Proof of Heisenberg's Error-Disturbance Relation for Individual Measurements
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 x̂t, rather than a prior observable x̂0, 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 X̂t 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 εX(x̂t) and the disturbance ηX(p̂0) 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.
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BibTeX
@misc{kosugi2026proof,
title = {{Proof of Heisenberg's Error-Disturbance Relation for Individual Measurements}},
author = {Seiji Kosugi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18211},
eprint = {2609.18211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18211},
}
RIS
TY - UNPB
TI - Proof of Heisenberg's Error-Disturbance Relation for Individual Measurements
AU - Seiji Kosugi
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18211
UR - https://arxiv.org/abs/2609.18211
AB - 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 x̂_t, rather than a prior observable x̂_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 X̂_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(x̂_t) and the disturbance η_X(p̂_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.
ER -