{
  "abstract": "We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\\varepsilon$, estimating an unknown state on $\\mathbb{C}^d$ of rank at most $r$ to trace norm error $\\varepsilon$ with constant success probability requires, and is achievable with, $$Θ\\left(\\frac{dr}{\\varepsilon^2}\\mathop{\\mathrm{max}}\\left\\{1,\\frac{r}{\\sqrt{t}}\\right\\}\\right)$$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\\sqrt{t}$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10514",
  "author_total": 2,
  "authors": [
    {
      "name": "Ashwin Nayak"
    },
    {
      "name": "Xingyu Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ajfkw0bpsa4pt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10514",
  "fulltext_url": null,
  "id": "ajfkw0bpsa4pt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10514",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7212108303",
  "pdf_url": "https://arxiv.org/pdf/2609.10514",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ajfkw0bpsa4pt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2609.10514",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10514"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements",
  "title_text": "Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/ajfkw0bpsa4pt",
  "venue": "arXiv",
  "versions": []
}