{
  "abstract": "Let $f:(0,\\infty)\\to(0,\\infty)$ be a nontrivial normalized operator monotone function and set $s=f'(1)$. We introduce the alternative-mean trace functional $$ \\altPhi_f(A,B) :=\\Tr(A\\nabla_s B) -\\Tr\\!\\left( f(A^{-1}\\sharp B)\\,A\\,f(A^{-1}\\sharp B) \\right). $$ on the positive definite cone. We prove that $\\altPhi_f$ is a quantum divergence in the sense of Bhatia--Gaubert--Jain whose diagonal Hessian induces a positive multiple of the Bures--Wasserstein Riemannian metric. We also establish the sharp comparison $$ s(1-s)d_{\\rm BW}(A,B)^2 \\le \\altPhi_f(A,B) \\le (1-s+s^2)d_{\\rm BW}(A,B)^2. $$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00034",
  "author_total": 3,
  "authors": [
    {
      "name": "Trung Dung Vuong"
    },
    {
      "name": "Hiroki Shudo"
    },
    {
      "name": "Hiroyuki Osaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8gm1xq2dg9yvj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00034",
  "fulltext_url": null,
  "id": "8gm1xq2dg9yvj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00034",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7206194247",
  "pdf_url": "https://arxiv.org/pdf/2609.00034",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8gm1xq2dg9yvj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "2609.00034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Alternative-mean trace divergences: geometry, data processing, and barycenters",
  "title_text": "Alternative-mean trace divergences: geometry, data processing, and barycenters",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/8gm1xq2dg9yvj",
  "venue": "arXiv",
  "versions": []
}