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<p>We construct and study the Kirkwood-Dirac (KD) representations naturally associated to the Fourier transform of finite abelian groups <i>G</i>. We identify all pure KD-positive states and all KD-real observables for these KD representations. We provide a necessary and sufficient condition ensuring that all KD-positive states are convex combinations of pure KD-positive states. We prove that for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2025_1614_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(G=\mathbb {Z}_{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mi>d</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, with <i>d</i> a prime power, this condition is satisfied. We provide examples of abelian groups where it is not. In those cases, the convex set of KD-positive states contains states outside the convex hull of the pure KD-positive states.</p>

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The Kirkwood-Dirac Representation Associated to the Fourier Transform for Finite Abelian Groups: Positivity

  • 期刊论文

  • 发表日期：

    2025年9月2日

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  • Stephan De Bièvre,
  • Christopher Langrenez,
  • Danylo Radchenko

摘要

We construct and study the Kirkwood-Dirac (KD) representations naturally associated to the Fourier transform of finite abelian groups G. We identify all pure KD-positive states and all KD-real observables for these KD representations. We provide a necessary and sufficient condition ensuring that all KD-positive states are convex combinations of pure KD-positive states. We prove that for \(G=\mathbb {Z}_{d}\) G = Z d , with d a prime power, this condition is satisfied. We provide examples of abelian groups where it is not. In those cases, the convex set of KD-positive states contains states outside the convex hull of the pure KD-positive states.