{
  "abstract": "We characterize the largest admissible side length of fixed-margin square Hankel submatrices generated by probabilistic realizations on a \\(d\\)-point carrier and quantum linear realizations on a \\(d\\)-dimensional Hilbert space, when prefix and suffix word lengths are bounded by \\(T\\). Using a four-symbol transition alphabet shared by all selected rows and columns, we show that, for fixed margins \\(λ_2>0\\) and \\(λ_1+λ_2<1\\), the corresponding side lengths satisfy \\(\\ln N_{\\mathsf Q,4}=Θ_{λ_1,λ_2}(\\min\\{d^2,T\\})\\) for completely positive trace-preserving (CPTP) quantum realizations and \\(\\ln N_{\\mathsf S,4}=Θ_{λ_1,λ_2}(\\min\\{d,T\\})\\) for probabilistic automata. Four unitary and four permutation transition maps attain the corresponding orders. At zero false-acceptance tolerance (\\(λ_2=0\\) with \\(0\\leqλ_1<1\\)), all four model classes satisfy \\(\\ln N=Θ(\\min\\{\\ln d,T\\})\\). The upper bounds follow from state-space packing and word counting, while the lower bounds combine explicit transition tuples with Grassmannian and positive-rate codebooks whose existence follows from the probabilistic method. These results establish sharp dimension--word-length tradeoffs for bounded-word Hankel tables.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21797",
  "author_total": 1,
  "authors": [
    {
      "name": "Yixin Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jgf85y6d17j8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21797",
  "fulltext_url": null,
  "id": "6jgf85y6d17j8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21797",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7222755362",
  "pdf_url": "https://arxiv.org/pdf/2610.21797",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jgf85y6d17j8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2610.21797",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21797"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Hankel Submatrices with Bounded Word Length: Dimension Tradeoffs in Probabilistic and Quantum Automata",
  "title_text": "Hankel Submatrices with Bounded Word Length: Dimension Tradeoffs in Probabilistic and Quantum Automata",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/6jgf85y6d17j8",
  "venue": "arXiv",
  "versions": []
}