{
  "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. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08745",
  "author_total": 3,
  "authors": [
    {
      "name": "Ayanava Dasgupta"
    },
    {
      "name": "Naqueeb Ahmad Warsi"
    },
    {
      "name": "Premanshu Chatterjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fgqa9rjd61641.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08745",
  "fulltext_url": null,
  "id": "fgqa9rjd61641",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08745",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7211985588",
  "pdf_url": "https://arxiv.org/pdf/2609.08745",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fgqa9rjd61641.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:48Z",
      "native_id": "2609.08745",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08745"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Uniqueness, Cramér-Rao Efficiency and Concentration Bounds for Quantum U-Statistics",
  "title_text": "Uniqueness, Cramér-Rao Efficiency and Concentration Bounds for Quantum U-Statistics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:58Z",
  "url": "https://research.draconitic.com/paper/fgqa9rjd61641",
  "venue": "arXiv",
  "versions": []
}