{
  "abstract": "Uncertainty relations are usually formulated as trade-offs between two or more observables. Yet the uncertainty of a single observable can already contain an intrinsically quantum contribution arising from its noncommutativity with the state. In the sense of Luo, such a contribution can be quantified by a quantum uncertainty $Q_ρ(A)$, with prominent examples including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed. We solve this fixed-state optimization problem by determining the largest coefficient $c_{\\rm opt}(ρ)$ for which $V_ρ(A)\\ge c_{\\rm opt}(ρ)Q_ρ(A)$ holds for every observable $A$, and show that it depends only on the smallest and largest eigenvalues of $ρ$. The same optimization principle extends to a new power-commutator family $\\frac{1}{2}|[ρ^s,A]|^2$ $(s\\ge 1/2)$, providing a direct measure of state--observable noncommutativity beyond the standard quantum-uncertainty framework. More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.26331",
  "author_total": 3,
  "authors": [
    {
      "name": "Haruki Yamashita"
    },
    {
      "name": "Aina Mayumi"
    },
    {
      "name": "Gen Kimura"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ngege8103022.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.26331",
  "fulltext_url": null,
  "id": "2ngege8103022",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.26331",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum Mechanics and Applications"
    ]
  ],
  "openalex": "W7162522258",
  "pdf_url": "https://arxiv.org/pdf/2605.26331",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ngege8103022.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2605.26331",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.26331"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Optimal Uncertainty Relations for a Single Observable",
  "title_text": "Optimal Uncertainty Relations for a Single Observable",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/2ngege8103022",
  "venue": "arXiv",
  "versions": []
}