{
  "abstract": "We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most $s$, we give an estimator using $O(s^2/\\varepsilon^2)$ copies, uniformly in the ambient dimension and reference rank. The estimator applies a random-purification channel and a covariant pure-state measurement, then rescales the observed amplitude before evaluating a weighted nuclear norm. Combined with the lower bound established independently in our first version and in concurrent work of Wang, this determines the worst-case complexity under rank bounds $r,s$ as $\\min\\{r,s\\}^2/\\varepsilon^2$ up to logarithmic factors in the smaller rank. The same estimator gives the upper bound $O(r(\\operatorname{tr}\\sqrtσ)^2/\\varepsilon^2)$ for a rank-$r$ reference $σ$. We combine spectral truncation with lower bounds obtained by embedding hard instances for the maximally mixed reference into spectral subspaces. For spectra $λ_i\\propto i^{-α}$ with fixed $1<α\\le2$, as $\\varepsilon\\downarrow0$ with $r\\ge C_α\\varepsilon^{-2/(α-1)}$, the bounds determine the complexity as $\\widetildeΘ_α(\\varepsilon^{-4/(α-1)})$. In particular, inverse-square spectra have accuracy exponent four. The lower-bound construction uses exact moment matching and an explicitly computable Schur measure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.01770",
  "author_total": 2,
  "authors": [
    {
      "name": "Gye Jin Lee"
    },
    {
      "name": "Sunghyeon Jo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0pt72rnz368wr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.01770",
  "fulltext_url": null,
  "id": "0pt72rnz368wr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.01770",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7172443293",
  "pdf_url": "https://arxiv.org/pdf/2608.01770",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0pt72rnz368wr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2608.01770",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.01770"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Fidelity Estimation to a Known Quantum State Is Nearly Quadratic in the Smaller Rank",
  "title_text": "Fidelity Estimation to a Known Quantum State Is Nearly Quadratic in the Smaller Rank",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:03Z",
  "url": "https://research.draconitic.com/paper/0pt72rnz368wr",
  "venue": "arXiv",
  "versions": []
}