Hudson定理在SU(1,1)离散系列中失效
Hudson's theorem fails for the SU(1,1) discrete series
- Università degli Studi di Palermo（巴勒莫大学）
- Université Paris Cité and Université de La Réunion（巴黎西岱大学和留尼汪大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
研究证明Hudson定理在SU(1,1)离散系列的弯曲相空间失效，Wigner正纯态集合大于相干轨道，并解析确定正性窗口受混合角限制。
中文摘要 AI 辅助
Hudson定理指出，玻色模的纯态具有非负Wigner函数当且仅当该态为高斯态。该定理支撑了将Wigner负性视为纯态非经典性的忠实标志这一解读。我们证明该命题在弯曲相空间上没有类似物。对于在双叶双曲面的上叶上实现的$SU(1,1)$正离散系列，Wigner正纯态集合严格大于Perelomov相干轨道。最低权重态与第一激发态的叠加在Bargmann指标$k=1$时，直到混合角$24.93^\circ$仍保持正性，且允许集合具有正体积，其最大宽度在双态方向上无法达到。我们解析地证明，控制正性的二次形式在远场退化为单一混合角。该退化将窗口限制为$\arctan(1/\sqrt{2k})$，并在中间双曲距离处保持阈值本身固定。
英文摘要
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.
