{
  "abstract": "Information distance and circuit complexity are both obtained by minimizing lengths, but they minimize over different objects. We make this distinction explicit for faithful one-mode Gaussian states. First, invariant-form uniqueness implies that no positive-definite quadratic gate cost can be invariant under the full adjoint action of the noncompact symplectic group; a positive Cartan majorant necessarily introduces additional reference data. The Uhlmann purification quotient realizes the Bures metric, and the radial covariance direction requires a system-ancilla coupling because system-only Gaussian unitaries preserve the Williamson eigenvalue. We then minimize fixed right-invariant quadratic norms on the minimal two-mode Gaussian gate algebra \\(\\mathfrak{sp}(4,\\mathbb R)\\). For the unweighted Frobenius norm, the quotient coefficients for radial and traceless covariance tangents are $G_0=[\\hbar^2(u-1)]^{-1}$ and $G_2=[\\hbar^2(3u-1)]^{-1}$, where $u=(2ν/\\hbar)^2$. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every $u$; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives $G_0/G_2=1+2(β/α)u/(u-1)$; matching Bures throughout the isotropic family would require the state-dependent relation $β/α=1/u$. At the Bures-Fisher determinant crossing \\(u=φ\\), pointwise matching is possible only by inserting $β/α=φ^{-1}$. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17574",
  "author_total": 1,
  "authors": [
    {
      "name": "Christian Kerskens"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/etbfkdqt64y2y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17574",
  "fulltext_url": null,
  "id": "etbfkdqt64y2y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17574",
  "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",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7213518923",
  "pdf_url": "https://arxiv.org/pdf/2609.17574",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/etbfkdqt64y2y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:19Z",
      "native_id": "2609.17574",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17574"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Gaussian Purification Quotients and Fixed Nielsen Penalties",
  "title_text": "Gaussian Purification Quotients and Fixed Nielsen Penalties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/etbfkdqt64y2y",
  "venue": "arXiv",
  "versions": []
}