{
  "abstract": "The Mandelstam-Tamm and Margolus-Levitin quantum speed limits bound how fast a state evolves, using the energy variance and the mean energy above the ground state. Speed limits on observables -- the change of an expectation value -- have so far used the variance (the Mandelstam-Tamm / quantum-Fisher-information lineage); the mean energy has not been brought to bear on the change of in a fixed state. We close this branch. First, a no-go theorem: there is no state-independent linear mean-energy bound on the time to change an observable's expectation by Delta = | - |; the optimal state-independent exponent of Delta is exactly two. Second, the corresponding sharp quadratic bound, T ( - E_0) >= C_* Delta^2 / sigma_A^2, for every time-independent Hamiltonian H (ground energy E_0), every bounded observable A with spectral spread sigma_A = (lambda_max - lambda_min)/2, and every pure or mixed state, with the dimension-independent constant C_* = 1/(8 sin x_*) = 0.172506267461..., where x_* is the smallest positive root of tan(x/2) = x. The bound is tight, approached but not attained by a near-ground two-level family. We then give the exact energy-time/swing trade-off curve of which C_* is the small-swing slope -- tight at every swing, the observable analog of the Giovannetti-Lloyd-Maccone curve for states -- show the constant survives for mixed states via joint convexity of the trace distance, sharpen it for bandwidth-limited generators and several observables at once, and show the quadratic law degrades to a linear one when the initial state is an eigenvector of the observable. It completes the (mean-energy x observable) corner of the speed-limit landscape and, being quadratic, is most constraining where the linear variance bounds are weakest. Its cleanest physical home is the autonomous quantum clock, where it gives a coherent mean-energy resolution floor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22658",
  "author_total": 1,
  "authors": [
    {
      "name": "Bryan Nasr"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3zdh2krge5av5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22658",
  "fulltext_url": null,
  "id": "3zdh2krge5av5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22658",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7204182069",
  "pdf_url": "https://arxiv.org/pdf/2608.22658",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3zdh2krge5av5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "2608.22658",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22658"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "A Margolus-Levitin speed limit for observables: mean energy bounds expectation-value change quadratically",
  "title_text": "A Margolus-Levitin speed limit for observables: mean energy bounds expectation-value change quadratically",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:41Z",
  "url": "https://research.draconitic.com/paper/3zdh2krge5av5",
  "venue": "arXiv",
  "versions": []
}