Kirkwood-Dirac classical states based on discrete Fourier transform: Representation with directed graph

• URL: http://arxiv.org/abs/2603.13863v1
• Date: Sat, 14 Mar 2026 09:44:07 GMT
• Title: Kirkwood-Dirac classical states based on discrete Fourier transform: Representation with directed graph
• Abstract summary: A quantum state is KD-classical if its KD-quasiprobability distribution forms a valid classical probability distribution with respect to two given bases, and KD-nonclassical otherwise.<n>We investigate the structural characteristics of the KD-classical state set when the transition matrix between two orthonormal bases takes the form of a discrete Fourier transform matrix.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Kirkwood-Dirac (KD) quasiprobability distribution is a fundamental representation for quantum states and has been widely applied in quantum metrology, quantum chaos, weak values in recent years. A quantum state is KD-classical if its KD-quasiprobability distribution forms a valid classical probability distribution with respect to two given bases, and KD-nonclassical otherwise, with the latter being closely associated with quantum advantages in various quantum processes. In this work, we investigate the structural characteristics of the KD-classical state set when the transition matrix between two orthonormal bases takes the form of a discrete Fourier transform (DFT) matrix. First, we adopt an alternative analytical approach to prove that the set of KD-classical states in a $p^r$-dimensional Hilbert space is the convex hull of KD-classical pure states--a conclusion that was recently established by De Bi{è}vre et al [Annales Henri Poincar{é}, 1-20, 2025]. Furthermore, we define a directed graph and use it to characterize KD-classical pure states in a Hilbert space of arbitrary dimension $d$. That is, the convex hull of KD-classical pure states along any path from the start vertex to the end vertex in this directed graph is exactly the intersection of the KD-classical state set and the linear space spanned by these path-associated KD-classical pure states. This general result not only yields the $p^r$-dimensional conclusion in a straightforward manner but also encompasses Theorem 2 in the existing work [J. Phys. A, 57, 435303, 2024], demonstrating its generality and inclusiveness.

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