Geometry of Kirkwood-Dirac classical states: A case study based on discrete Fourier transform

The characterization of Kirkwood-Dirac (KD) classicality or non-classicality is very important in quantum information processing. In general, the set of KD classical states with respect to two bases is not a convex polytope[J. Math. Phys. \textbf{65} 072201 (2024)], which makes us interested in finding out in which circumnstances they do form a polytope. In this paper, we focus on the characterization of KD classicality of mixed states for the case where the transition matrix between two bases is a discrete Fourier transform (DFT) matrix in Hilbert space with dimensions $p^2$ and $pq$, respectively, where $p, q$ are prime. For the two particular cases we investigate, the sets of extremal points are finite, implying that the set of KD classical states we characterize forms a convex polytope. We show that for $p^2$ dimensional system, the set $\rm{KD}_{\mathcal{A},\mathcal{B}}^+$ is a convex hull of the set $\rm {pure}({\rm {KD}_{\mathcal{A},\mathcal{B}}^+})$ based on DFT, where $\rm{KD}_{\mathcal{A},\mathcal{B}}^+$ is the set of KD classical states with respect to two bases and $\rm {pure}({\rm {KD}_{\mathcal{A},\mathcal{B}}^+})$ is the set of all the rank-one projectors of KD classical pure states with respect to two bases. In $pq$ dimensional system, we believe that this result also holds. Unfortunately, we do not completely prove it, but some meaningful conclusions are obtained about the characterization of KD classicality.