SU(1,1) 态的维格纳负性与星秩
Wigner negativity and stellar rank for SU(1,1) states
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中文总结 AI 辅助
研究SU(1,1)态的量子特性，通过构建全协变准概率分布族填补相关研究空白，得出维格纳函数对特定相干态严格为正等结果，还研究了星秩及引入广义多极层次结构，提供了量化该系统量子资源的工具包。
中文摘要 AI 辅助
具有SU(1,1)动力学对称性的系统的准概率分布，尽管在双光子物理、压缩态和非线性干涉测量中起核心作用，但受到的关注却出奇地少。在此，我们通过构建一族在双叶双曲面或通过球极投影在庞加莱单位圆盘上定义的全协变s序准概率分布来填补这一空白。关键结果是，维格纳函数对所有佩雷洛莫夫SU(1,1)相干态严格为正，这与SU(2)情况形成鲜明对比。这种正性赋予维格纳负性明确的操作意义：任何负体积都是真正量子行为的直接标志。我们进一步研究了通过胡西米Q函数的零点定义的SU(1,1)态的星秩，并展示了它与维格纳负性作为这种情况下非经典性的几何适配见证的比较。我们还通过双曲面上密度算符的调和展开引入了广义多极层次结构，为探测量子性提供了一个补充框架。这为表征和量化SU(1,1)系统中的量子资源提供了一个全面的工具包。
英文摘要
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 $s$-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 $Q$-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.
