{
  "abstract": "Planar $k$-uniform states have maximally mixed reductions on every $k$-site interval of a ring. We introduce planar $k$-purity---the mean purity of those intervals---as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of $n$ parties with local dimension $p$, the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every $1\\le k\\le\\lfloor n/2\\rfloor$. In the balanced qubit case, $k=\\lfloor n/2\\rfloor$ and $N=2^n$, the variance is asymptotic to $20/(3nN^2)$ for even $n$ and $6/(nN^2)$ for odd $n$. A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through $n=10$ validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately $2.83$ times larger at $n=10$. Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28914",
  "author_total": 1,
  "authors": [
    {
      "name": "Kourosh Mirsohi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dkha1yqy9q3q7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28914",
  "fulltext_url": null,
  "id": "dkha1yqy9q3q7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28914",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2608.28914",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dkha1yqy9q3q7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "2608.28914",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28914"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Exact Haar Statistics of Planar $k$-Purity in Multipartite Quantum Systems",
  "title_text": "Exact Haar Statistics of Planar k-Purity in Multipartite Quantum Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/dkha1yqy9q3q7",
  "venue": "arXiv",
  "versions": []
}