{
  "abstract": "In this work, we consider the fundamental task of estimating the purity of an unknown state to within $\\textit{multiplicative}$ error $\\varepsilon$. When one can perform general collective measurements, $Θ\\left(\\frac{\\sqrt{d}}{\\varepsilon^2} + \\frac{d}{\\varepsilon}\\right)$ copies are known to be necessary and sufficient for this problem [AISW20]. However, implementing collective measurements on such a large number of copies can be experimentally demanding, and we thus aim to characterize the copy complexity of this problem with incoherent measurements. In this setting, the only non-trivial result is a non-adaptive algorithm that uses $O\\left( \\frac{d}{\\varepsilon^2} + \\frac{d^2}{\\varepsilon} \\right)$ copies [PTTW26], which is polynomially larger than the best-known lower bound. Our first result is a new algorithm performing non-adaptive incoherent measurements that succeeds using $O\\left( \\frac{d}{\\varepsilon^2} + \\frac{d^{3/2}}{\\varepsilon} \\right)$ copies, improving on the latter term in the copy complexity. Moreover, we show that for any algorithm restricted to non-adaptive measurements, the above copy complexity is optimal. We also develop a new adaptive estimator for the purity of a state that improves on the above complexity in the high-precision regime, i.e., for $\\varepsilon = o(1/d)$. We also show that the non-adaptive and adaptive estimators together yield the optimal complexity for incoherent purity estimation; in particular, we show that the copy complexity of this problem is $$ Θ\\left(\\min\\left\\{ \\frac{d}{\\varepsilon^2} + \\frac{d^{3/2}}{\\varepsilon}, \\frac{d^2}{\\varepsilon} + \\frac{\\sqrt{d}}{\\varepsilon^2}\\right\\} \\right). $$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40248",
  "author_total": 2,
  "authors": [
    {
      "name": "Junseo Lee"
    },
    {
      "name": "Chirag Wadhwa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c2bpea33q56kp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40248",
  "fulltext_url": null,
  "id": "c2bpea33q56kp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40248",
  "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 Information and Cryptography"
    ]
  ],
  "openalex": "W7215045422",
  "pdf_url": "https://arxiv.org/pdf/2609.40248",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c2bpea33q56kp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40248",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40248"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Optimal Purity Estimation with Incoherent Measurements",
  "title_text": "Optimal Purity Estimation with Incoherent Measurements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/c2bpea33q56kp",
  "venue": "arXiv",
  "versions": []
}