Proof of Heisenberg's Error-Disturbance Relation for Individual Measurements

arXiv:2609.18211v1 Announce Type: new 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 $\epsilon_{X}(\hat{x}_t)$ and the disturbance $\eta_{X}(\hat{p}_0)$ 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: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 $\epsilon_{X}(\hat{x}_t)$ and the disturbance $\eta_{X}(\hat{p}_0)$ 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.

This paper builds upon the preliminary framework presented in arXiv:1504.03779

arXiv:2609.18211 [quant-ph]

(or arXiv:2609.18211v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.18211 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Proof of Heisenberg's Error-Disturbance Relation for Individual Measurements

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