Tight Quasiprobabilistic Entropic Uncertainty Relations for Discrete Quantum Systems
Abstract
It was conjectured by Wehrl and proven by Lieb that the Shannon differential entropy of the Husimi function of a continuous-variable quantum state is lower bounded by the value it takes on coherent states. This corroborates the intuition, familiar from Heisenberg's uncertainty relations, that coherent states are minimal uncertainty states. The analogous statement for the Wigner function would be that the Wigner differential entropy of states with a positive Wigner function is bounded below by the entropy of pure Gaussian states. This was conjectured a decade ago, but recently disproven. Here, we prove the analogous statement for Kirkwood-Dirac distributions with mutually unbiased bases and for the Gross-Wigner distributions. For these discrete systems, the entropy-minimizing role is played by the pure states with positive quasiprobability distributions. Further, we prove that Kirkwood-Dirac distributions with large nonpositivity have large uncertainty.
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BibTeX
@misc{thio2026tight,
title = {{Tight Quasiprobabilistic Entropic Uncertainty Relations for Discrete Quantum Systems}},
author = {Jonathan J. Thio and Stephan De Bièvre and Crispin H. W. Barnes and David R. M. Arvidsson-Shukur},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04009},
eprint = {2611.04009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04009},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tight Quasiprobabilistic Entropic Uncertainty Relations for Discrete Quantum Systems
AU - Jonathan J. Thio
AU - Stephan De Bièvre
AU - Crispin H. W. Barnes
AU - David R. M. Arvidsson-Shukur
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.04009
UR - https://arxiv.org/abs/2611.04009
AB - It was conjectured by Wehrl and proven by Lieb that the Shannon differential entropy of the Husimi function of a continuous-variable quantum state is lower bounded by the value it takes on coherent states. This corroborates the intuition, familiar from Heisenberg's uncertainty relations, that coherent states are minimal uncertainty states. The analogous statement for the Wigner function would be that the Wigner differential entropy of states with a positive Wigner function is bounded below by the entropy of pure Gaussian states. This was conjectured a decade ago, but recently disproven. Here, we prove the analogous statement for Kirkwood-Dirac distributions with mutually unbiased bases and for the Gross-Wigner distributions. For these discrete systems, the entropy-minimizing role is played by the pure states with positive quasiprobability distributions. Further, we prove that Kirkwood-Dirac distributions with large nonpositivity have large uncertainty.
ER -