Uniqueness, Cramér-Rao Efficiency and Concentration Bounds for Quantum U-Statistics
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/n2). 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 χ2-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.
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BibTeX
@misc{dasgupta2026uniqueness,
title = {{Uniqueness, Cramér-Rao Efficiency and Concentration Bounds for Quantum U-Statistics}},
author = {Ayanava Dasgupta and Naqueeb Ahmad Warsi and Premanshu Chatterjee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08745},
eprint = {2609.08745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08745},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniqueness, Cramér-Rao Efficiency and Concentration Bounds for Quantum U-Statistics
AU - Ayanava Dasgupta
AU - Naqueeb Ahmad Warsi
AU - Premanshu Chatterjee
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.08745
UR - https://arxiv.org/abs/2609.08745
AB - 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.
ER -