Characterizing an Uncertainty Diagram and Kirkwood–Dirac Nonclassicality Based on Discrete Fourier Transform

In this paper, we investigate an uncertainty diagram and Kirkwood–Dirac (KD) nonclassicality based on discrete Fourier transform (DFT) in a d-dimensional system. We first consider the uncertainty diagram of the DFT matrix, which is a transition matrix from basis [Formula: see text] to basis [Formula: see text]. Here, the bases [Formula: see text] are not necessarily completely incompatible. We show that for the uncertainty diagram of the DFT matrix, there is no “hole” in the region of the [Formula: see text] plane above and on the line [Formula: see text]. Then, we present where the holes are in the region strictly below the line and above the hyperbola [Formula: see text]. Finally, we provide an alternative proof of the conjecture about KD nonclassicality based on DFT.