{
  "abstract": "Toroidal discrete Wigner functions (DWFs) for finite-dimensional systems are non-unique. We completely parameterize and topologically classify the labeled single-qudit ${d\\times d}$ representations whose phase-point operators are Hermitian, unit-trace, Hilbert--Schmidt orthogonal, and Weyl--Heisenberg covariant, determining the independent parameter freedom in every dimension. Our previously established stencil theorem expresses this family as descents of a doubled ${2d\\times2d}$ parent Wigner representation. Here, we solve its projected-stencil admissibility conditions explicitly to obtain the full parameter space of this family. In the symplectic-Fourier representation, admissibility fixes the modulus, leaving phase data on a ${d\\times d}$ base cell satisfying a parity-dependent twisted-oddness relation. This gives the parameter space ${(S^1)^{(d^2-1)/2}}$ for odd ${d}$ and ${(S^1)^{(d^2-4)/2}\\times\\mathbb{Z}_{2}^{3}}$ for even ${d}$. Canonical horizontal and vertical marginals reduce these spaces to ${(S^1)^{(d-1)^2/2}}$ and ${(S^1)^{d\\,(d-2)/2}\\times\\mathbb{Z}_{2}}$, respectively. The same phase data parameterize validity-preserving rephasings of the doubled Weyl--Heisenberg displacement operators. Requiring these operators to have order dividing the Hilbert-space dimension ${d}$ replaces each ${S^1}$ factor by a discrete ${\\mathbb{Z}_{d}}$ factor. The associated symplectic-Fourier characteristic functions encode the same operator information, with pointwise magnitudes independent of the valid stencil convention and hence providing convention-independent information about the represented operator within this family. This classification separates the freedom intrinsic to DWF validity from that selected by marginal and displacement-algebra requirements and makes explicit the structural distinction between odd and even dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39119",
  "author_total": 2,
  "authors": [
    {
      "name": "Lucky K. Antonopoulos"
    },
    {
      "name": "Nicolas C. Menicucci"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cnwnvnvsbvdg8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39119",
  "fulltext_url": null,
  "id": "cnwnvnvsbvdg8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39119",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum Mechanics and Applications"
    ]
  ],
  "openalex": "W7215100574",
  "pdf_url": "https://arxiv.org/pdf/2609.39119",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cnwnvnvsbvdg8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2609.39119",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39119"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Complete parameterization and parameter-space topology of discrete Wigner representations on $d \\times d$ phase space",
  "title_text": "Complete parameterization and parameter-space topology of discrete Wigner representations on d × d phase space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/cnwnvnvsbvdg8",
  "venue": "arXiv",
  "versions": []
}