In this paper, we investigate the Kirkwood-Dirac nonclassicality and uncertainty diagram based on discrete Fourier transform (DFT) in a $d$ dimensional system. The uncertainty diagram of complete incompatibility bases $\mathcal { A},\mathcal { B}$ are characterized by De Bi\`{ e}vre [arXiv: 2207.07451]. We show that for the uncertainty diagram of the DFT matrix which is a transition matrix from basis $\mathcal { A}$ to basis $\mathcal { B}$, there is no ``hole" in the region of the $(n_{ \mathcal { A}}, n_{ \mathcal { B}})$-plane above and on the line $n_{ \mathcal { A}}+n_{ \mathcal { B}}\geq d+1$, whether the bases $\mathcal { A},\mathcal { B}$ are not complete incompatible bases or not. Then we present that the KD nonclassicality of a state based on the DFT matrix can be completely characterized by using the support uncertainty relation $n_{ \mathcal { A}}(\psi)n_{ \mathcal { B}}(\psi)\geq d$, where $n_{ \mathcal { A}}(\psi)$ and $n_{ \mathcal { B}}(\psi)$ count the number of nonvanishing coefficients in the basis $\mathcal { A}$ and $\mathcal { B}$ representations, respectively. That is, a state $|\psi\rangle$ is KD nonclassical if and only if $n_{ \mathcal { A}}(\psi)n_{ \mathcal { B}}(\psi)> d$, whenever $d$ is prime or not. That gives a positive answer to the conjecture in [Phys. Rev. Lett. \textbf{ 127}, 190404 (2021)].
Categories: quant-ph