The Kirkwood-Dirac representation associated to the Fourier transform for finite abelian groups: positivity
arXiv:2501.12252 · doi:10.1007/s00023-025-01614-7
Abstract
We construct and study the Kirkwood-Dirac (KD) representations naturally associated to the Fourier transform of finite abelian groups . 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 , with 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.
arXiv comment17 pages, references added, typos corrected
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