{
  "abstract": "Heisenberg originally envisioned the error-disturbance relation under the premise that the state after measurement must remain consistent with the Kennard-Robertson uncertainty relation. This implies that the measurement error must be formulated via a posterior observable $\\hat{x}_t$, rather than a prior observable $\\hat{x}_0$, because the posterior measurement error determines the post-measurement uncertainty of the electron. Building upon the preliminary conceptual foundation reported in arXiv:1504.03779, this paper presents a rigorous derivation of the error-disturbance principle formulated from the unitary transformation equations of observables. Each readout $X$ of a posterior probe observable $\\hat{X}_t$ completes a single measurement event, producing a specific conditional object state. The Kennard-Robertson uncertainty relation must hold for these states. Specifically, we show that the uncertainty relation between the error $ε_{X}(\\hat{x}_t)$ and the disturbance $η_{X}(\\hat{p}_0)$ holds strictly at the level of individual measurements, characterized by the specific readout $X$. We also verify that conventional error-disturbance relations, where the errors are evaluated by averaging over the unconditioned state, hold true. Our results provide a refined theoretical basis for understanding the fundamental trade-off in individual measurement outcomes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18211",
  "author_total": 1,
  "authors": [
    {
      "name": "Seiji Kosugi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/80qc7v0w11pam.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18211",
  "fulltext_url": null,
  "id": "80qc7v0w11pam",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18211",
  "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": "W7213500093",
  "pdf_url": "https://arxiv.org/pdf/2609.18211",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/80qc7v0w11pam.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:01Z",
      "native_id": "2609.18211",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18211"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Proof of Heisenberg's Error-Disturbance Relation for Individual Measurements",
  "title_text": "Proof of Heisenberg's Error-Disturbance Relation for Individual Measurements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:01Z",
  "url": "https://research.draconitic.com/paper/80qc7v0w11pam",
  "venue": "arXiv",
  "versions": []
}