Uniqueness and Cramér-Rao Efficiency of Quantum U-Statistics

• URL: http://arxiv.org/abs/2609.08745v1
• Date: Tue, 08 Sep 2026 13:37:19 GMT
• Title: Uniqueness and Cramér-Rao Efficiency of Quantum U-Statistics
• Abstract summary: We study unbiased estimation of scalar-valued functionals of quantum states from independent copies.<n>We prove that, among unbiased permutation-invariant estimators, the quantum U-statistic is the unique extension to an arbitrary number of copies.
• Score: 8.156494881838947
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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