Hermitian Kirkwood-Dirac real operators for discrete Fourier transformations

TL;DR

This paper investigates the structure of quantum states with positive Kirkwood-Dirac distributions under discrete Fourier transform bases, proving they can be expressed as real linear combinations of pure positive states.

Contribution

It extends previous results by showing that for any finite dimension DFT matrix, KD positive states can be decomposed into real linear combinations of pure KD positive states.

Findings

KD positive states are convex combinations of pure states for prime dimension DFT matrices.

For any finite dimension DFT matrix, KD positive states are real linear combinations of pure KD positive states.

The work generalizes the structure of KD positive states beyond prime dimensions.

Abstract

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.