发表机构
Indian Statistical Institute（印度统计研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明量子U-统计量是无偏置换不变估计量的唯一扩展，其方差主导项达到量子Cramér-Rao极限，实现渐近效率，并应用于Bures χ²散度的估计。
AI 中文摘要
我们研究基于独立副本的量子态标量值多项式泛函的无偏估计。我们建立了置换不变有限副本可观测量的边缘分布与泛函梯度之间的等价关系。随后我们证明，在无偏置换不变估计量中，量子U-统计量是到任意副本数的唯一扩展。我们进一步推导出一个普适的方差展开式，其中主导的$1/n$项由泛函梯度的方差决定，而高阶贡献为$O(1/n^2)$阶。该主导方差与多参数量子Cramér-Rao极限一致，确立了量子U-统计量的渐近效率。我们还刻画了一阶梯度方差为零时的高阶标度行为。作为应用，我们分析了Bures $\chi^2$-散度，并证明参考态上的谱下界是有限方差估计的充分但非必要条件。
英文摘要
We study unbiased estimation of scalar-valued polynomial functionals of quantum states from independent copies. We establish an equivalence between the first-order marginal of a permutation-invariant finite-copy observable and the functional gradient. We then prove that, among unbiased permutation-invariant estimators, the quantum U-statistic is the unique extension to an arbitrary number of copies. We further derive a universal variance expansion in which the leading $1/n$ term is determined by the variance of the functional gradient, while higher-order contributions are of order $O(1/n^2)$. This leading variance coincides with the multiparameter quantum Cramér--Rao limit, establishing asymptotic efficiency of quantum U-statistics. We also characterize the higher-order scaling at points where the variance of the first-order gradient vanishes. As an application, we analyze the Bures $χ^2$-divergence and show that a spectral lower bound on the reference state is sufficient but not necessary for bounded-variance estimation. Beyond asymptotic variance, we establish variance-sensitive exponential concentration bounds, deriving a closed-form Bernstein-type inequality to capture finite-sample tail behaviour, and establish the Moderate Deviation Principle to characterize the intermediate asymptotic regime.
Comments56 pages, 2 figures, 1 table; Major Revision: Added two new figures, an algorithm, and a new Section V on variance-sensitive Bernstein-type probability concentration bounds; Abstract and Introduction have been updated to reflect these additions
