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

Proposition: Uniform instrument transfer

Complete binary instruments and quantitative transfer › Complete-channel rather than record-only errors · lines 497–536 · proof included or linked

IV-B@1#prop:instrument-error, IV-B@1#eq:prob-error, IV-B@1#eq:instrument-error, IV-B@1#eq:duhamel

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For two unitaries $U, V$ , let $I_{U}, I_{V}$ be the channels defined by (eq:kraus)–(eq:ramsey-channel). Then

∣ p_{s} (U; ρ) - p_{s} (V; ρ) ∣ d_{⋄} (I_{U}, I_{V}) \leq \frac{1}{2} ∥ U - V ∥, \leq min {1, ∥ U - V ∥ / 2} .

For bounded self-adjoint $A, B$ ,

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

These estimates survive compatible CPTP pre- and post-composition.

Proof. The probability bound follows from (eq:ramsey-prob). For the complete bound, define isometries

V_{U} ψ = s \sum ∣ s ⟩ \otimes K_{s} (U) ψ .

Then

∥ V_{U} - V_{V} ∥ = ∥ U - V ∥ / 2 .

For any reference extension and trace-class input $Z$ ,

∥ V_{U} Z V_{U}^{*} - V_{V} Z V_{V}^{*} ∥_{1} \leq 2 ∥ V_{U} - V_{V} ∥ ∥ Z ∥_{1} .

Pinching the first factor produces the recorded channels and cannot increase the norm. This proves (eq:instrument-error). Differentiating $e^{i (τ - s) A} e^{i s B}$ and integrating proves (eq:duhamel); negative $τ$ follows by reversal. CPTP composition is a contraction for the complete comparison norm.

Displayed formulas in this passage

IV-B@1#eq:prob-error · line 501
$∣ p_{s} (U; ρ) - p_{s} (V; ρ) ∣ d_{⋄} (I_{U}, I_{V}) \leq \frac{1}{2} ∥ U - V ∥, \leq min {1, ∥ U - V ∥ / 2} .$
IV-B@1#eq:instrument-error · line 501
$∣ p_{s} (U; ρ) - p_{s} (V; ρ) ∣ d_{⋄} (I_{U}, I_{V}) \leq \frac{1}{2} ∥ U - V ∥, \leq min {1, ∥ U - V ∥ / 2} .$
IV-B@1#eq:duhamel · line 508
$e^{i τ A} - e^{i τ B} \leq ∣ τ ∣ ∥ A - B ∥ .$
unlabeled · line 519
$V_{U} ψ = s \sum ∣ s ⟩ \otimes K_{s} (U) ψ .$
unlabeled · line 523
$∥ V_{U} - V_{V} ∥ = ∥ U - V ∥ / 2 .$
unlabeled · line 527
$∥ V_{U} Z V_{U}^{*} - V_{V} Z V_{V}^{*} ∥_{1} \leq 2 ∥ V_{U} - V_{V} ∥ ∥ Z ∥_{1} .$

Proposition: Uniform instrument transfer

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