{
  "abstract": "The main result of the paper states that if $0\\neq v\\in\\mathbb C^m\\otimes\\mathbb C^n$, $d=\\min\\{m,n\\}$, and $p=SR(v)$, then for every $1\\le q\\le d$, \\begin{equation*} \\min\\left\\{OSR(\\mathbf U): \\begin{array}{l} \\mathbf U\\in\\mathbb M_m\\otimes\\mathbb M_n\\text{ is unitary}, SR(\\mathbf Uv)=q \\end{array}\\right\\} = \\left\\lceil\\max\\left\\{\\frac pq,\\frac qp\\right\\}\\right\\rceil . \\end{equation*} Here $OSR(\\mathbf U)$ denotes the operator Schmidt rank of $\\mathbf U$. Furthermore, if $\\mathbf U\\in\\mathbb M_m\\otimes\\mathbb M_n$ is unitary and $OSR(\\mathbf U)\\le k$, then, for every unit vector $x\\in\\mathbb C^m\\otimes\\mathbb C^n$ and every $1\\le r\\le d$, \\begin{equation*} F_{\\min\\{kr,d\\}}(\\mathbf Ux)\\ge F_r(x) \\quad\\text{and}\\quad F_{\\min\\{kr,d\\}}(x)\\ge F_r(\\mathbf Ux), \\end{equation*} where $F_r(x)$ denotes the sum of the $r$ largest squared Schmidt coefficients of $x$. This result is stronger than rank reachability as it controls the entire cumulative Schmidt profile. We also establish a majorization relation and use it to obtain sharp bounds on Schatten-norm distortion and on the change of Rényi entropies under unitaries with bounded operator Schmidt rank.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09794",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohsen Kian"
    },
    {
      "name": "Mohammad Sal Moslehian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5fnt9gyfxbv3r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09794",
  "fulltext_url": null,
  "id": "5fnt9gyfxbv3r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09794",
  "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-ph"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7220893529",
  "pdf_url": "https://arxiv.org/pdf/2610.09794",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5fnt9gyfxbv3r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:37Z",
      "native_id": "2610.09794",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09794"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Operator Schmidt Rank and the Geometry of Schmidt Profiles",
  "title_text": "Operator Schmidt Rank and the Geometry of Schmidt Profiles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/5fnt9gyfxbv3r",
  "venue": "arXiv",
  "versions": []
}