{
  "abstract": "Inner products of random matrices and random quantum states---such as orbital inner products, expectation values, overlaps, and cyclic Bargmann invariants---are shown to follow from two underlying mechanisms: the diagonal of an induced random state is Dirichlet distributed, and the expectation value of a random pure state in a rank-one or two-by-two operator is a numerical shadow on an elliptical disk. For states induced by an environment of dimension $K$, the expectation value of any observable is a Curry--Schoenberg spline with each eigenvalue repeated $K$ times, and the vector of moments has an explicit density on a simplex; for a qutrit, the variance law is obtained in finite form. The overlap of two random qubit states attains its mode at its mean for every rotation-invariant ensemble, but for Hilbert--Schmidt qutrits the density at the mean has a strictly negative derivative, so mode and mean separate; for large environments, the centred overlap converges to a variance-gamma law rather than a Gaussian. For two noncommuting qubit observables, the joint law of the two standard deviations is derived for all $K$, and the probability of coming within a small distance of the optimal sum of variances scales as a power of that distance, with a different exponent when the Bloch axes are orthogonal. For three Haar-random qubit states, the exact density of the Bargmann invariant is given in terms of a complete elliptic integral, together with its local behaviour and elementary marginals. In every dimension, the probability that its real part is negative---certifying basis-independent coherence---is computed, along with all mixed moments. For cycles of arbitrary length, the invariant is shown to be driven by an exact three-parameter Markov chain, and the logarithm of its modulus obeys a law of large numbers and a central limit theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25196",
  "author_total": 1,
  "authors": [
    {
      "name": "Lin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40fkqvng5bgf4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25196",
  "fulltext_url": null,
  "id": "40fkqvng5bgf4",
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  "landing_url": "https://arxiv.org/abs/2610.25196",
  "language": null,
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  "metadata_licence": "CC0-1.0",
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      "math-ph"
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      "3107|Quantum Mechanics and Applications"
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  "openalex": "W7223014971",
  "pdf_url": "https://arxiv.org/pdf/2610.25196",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40fkqvng5bgf4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2610.25196",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25196"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "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": "Exact probability laws of random inner products: Dirichlet splines, overlaps, uncertainty, and Bargmann invariants",
  "title_text": "Exact probability laws of random inner products: Dirichlet splines, overlaps, uncertainty, and Bargmann invariants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/40fkqvng5bgf4",
  "venue": "arXiv",
  "versions": []
}