{
  "abstract": "We address the problem of testing whether an unknown $n$-qudit state $ρ$ is $a$-close to a product state or $b$-far away from any product state, as measured in terms of state overlap. We provide a time-efficient algorithm to solve this problem that requires an $n$-independent number of copies of the unknown state. Our random coloring argument shows that for any unknown state there always exists a partition of $[n]$ into $q$ parts such that the square of the overlap with the closest product state with respect to this partition is only an additive factor $O(1/q)$ larger. This reduces the problem to tolerant testing of $q$ parties, with potentially growing local dimensions. Combining this insight with blockwise spectral projection arguments we can show that the natural $k$-copy generalization of Harrow & Montanaro's product state test provides an efficient tolerant tester. We use the same random coloring and blockwise spectral projection techniques to obtain a substantially improved algorithm for closest product state learning. In particular we give an algorithm that takes in $\\widetilde{O}\\big((nd)^2\\big)\\, 2^{\\widetilde{O}(1/\\varepsilon^8)}$ copies of the unknown state and produces an $\\varepsilon$-approximately optimal product state. The key technical components of this learner are a qudit variant of the Bakshi et al. high-fidelity product state learning algorithm and a sampling technique based on Werner's optimal cloning channel.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01979",
  "author_total": 3,
  "authors": [
    {
      "name": "Zongbo Bao"
    },
    {
      "name": "Jonas Helsen"
    },
    {
      "name": "Tuyen Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40jp4d97h3rdc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01979",
  "fulltext_url": null,
  "id": "40jp4d97h3rdc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01979",
  "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": "W7216309937",
  "pdf_url": "https://arxiv.org/pdf/2610.01979",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40jp4d97h3rdc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.01979",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01979"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Fully tolerant product state testing and closest product state learning",
  "title_text": "Fully tolerant product state testing and closest product state learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:03Z",
  "url": "https://research.draconitic.com/paper/40jp4d97h3rdc",
  "venue": "arXiv",
  "versions": []
}