{
  "abstract": "We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A\\|B)=2\\int a\\log(a/b)\\,\\mathrm{d}μ_{A,B}(a,b)$, where $μ_{A,B}$ denotes Heinävaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations of relative entropy from scalar equalities. Scalar inequalities can be used in the same way to derive Pinsker-type bounds. We further show that the testing curve $t\\mapsto \\mathrm{tr}[A-tB]_+$ is piecewise affine exactly when $A$ and $B$ commute, which is equivalent to their testing region being a polygon. This gives a characterization of noncommutativity through the curvature. As further applications, we derive inequalities for relative entropy variance and the loss of relative entropy under positive trace-preserving maps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12547",
  "author_total": 1,
  "authors": [
    {
      "name": "Lukas Schmitt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fhpx3ndnxafz2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12547",
  "fulltext_url": null,
  "id": "fhpx3ndnxafz2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12547",
  "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.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7212831325",
  "pdf_url": "https://arxiv.org/pdf/2609.12547",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fhpx3ndnxafz2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:07Z",
      "native_id": "2609.12547",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12547"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Relative entropy representations via tracial joint spectral measures",
  "title_text": "Relative entropy representations via tracial joint spectral measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/fhpx3ndnxafz2",
  "venue": "arXiv",
  "versions": []
}