{
  "abstract": "Suppose that $F_E$ is the face of the convex set of all $m\\otimes n$ bi-partite states which consists of states with ranges contained in a subspace $E$. By the result in [Phys. Rev. A {\\bf 112} (2025), 032426], there exist Schmidt number $\\ell$ witnesses outside of the face $F_{E^\\perp}$ if and only if $\\ell\\leμ$, where $μ$ is the minimum of Schmidt ranks of nonzero vectors in $E$. In this paper, we show that every neighborhood of the projection state located at the center of $F_{E^\\perp}$ contains Schmidt number $\\ell$ witnesses if and only if $\\ell\\geμ$. In order to understand the local shape of the convex sets of $k$-blockpositive matrices around this projection state, we introduce principal supporting hyperplanes, and show that they correspond to vectors in $E$ with Schmidt rank $\\le k$. Using techniques from algebraic geometry, we show that the inclusions between these convex sets are detected by principal supporting hyperplanes through the projection state for generic subspaces $E$, while for non-generic subspaces they may not be detected by any supporting hyperplane.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14199",
  "author_total": 2,
  "authors": [
    {
      "name": "Young-Hoon Kiem"
    },
    {
      "name": "Seung-Hyeok Kye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7p31t2srjd2ar.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14199",
  "fulltext_url": null,
  "id": "7p31t2srjd2ar",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14199",
  "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": "W7203648204",
  "pdf_url": "https://arxiv.org/pdf/2608.14199",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7p31t2srjd2ar.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:22Z",
      "native_id": "2608.14199",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14199"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Local geometry for Schmidt number witnesses",
  "title_text": "Local geometry for Schmidt number witnesses",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:44Z",
  "url": "https://research.draconitic.com/paper/7p31t2srjd2ar",
  "venue": "arXiv",
  "versions": []
}