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arXiv:2609.07242

Hudson's theorem fails for the SU(1,1) discrete series

Chon-Fai Kam

quant-phSubmitted Sep 7, 2026• Announced Sep 8, 2026• 0 citations
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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 $24.93^\circ$ 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 $\arctan(1/\sqrt{2k})$ and leaves the threshold itself fixed at intermediate hyperbolic distance.

Comments
10 pages, 1 figure, 3 tables. Submitted to Physical Review A. Code and data: doi:10.5281/zenodo.22310147

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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)SU(1,1)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∘24.93^\circ24.93∘ at Bargmann index k=1k=1k=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)\arctan(1/\sqrt{2k})arctan(1/2k​) and leaves the threshold itself fixed at intermediate hyperbolic distance.
We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.
According to Hudson's theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. Here, we make a step toward the extension of this theorem to mixed quantum states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity of a state, its purity, and the purity of the Gaussian state characterized by the same covariance matrix. Although our bounds are not tight, they permit us to visualize the set of states with positive Wigner functions.
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Quasiprobability distributions for systems endowed with SU(1,1) dynamical symmetry have received surprisingly little attention, despite the central role of this symmetry in two-photon physics, squeezed states, and nonlinear interferometry. Here, we fill this gap by constructing a full covariant family of sss-ordered quasiprobability distributions defined on the two-sheeted hyperboloid, or equivalently, on the Poincaré unit disk via stereographic projection. A key result is that the Wigner function is strictly positive for all Perelomov SU(1,1) coherent states, in sharp contrast to the SU(2) case. This positivity endows Wigner negativity with an unambiguous operational meaning: any negative volume is a direct signature of genuinely quantum behavior. We further examine the stellar rank of SU(1,1) states, defined through the zeros of the Husimi QQQ-function, and show how it compares with Wigner negativity as a geometry-adapted witness of nonclassicality in this setting. We further introduce a hierarchy of generalized multipoles through a harmonic expansion of the density operator on the hyperboloid, providing a complementary framework for probing quantumness. This offers a comprehensive toolkit for characterizing and quantifying quantum resources in SU(1,1) systems.