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

RESUMO

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.