AI 中文总结
该研究构建离散柱面相空间态的零点层级，发现缠绕相干态具无穷多零点却最具量子性，且零点层级与局域化层级未必同步，为量子计算相关研究提供新视角。
AI 中文摘要
离散柱面相空间为扭曲光子、约瑟夫森结及某些玻色子编码提供了自然框架。在此背景下，我们针对离散柱面相空间上态的恒星分布构建了零点层级，该层级与量子计算的实用性相关。轨道角动量（OAM）和相位本征态无零点，处于层级底层，与平面对相空间上的相干态具有相同数学性质，二者均具有零秩和正维格纳分布。这揭示了柱面的结构特性：其相干态——缠绕相干态——却是最具量子性的，带有无穷多零点，因此“相干态”并非在所有相空间几何中都对应经典性。我们确定了保持或改变零点数的操作，可明确获取层级的每一级。随后我们探究该层级是否也支配局域化：利用$L^r$范数、逆参与率和魏尔熵作为胡西米密度扩展的互补度量，我们发现两个层级未必同步：缠绕相干态即便带有无穷多零点，仍可比OAM本征态更局域化。
英文摘要
The discrete cylinder phase space offers a natural setting for twisted photons, Josephson junctions, and certain bosonic codes. In that context, we build a number-of-zeros hierarchy for the stellar distribution of states on the discrete cylinder phase space, tied to usefulness for quantum computation. The orbital angular momentum (OAM) and phase eigenstate, having no zeros, sits at the bottom of the hierarchy and share mathematical properties with the coherent state on the plane phase space, both sharing zero rank and a positive Wigner distribution. This reveals a structural peculiarity of the cylinder: its coherent state---the wrapped coherent state---is instead the most quantum, carrying infinitely many zeros, so "coherent state" need not signal classicality across different phase-space geometries. We identify the operations that preserve or change the zero count, giving explicit access to every rank of the hierarchy. We then ask whether this hierarchy also governs localization. Using the $L^r$ norm, the inverse participation ratio, and the Wehrl entropy as complementary measures of spread in the Husimi density, we find that the two hierarchies need not track one another: the wrapped coherent state can be more localized than the OAM eigenstate even though it carries infinitely many zeros.
