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

2025/01/21 by Stephan De Bièvre, De Bièvre, Stephan, Christopher Langrenez +3 · 3 citations
Computer Science · Mathematics · #Algebraic and Geometric Analysis #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Quantum Physics (quant-ph) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2501.12252

openalex publication_date 2025/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• What is special about the Kirkwood-Dirac distributions? Only they produce natural conditional expectations
• Almost no experiments have classical Kirkwood-Dirac representations
• Characterizing the Kirkwood-Dirac positivity on second countable LCA groups
• Kirkwood-Dirac Nonpositivity is a Necessary Resource for Quantum Computing

Related

• Characterizing the Kirkwood-Dirac positivity on second countable LCA groups
• Hermitian Kirkwood-Dirac real operators for discrete Fourier transformations
• Fourier transforms from strongly complementary observables
• What is special about the Kirkwood-Dirac distributions? Only they produce natural conditional expectations
• Convex roofs witnessing Kirkwood-Dirac nonpositivity