Characterizing Kirkwood-Dirac nonclassicality and uncertainty diagram based on discrete Fourier transform

Abstract

In this paper, we investigate the Kirkwood-Dirac nonclassicality and uncertainty diagram based on discrete Fourier transform (DFT) in a $d$ dimensional system. The uncertainty diagram of complete incompatibility bases $\mathcal{A},\mathcal{B}$ are characterized by De Bièvre [arXiv: 2207.07451]. We show that for the uncertainty diagram of the DFT matrix which is a transition matrix from basis $\mathcal{A}$ to basis $\mathcal{B}$, there is no ``hole" in the region of the $(n_{\mathcal{A}},n_{\mathcal{B}})$-plane above and on the line $n_{\mathcal{A}}+n_{\mathcal{B}}\ge d+1$, whether the bases $\mathcal{A},\mathcal{B}$ are not complete incompatible bases or not. Then we present that the KD nonclassicality of a state based on the DFT matrix can be completely characterized by using the support uncertainty relation $n_{\mathcal{A}}(\psi )n_{\mathcal{B}}(\psi )\ge d$, where $n_{\mathcal{A}}(\psi )$ and $n_{\mathcal{B}}(\psi )$ count the number of nonvanishing coefficients in the basis $\mathcal{A}$ and $\mathcal{B}$ representations, respectively. That is, a state $|\psi ⟩$ is KD nonclassical if and only if $n_{\mathcal{A}}(\psi )n_{\mathcal{B}}(\psi )>d$, whenever $d$ is prime or not. That gives a positive answer to the conjecture in [Phys. Rev. Lett. \textbf{127}, 190404 (2021)].

References in corpus (11)

• Measuring measurement
• Anomalous Weak Values Are Proofs of Contextuality
• Process tomography of ion trap quantum gates
• Weak Values are Interference Phenomena
• Quantum theory of successive projective measurements
• Negative quasiprobabilities enhance phase estimation in quantum-optics experiment
• Quantifying quantum coherence via nonreal Kirkwood-Dirac quasiprobability
• Kirkwood-Dirac nonclassicality, support uncertainty and complete incompatibility
• Conditions tighter than noncommutation needed for nonclassicality
• Classification of incompatibility for two orthonormal bases
• Uncertainty relations for the support of quantum states

Cited by in corpus (3)

• Convex roofs witnessing Kirkwood-Dirac nonpositivity
• Geometry of Kirkwood-Dirac classical states: A case study based on discrete Fourier transform
• Almost no experiments have classical Kirkwood-Dirac representations