{
  "abstract": "We study estimation of the von Neumann entropy of a $d$-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let $κ_ν\\in[1,d+1]$ denote the normalized trace-zero frame spread; $κ_ν=1$ for exact complex projective $2$-designs. For any fixed rank-one measurement and $n\\le c_2d^2/κ_ν$, the minimax mean-squared error is at least $c_1\\log^2\\{d^2/(κ_νn)\\}$. For bounded frame spread and every fixed $0<a<2$, the minimax risk is of order $\\log^2 d$ throughout $n\\lesssim d^{2-a}$; hence fixed uniform accuracy requires $Ω(d^2)$ observations. Existing tomography bounds give bounded risk from $O(d^3\\log^3 d)$ observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00270",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinyu Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0fe7zzf7pv372.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00270",
  "fulltext_url": null,
  "id": "0fe7zzf7pv372",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00270",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.00270",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0fe7zzf7pv372.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.00270",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00270"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "A Frame-Spread Lower Bound for Quantum Entropy Estimation under Fixed Rank-One Measurements",
  "title_text": "A Frame-Spread Lower Bound for Quantum Entropy Estimation under Fixed Rank-One Measurements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:14Z",
  "url": "https://research.draconitic.com/paper/0fe7zzf7pv372",
  "venue": "arXiv",
  "versions": []
}