Hermitian Kirkwood-Dirac real operators for discrete Fourier transformations

The Kirkwood-Dirac (KD) distribution is a quantum state representation that relies on two chosen fixed orthonormal bases, or alternatively, on the transition matrix of these two bases. In recent years, it has been discovered that the KD distribution has numerous applications in quantum information science. The presence of negative or nonreal KD distributions may indicate certain quantum features or advantages. If the KD distribution of a quantum state consists solely of positive or zero elements, the state is called a KD positive state. Consequently, a crucial inquiry arises regarding the determination of whether a quantum state is KD positive when subjected to various physically relevant transition matrices. When the transition matrix is discrete Fourier transform (DFT) matrix of dimension $p$ [\href{https://doi.org/10.1063/5.0164672} {J. Math. Phys. 65, 072201 (2024)}] or $p^{2}$ [\href{https://dx.doi.org/10.1088/1751-8121/ad819a} {J. Phys. A: Math. Theor. 57 435303 (2024)}] with $p$ being prime, it is proved that any KD positive state can be expressed as a convex combination of pure KD positive states. In this work, we prove that when the transition matrix is the DFT matrix of any finite dimension, any KD positive state can be expressed as a real linear combination of pure KD positive states.