{
  "abstract": "Symmetric informationally complete positive operator-valued measures (SIC-POVMs, or SICs) are optimal minimal measurements for quantum state tomography. Problem~1 of Horodecki, Rudnicki and Życzkowski's open problems in quantum information asks for SICs in infinitely many dimensions. By work of Appleby--Flammia--Kopp and Radchenko--Wheeler, ghost SICs, which satisfy the SIC equations with Hermiticity up to parity, exist unconditionally for $d\\ge4$. We study the passage from ghosts to SICs. We compute all isolated points of the complexified SIC variety for $d=4,5,6$; exactly half are SICs. We prove that no symmetry of the SIC equations maps a ghost to a SIC, so the step is purely arithmetic, and that a SIC exists in dimension $d$ whenever the ghost overlap units satisfy a place law: each Galois conjugate is real or unimodular according to the infinite place of the real quadratic field $F$ below it. If the unit field is abelian over $F$, the place law is the Stark relation $c_-=κ$; equivalently, a Galois conjugate of an associated near-group fusion category is unitary exactly when it is pseudo-unitary. For $d$ a power of a prime $p\\equiv5\\pmod 6$ or of $3$, an infinite family, we prove that under the algebraic part of Stark's conjecture $c_-=κ$ is equivalent to the triviality of $κ$ on the Hilbert-class-field part of the unit field. That condition involves no complex conjugation, and it follows from a sign-free multiplet hypothesis. We state an explicit $GL_2$ reciprocity law for the squared ghost overlaps that implies $c_-=κ$, and we prove it for the computed algebraic overlaps in $d=5,7,11,15$ by Artin reciprocity and a check in every ray class. Zauner's conjecture remains open: the algebraic part of Stark's conjecture and the multiplet hypothesis are not proved here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01524",
  "author_total": 1,
  "authors": [
    {
      "name": "Hassan Khodaiemehr"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fng44ngw7ycfq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01524",
  "fulltext_url": null,
  "id": "fng44ngw7ycfq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01524",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01524",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fng44ngw7ycfq.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.01524",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01524"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "From Ghost SICs to SIC-POVMs: A Galois Reduction of Zauner's Conjecture",
  "title_text": "From Ghost SICs to SIC-POVMs: A Galois Reduction of Zauner's Conjecture",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/fng44ngw7ycfq",
  "venue": "arXiv",
  "versions": []
}