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

Lemma: Ramsey identity and record law

Complete binary instruments and quantitative transfer › The corrected Ramsey formula · lines 420–450 · proof included or linked

IV-B@1#lem:ramsey, IV-B@1#eq:ramsey-prob

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Equation (eq:ramsey-channel) is CPTP and retains every untouched reference. Its outcome probabilities are

p_{s} (A; ρ) = \frac{1}{2} + \frac{s}{2} Re (e^{- i θ} Tr (ρ e^{i τ A})) .

It is implemented by preparing a probe in $∣ + ⟩$ , applying

W_{A} = ∣ 0 ⟩ ⟨ 0 ∣ \otimes I + ∣ 1 ⟩ ⟨ 1 ∣ \otimes U_{A},

and destructively reading the probe in the basis

∣ s_{θ} ⟩ = \frac{∣ 0 ⟩ + s e ^{i θ} ∣ 1 ⟩}{2} .

Proof. Direct expansion gives $\sum_{s} K_{s}^{*} K_{s} = I$ , which proves complete positivity and trace preservation. The same expansion gives (eq:ramsey-prob). Moreover,

(⟨ s_{θ} ∣ \otimes I) W_{A} (∣ + ⟩ \otimes I) = K_{s} (A) .

Tensoring these operator identities with any reference identity proves the complete statements.

Displayed formulas in this passage

IV-B@1#eq:ramsey-prob · line 424
$p_{s} (A; ρ) = \frac{1}{2} + \frac{s}{2} Re (e^{- i θ} Tr (ρ e^{i τ A})) .$
unlabeled · line 431
$W_{A} = ∣ 0 ⟩ ⟨ 0 ∣ \otimes I + ∣ 1 ⟩ ⟨ 1 ∣ \otimes U_{A},$
unlabeled · line 435
$∣ s_{θ} ⟩ = \frac{∣ 0 ⟩ + s e ^{i θ} ∣ 1 ⟩}{2} .$
unlabeled · line 445
$(⟨ s_{θ} ∣ \otimes I) W_{A} (∣ + ⟩ \otimes I) = K_{s} (A) .$

Nearby in this section

section: The corrected Ramsey formula