{
  "abstract": "We study when local reduced density operators, viewed as quantum marginals, can be assembled into a global quantum state with a prescribed Markov structure. The starting point is a canonical logarithmic construction $T(\\mathcal R)$, the noncommutative analogue of the junction-tree formula for decomposable graphical models. Unlike in the classical case, this formal construction may fail: noncommutativity can prevent it from being a normalized state with the prescribed marginals. We prove that this obstruction is captured exactly by a trace condition. For two overlapping marginals, and for clique marginals on a chordal graph, the condition ${\\rm Tr}(T(\\mathcal R))=1$ is equivalent to the existence of a quantum Markov completion. When it exists, the completion is unique, equal to $T(\\mathcal R)$, and selected by the maximum entropy principle. In the two-clique case, we also give an equivalent conditional reconstruction characterization: the two natural one-sided sandwich reconstructions agree if and only if the trace condition holds. We introduce the global quantum information $g{\\rm I}(\\mathcal{G})_ρ$ associated with a chordal graph $\\mathcal{G}$ and show that it is a relative-entropy discrepancy from $ρ$ to the logarithmic candidate, with a trace correction when the candidate is not normalized. We also prove an intersection property for strictly positive quantum conditional independence. Three-qubit Pauli examples illustrate how the quantum obstructions are real: local consistency, feasibility, Markov feasibility, and maximum entropy can all separate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.19453",
  "author_total": 2,
  "authors": [
    {
      "name": "Steffen Lauritzen"
    },
    {
      "name": "Piotr Zwiernik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8vxspfnfc7cre.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.19453",
  "fulltext_url": null,
  "id": "8vxspfnfc7cre",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.19453",
  "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",
      "math.PR"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum Mechanics and Applications"
    ]
  ],
  "openalex": "W7161772574",
  "pdf_url": "https://arxiv.org/pdf/2605.19453",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8vxspfnfc7cre.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2605.19453",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.19453"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The Markov Marginal Problem for Density Operators",
  "title_text": "The Markov Marginal Problem for Density Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/8vxspfnfc7cre",
  "venue": "arXiv",
  "versions": []
}