Hudson's theorem fails for the SU(1,1) discrete series
Abstract
Hudson's theorem states that a pure state of a bosonic mode has a non-negative Wigner function if and only if it is Gaussian. It underwrites the reading of Wigner negativity as a faithful signature of pure-state non-classicality. We show the statement has no analogue on curved phase space. For the positive discrete series of SU(1,1), realised on the upper sheet of a two-sheeted hyperboloid, the Wigner-positive pure states form a strictly larger set than the Perelomov coherent orbit. Superpositions of the lowest weight state with the first excited state stay positive up to a mixing angle of 24.93∘ at Bargmann index k=1, and the admissible set has positive volume, with a maximal width that is not attained in the two-state direction. We show analytically that the quadratic form controlling positivity degenerates in the far field onto a single mixing angle. That degeneracy bounds the window by arctan(1/√(2k)) and leaves the threshold itself fixed at intermediate hyperbolic distance.
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BibTeX
@misc{kam2026hudsons,
title = {{Hudson's theorem fails for the SU(1,1) discrete series}},
author = {Chon-Fai Kam},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07242},
eprint = {2609.07242},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07242},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hudson's theorem fails for the SU(1,1) discrete series
AU - Chon-Fai Kam
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07242
UR - https://arxiv.org/abs/2609.07242
AB - Hudson's theorem states that a pure state of a bosonic mode has a non-negative Wigner function if and only if it is Gaussian. It underwrites the reading of Wigner negativity as a faithful signature of pure-state non-classicality. We show the statement has no analogue on curved phase space. For the positive discrete series of SU(1,1), realised on the upper sheet of a two-sheeted hyperboloid, the Wigner-positive pure states form a strictly larger set than the Perelomov coherent orbit. Superpositions of the lowest weight state with the first excited state stay positive up to a mixing angle of 24.93^∘ at Bargmann index k=1, and the admissible set has positive volume, with a maximal width that is not attained in the two-state direction. We show analytically that the quadratic form controlling positivity degenerates in the far field onto a single mixing angle. That degeneracy bounds the window by arctan(1/√(2k)) and leaves the threshold itself fixed at intermediate hyperbolic distance.
ER -