Characterizing Kirkwood-Dirac positive states based on discrete Fourier transform

Kirkwood-Dirac (KD) distribution is helpful to describe nonclassical phenomena and quantum advantages, which have been linked with nonpositive entries of KD distribution. Suppose that A and B are the eigenprojectors of the two eigenbases of two observables and the discrete Fourier transform (DFT) matrix is the transition matrix between the two eigenbases. In a system with prime dimension, the set EKD+ of KD positive states based on the DFT matrix is convex combinations of A and B. That is, EKD+= conv(A) [arXiv:2306.00086]. In this paper, we generalize the result. That is, in a d-dimensional system, EKD+= conv(Ω) for d=p² and $d=pq$, where $p, q$ are prime and Ω is the set of projectors of all the pure KD positive states.