The Kirkwood-Dirac representation associated to the Fourier transform for finite abelian groups: positivity

• URL: http://arxiv.org/abs/2501.12252v1
• Date: Tue, 21 Jan 2025 16:16:55 GMT
• Title: The Kirkwood-Dirac representation associated to the Fourier transform for finite abelian groups: positivity
• Authors: Stephan De Bièvre, Christopher Langrenez, Danylo Radchenko,
• Abstract summary: We construct and study the Kirkwood-Dirac 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.
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• Abstract: 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 $G=\Z_{d}$, 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.

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