{
  "abstract": "We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least $(d/n)\\log^2\\{n/(4d)\\}$ when $d\\ge C$ and $n\\ge Cd$, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius $r$ around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by $\\varepsilon<1$ from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most $d^3r^2/n+d^4/n^2+d^6r^6$. Lower bounds under the same frame control yield the local minimax rate $d^3r^2/n+d^4/n^2$ when $n\\ge Cd^2$ and $r$ lies in an explicit matching range. For every fixed upper bound on $\\varepsilon$, approximation changes only the constants, not the powers of $d,n,r$. At the critical radius $r=n^{-1/2}$, the local risk is asymptotically negligible relative to the global risk when $n\\log^2(n/d)\\gg d^3$. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29485",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinyu Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b7yyx0grtcyh7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29485",
  "fulltext_url": null,
  "id": "b7yyx0grtcyh7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29485",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7214300192",
  "pdf_url": "https://arxiv.org/pdf/2609.29485",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b7yyx0grtcyh7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2609.29485",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29485"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Local and Global Risk Bounds for Quantum Entropy Estimation under Projective-Design Measurements",
  "title_text": "Local and Global Risk Bounds for Quantum Entropy Estimation under Projective-Design Measurements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:21Z",
  "url": "https://research.draconitic.com/paper/b7yyx0grtcyh7",
  "venue": "arXiv",
  "versions": []
}