AI 中文总结
该研究证明可通过单快照系综的概率分布确定Laughlin准空穴的量子几何，实现度量、相位等的有限次测量重构，达到多参数信息界。
AI 中文摘要
一个基下测得的概率分布能否确定复杂量子几何？对于格点Laughlin准空穴，答案是肯定的。一个通用参考位置处的精确占据律，结合已知的解析准空穴因子，可确定完整的复Gram核。有限快照系综无需制备该族的另一个成员即可估计此核，进而确定有限距离重叠、Bargmann相位、量子度量和Berry曲率。相同的解析结构将该族限制在精确射影子空间内，其维度随粒子数最多线性增长。对Nielsen-Cirac-Sierra态的精确枚举证明了度量和几何相位的有限次测量重构，此外，Rényi-2散度设定了重构的统计范围，在非简并点，占据读出也达到了局域单副本多参数信息界。
英文摘要
Can a probability distribution measured in one basis determine complex quantum geometry? The answer is affirmative for a lattice Laughlin quasihole. The exact occupation law at one generic reference position, together with the known analytic quasihole factor, fixes the complete complex Gram kernel. A finite snapshot ensemble estimates this kernel without preparing another member of the family, and thereby determines finite-distance overlaps, Bargmann phases, the quantum metric, and the Berry curvature. The same analytic structure confines the family to an exact projective subspace whose dimension grows at most linearly with particle number. Exact enumeration of a Nielsen-Cirac-Sierra state demonstrates finite-shot reconstruction of both the metric and a geometric phase. Moreover, a Rényi-2 divergence sets the statistical range of the reconstruction while, at nondegenerate points, occupation readout also attains the local single-copy multiparameter information bound.
Comments3 pages, 1 figure; Supplemental Material appended
