Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1309.1966

Uncertainty Relations for Accurate Measurements

Seiji Kosugi

Published Sep 8, 2013• 0 citations
arXivHTMLPDF

Abstract

Our investigation of the results of the neutron spin experiment by Ehhart et al. demonstrates that their results cannot be understood in accordance with common sense. For example, their results obtained with different measurement errors are equal to each other for a same neutron state. This is because their measurement values have a large systematic error due to adopting an incorrect measurement-value operator, which is the operator that gives the measurement values of the measured observable. They asserted that their results confirmed the validity of a universally valid uncertainty relation proposed by Ozawa. Therefore, their results demonstrate unexpectedly that Ozawa's uncertainty relation holds true for incorrect measurements. In addition, we have proved this fact theoretically. The measurement-value operator must be defined in order that the systematic error of measurement values predicted by the operator is zero. Such a measurement is called an accurate measurement. Two kind of uncertainty relations are derived for accurate measurements of a pre-measurement observable $\hat{x}_0$. In addition, we have presented an uncertainty relation for accurate measurements of a post-measurement observable $\hat{x}_t$. Heisenberg's original uncertainty relation for measurement processes should be interpreted as this relation between the resolution, that is, the measurement error of the post-measurement observable $\hat{x}_t$ and the disturbance of an object observable $\hat{y}_0$.

DOI
10.48550/arxiv.1309.1966

Related papers

Proof of Heisenberg's error-disturbance principle

Seiji Kosugi

In Heisenberg's error-disturbance relation for electron position measurement, the measurement error must be the one that determines the uncertainty in the electron position just after the measurement. It is the resolution $ε(x_t)$ i.e., the measurement error of the post-measurement observable $\hat{x}_t$, not the precision $ε(x_0)$ i.e., that of the pre-measurement observable $\hat{x}_0$ that determines the uncertainty of the observable $\hat{x}_t$. Therefore, Heisenberg's relation must be interpreted as one between the resolution $ε(x_t)$ and the disturbance $η(y_0)$. The magnitude of the disturbance $η(y_0)$ is independent of the definition of the measurement value of the observable $\hat{x}_t$. Heisenberg's error-disturbance relation is proven to hold true in general, when the measurement value of $\hat{x}_t$ is defined in order to minimize the resolution $ε(x_t)$.

arXiv:1504.03779•Apr 15, 2015•0 citations

Aspects of universally valid Heisenberg uncertainty relation

Kazuo Fujikawa, Koichiro Umetsu

A numerical illustration of a universally valid Heisenberg uncertainty relation, which was proposed recently, is presented by using the experimental data on spin-measurements by J. Erhart et al. [Nat. Phys. 8, 185 (2012)]. This uncertainty relation is closely related to a modified form of the Arthurs–Kelly uncertainty relation, which is also tested by the spin-measurements. The universally valid Heisenberg uncertainty relation always holds, but both the modified Arthurs–Kelly uncertainty relation and the Heisenberg error–disturbance relation proposed by Ozawa, which was analyzed in the original experiment, fail in the present context of spin-measurements, and the cause of their failure is identified with the assumptions of unbiased measurement and disturbance. It is also shown that all the universally valid uncertainty relations are derived from Robertson's relation and thus the essence of the uncertainty relation is exhausted by Robertson's relation, as is widely accepted.

arXiv:1211.1828•Progress of Theoretical and Experimental Physics•Jan 1, 2013•9 citations

Comment on "Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements"

Y. Kurihara

In this comment we show that the experimental results for a universally valid uncertainty relation in ref.[J. Erhart et al. Nat. Phys.(2012)10.1038/nphys2194]cannot be justified. The experiments cannot be recognized to establish a violation of the Heisenberg-type uncertainty relation suggested by Ozawa[Phys. Lett. A318(2003)21, Phys. Rev. A67(2003)042105].

arXiv:1201.5151•Jan 24, 2012•1 citation

Violation of Heisenberg's error-disturbance uncertainty relation in neutron-spin measurements

Georg Sulyok, Stephan Sponar, Jacqueline Erhart +3 more

In its original formulation, Heisenberg's uncertainty principle dealt with the relationship between the error of a quantum measurement and the thereby induced disturbance on the measured object. Meanwhile, Heisenberg's heuristic arguments have turned out to be correct only for special cases. An alternative universally valid relation was derived by Ozawa in 2003. Here, we demonstrate that Ozawa's predictions hold for projective neutron-spin measurements. The experimental inaccessibility of error and disturbance claimed elsewhere has been overcome using a tomographic method. By a systematic variation of experimental parameters in the entire configuration space, the physical behavior of error and disturbance for projective spin-$\frac{1}{2}$ measurements is illustrated comprehensively. The violation of Heisenberg's original relation, as well as the validity of Ozawa's relation become manifest. In addition, our results conclude that the widespread assumption of a reciprocal relation between error and disturbance is not valid in general.

