Kirkwood-Dirac classical pure states

Kirkwood-Dirac (KD) distribution is a representation of quantum states. Recently, KD distribution has been found applications in many areas such as in quantum metrology, quantum chaos and foundations of quantum theory. KD distribution is a quasiprobability distribution, and negative or nonreal elements may signify quantum advantages in certain tasks. A quantum state is called KD classical if its KD distribution is a probability distribution. Since most quantum information processings use pure states as ideal resources, then a key problem is to determine whether a quantum pure state is KD classical. In this paper, we provide some characterizations for the general structure of KD classical pure states. As an application of our results, we prove a conjecture raised by De Bièvre (2021) [56] which finds out all KD classical pure states for discrete Fourier transformation.