Characterizing Kirkwood-Dirac positive states based on discrete Fourier transform

• URL: http://arxiv.org/abs/2404.09399v1
• Date: Mon, 15 Apr 2024 01:09:21 GMT
• Title: Characterizing Kirkwood-Dirac positive states based on discrete Fourier transform
• Authors: Ying-Hui Yang, Shuang Yao, Shi-Jiao Geng, Xiao-Li Wang, Pei-Ying Chen,
• Abstract summary: Kirkwood-Dirac (KD) distribution is helpful to describe nonclassical phenomena and quantum advantages. In this paper, we generalize the result to a $d$-dimensional system.
• Score: 7.9992601246096315
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Kirkwood-Dirac (KD) distribution is helpful to describe nonclassical phenomena and quantum advantages, which have been linked with nonpositive entries of KD distribution. Suppose that $\mathcal{A}$ and $\mathcal{B}$ are the eigenprojectors of the two eigenbases of two observables and the discrete Fourier transform (DFT) matrix is the transition matrix between the two eigenbases. In a system with prime dimension, the set $\mathcal{E}_{KD+}$ of KD positive states based on the DFT matrix is convex combinations of $\mathcal{A}$ and $\mathcal{B}$. That is, $\mathcal{E}_{KD+}={\rm conv}(\mathcal{A}\cup\mathcal{B})$ [arXiv:2306.00086]. In this paper, we generalize the result. That is, in a $d$-dimensional system, $\mathcal{E}_{KD+}={\rm conv}(\Omega)$ for $d=p^{2}$ and $d=pq$, where $p, q$ are prime and $\Omega$ is the set of projectors of all the pure KD positive states.

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