}$ and integrating. Formula (eq:prob) gives the factor $1/2$ .

Displayed formulas in this passage

IV@1#eq:prob · line 543
$p_{+} (Ψ) = \frac{1}{2} + \frac{1}{2} Re (e^{- i θ} ⟨ Ψ, e^{i τ A} Ψ ⟩) .$
IV@1#eq:Taylor · line 550
$⟨ Ψ, e^{i τ A} Ψ ⟩ - 1 - i τ ⟨ Ψ, A Ψ ⟩ \leq \frac{τ ^{2}}{2} ∥ A Ψ ∥^{2} .$
unlabeled · line 557
$e^{i τ A} - e^{i τ B} \leq ∣ τ ∣ ∥ A - B ∥,$

Lemma: Instrument, calibration, and remainder

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