{
  "abstract": "The von Neumann, Tsallis, and Rényi entropies are fundamental measures of quantum information. A common estimation strategy multiplies the worst-case costs of spectral transformation and statistical readout, leaving gaps to query lower bounds. We reduce this overhead with multi-level algorithms that use local normalization and precision allocation, while variable-time estimation accounts for the probability of reaching expensive spectral tests. With controlled purified access and its inverse, we obtain bounds for additive error $\\varepsilon$, rank upper bound $R$, and fixed order $α$. Tsallis entropy estimation has near-optimal rank-independent query complexity $\\widetilde O(\\varepsilon^{-1/(α-1)})$ for $1<α<2$, when the allowed dimension and rank accommodate the lower-bound instances, and $\\widetilde O(1/\\varepsilon)$ for $α\\ge2$, where $\\widetilde O$ suppresses logarithmic factors. The first saves a factor $1/\\varepsilon$; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from $\\widetilde O(R/\\varepsilon^2)$ to $\\widetilde O(R/\\varepsilon)$ and obtain $\\widetilde O(R/\\varepsilon)$ for noninteger Rényi orders $1<α<3$, with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using $\\widetilde O(R/\\varepsilon)$ queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40179",
  "author_total": 5,
  "authors": [
    {
      "name": "Junxiang Huang"
    },
    {
      "name": "Chenyang Li"
    },
    {
      "name": "Lu-Fan Zhang"
    },
    {
      "name": "Yusen Wu"
    },
    {
      "name": "Yukun Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dp6nfmz2ww0vg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40179",
  "fulltext_url": null,
  "id": "dp6nfmz2ww0vg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40179",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7215122823",
  "pdf_url": "https://arxiv.org/pdf/2609.40179",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dp6nfmz2ww0vg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40179",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40179"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Breaking the Multiplicative Overhead in Quantum Entropy Estimation",
  "title_text": "Breaking the Multiplicative Overhead in Quantum Entropy Estimation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/dp6nfmz2ww0vg",
  "venue": "arXiv",
  "versions": []
}