{
  "abstract": "We prove that, for an $n$-qubit system of dimension $d=2^n$, every state satisfying $\\operatorname{Tr}(ρ^2)\\le 1/(d-a_\\ast)$, with $a_\\ast=0.458327\\cdots$, lies inside the stabilizer polytope and is therefore magic-free. Combining this result with general geometric properties of high-dimensional polytopes, we establish quantitative estimates for the Hilbert--Schmidt inradius and volume radius of the stabilizer polytope, and use them to characterize the typicality of magic in random induced states obtained by tracing out a $k$-dimensional subsystem from a $d\\times k$-dimensional Haar-random pure state. We prove a sharp phase transition in the probability of such states having magic, whose transition dimension $k_\\star$ is bounded between $Ω(d^2/\\log^2d)$ and $\\mathcal{O}(d^2)$. We further prove that the number of facets of the stabilizer polytope lies between $\\exp[Ω(d^2/\\log^2 d)]$ and $\\exp[\\mathcal{O}(d^2\\log^2 d)]$ employing a result of Bourgain and Milman in convex geometry, substantially improving upon the previous quasipolynomial lower bound and implying that doubly-exponentially many linear inequalities in the number of qubits are required for an exact description of the magic-free region. Overall, our results reveal the near-extremal geometry of the stabilizer polytope and provide a quantitative foundation for understanding the typicality, robustness, and detectability of magic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03944",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhenhuan Liu"
    },
    {
      "name": "Zi-Wen Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6radrnrgqsca2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03944",
  "fulltext_url": null,
  "id": "6radrnrgqsca2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03944",
  "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": "W7208722847",
  "pdf_url": "https://arxiv.org/pdf/2609.03944",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6radrnrgqsca2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:52Z",
      "native_id": "2609.03944",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03944"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "On the geometry and typicality of quantum magic",
  "title_text": "On the geometry and typicality of quantum magic",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:18Z",
  "url": "https://research.draconitic.com/paper/6radrnrgqsca2",
  "venue": "arXiv",
  "versions": []
}