{
  "abstract": "The quantum data processing inequality asserts that quantum relative entropy is monotonically non-increasing under quantum channels. Here, we promote this fundamental inequality to an exact equality via boundary and surface Poisson integral representations across a complex strip. The relative-entropy loss is resolved into two independent, nonnegative physical mechanisms: an operational Petz recovery component governing state reconstructibility, and a boundary term measuring the mismatch of relative modular dynamics. We further express the loss as the measured relative entropy between the original state and its averaged Petz recovery state plus a nonnegative variational remainder. For the Belavkin--Staszewski (BS) loss, analogous integral representations yield a decomposition into negative log Uhlmann fidelity with an averaged reconstruction plus a nonnegative variational remainder. Furthermore, we establish recovery bounds at any prescribed modular parameter, including the canonical unrotated Petz map, with optimal square-root scaling in the relative entropy loss. The prefactors depend on distinct-eigenvalue counts or logarithmic spectral quantities, improving upon the inverse-power dependence on small eigenvalues in previous bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33815",
  "author_total": 1,
  "authors": [
    {
      "name": "Tai-Hsuan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9w4mb5hb4byba.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33815",
  "fulltext_url": null,
  "id": "9w4mb5hb4byba",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33815",
  "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",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7214715298",
  "pdf_url": "https://arxiv.org/pdf/2609.33815",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9w4mb5hb4byba.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2609.33815",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33815"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum data processing equality",
  "title_text": "Quantum data processing equality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/9w4mb5hb4byba",
  "venue": "arXiv",
  "versions": []
}