{
  "abstract": "In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\\C^{d_1}\\otimes\\cdots\\otimes\\C^{d_p}$, a nonzero vector is a \\emph{product state} if it can be written as $\\ket{φ_1}\\otimes\\cdots\\otimes\\ket{φ_p}$ with $\\ket{φ_j}\\in\\C^{d_j}\\setminus\\{0\\}$. An \\emph{unextendible product basis} (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lovász. Let $f_m(d_1,\\ldots,d_p)$ be the minimum cardinality of a UPB in the space, and let \\(f_{LB}(d_1,\\ldots,d_p)=1+\\sum_{j=1}^{p}(d_j-1)\\) be the natural lower bound. Alon and Lovász determined exactly when \\(f_m\\) attains the lower bound \\(f_{LB}\\), but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with $p\\ge3$, whenever the natural lower bound $f_{LB}$ is unattainable, the true minimum is exactly $f_{LB}+1$. Precisely, if the number of even local dimensions $d_i$ is positive and even, and at least one local dimension is greater than two, then \\(f_m(d_1,\\ldots,d_p)=f_{LB}(d_1,\\ldots,d_p)+1\\). The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum size problem for UPBs in all finite quantum systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05657",
  "author_total": 1,
  "authors": [
    {
      "name": "Chenhao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f0zz6vscbf6yp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05657",
  "fulltext_url": null,
  "id": "f0zz6vscbf6yp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05657",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7211995954",
  "pdf_url": "https://arxiv.org/pdf/2609.05657",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f0zz6vscbf6yp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2609.05657",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05657"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Minimum Cardinalities of Multipartite Unextendible Product Bases",
  "title_text": "Minimum Cardinalities of Multipartite Unextendible Product Bases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/f0zz6vscbf6yp",
  "venue": "arXiv",
  "versions": []
}