arXiv:1305.7251•Physical Review A•Aug 14, 2013•105 citations

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

Seiji Kosugi

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.

quant-pharXiv:2609.18211•Sep 17, 2026•0 citations

Error-disturbance uncertainty relations in neutron spin measurements

Stephan Sponar

Heisenberg’s uncertainty principle in a formulation of uncertainties, intrinsic to any quantum system, is rigorously proven and demonstrated in various quantum systems. Nevertheless, Heisenberg’s original formulation of the uncertainty principle was given in terms of a reciprocal relation between the error of a position measurement and the thereby induced disturbance on a subsequent momentum measurement. However, a naive generalization of a Heisenberg-type error-disturbance relation for arbitrary observables is not valid. An alternative universally valid relation was derived by Ozawa in 2003. Though universally valid, Ozawa’s relation is not optimal. Recently, Branciard has derived a tight error-disturbance uncertainty relation (EDUR), describing the optimal trade-off between error and disturbance under certain conditions. Here, we report a neutron-optical experiment that records the error of a spin-component measurement, as well as the disturbance caused on another spin-component to test EDURs. We demonstrate that Heisenberg’s original EDUR is violated, and Ozawa’s and Branciard’s EDURs are valid in a wide range of experimental parameters, as well as the tightness of Branciard’s relation.

International Journal of Quantum Information•May 21, 2016•0 citations
Our investigation of the results of the neutron spin experiment by Ehhart et al. demonstrates that their results cannot be understood in accordance with common sense. For example, their results obtained with different measurement errors are equal to each other for a same neutron state. This is because their measurement values have a large systematic error due to adopting an incorrect measurement-value operator, which is the operator that gives the measurement values of the measured observable. They asserted that their results confirmed the validity of a universally valid uncertainty relation proposed by Ozawa. Therefore, their results demonstrate unexpectedly that Ozawa's uncertainty relation holds true for incorrect measurements. In addition, we have proved this fact theoretically. The measurement-value operator must be defined in order that the systematic error of measurement values predicted by the operator is zero. Such a measurement is called an accurate measurement. Two kind of uncertainty relations are derived for accurate measurements of a pre-measurement observable x^0\hat{x}_0x^0​. In addition, we have presented an uncertainty relation for accurate measurements of a post-measurement observable x^t\hat{x}_tx^t​. Heisenberg's original uncertainty relation for measurement processes should be interpreted as this relation between the resolution, that is, the measurement error of the post-measurement observable x^t\hat{x}_tx^t​ and the disturbance of an object observable y^0\hat{y}_0y^​0​.
In Heisenberg's error-disturbance relation for electron position measurement, the measurement error must be the one that determines the uncertainty in the electron position just after the measurement. It is the resolution ε(xt)ε(x_t)ε(xt​) i.e., the measurement error of the post-measurement observable x^t\hat{x}_tx^t​, not the precision ε(x0)ε(x_0)ε(x0​) i.e., that of the pre-measurement observable x^0\hat{x}_0x^0​ that determines the uncertainty of the observable x^t\hat{x}_tx^t​. Therefore, Heisenberg's relation must be interpreted as one between the resolution ε(xt)ε(x_t)ε(xt​) and the disturbance η(y0)η(y_0)η(y0​). The magnitude of the disturbance η(y0)η(y_0)η(y0​) is independent of the definition of the measurement value of the observable x^t\hat{x}_tx^t​. Heisenberg's error-disturbance relation is proven to hold true in general, when the measurement value of x^t\hat{x}_tx^t​ is defined in order to minimize the resolution ε(xt)ε(x_t)ε(xt​).
A numerical illustration of a universally valid Heisenberg uncertainty relation, which was proposed recently, is presented by using the experimental data on spin-measurements by J. Erhart et al. [Nat. Phys. 8, 185 (2012)]. This uncertainty relation is closely related to a modified form of the Arthurs–Kelly uncertainty relation, which is also tested by the spin-measurements. The universally valid Heisenberg uncertainty relation always holds, but both the modified Arthurs–Kelly uncertainty relation and the Heisenberg error–disturbance relation proposed by Ozawa, which was analyzed in the original experiment, fail in the present context of spin-measurements, and the cause of their failure is identified with the assumptions of unbiased measurement and disturbance. It is also shown that all the universally valid uncertainty relations are derived from Robertson's relation and thus the essence of the uncertainty relation is exhausted by Robertson's relation, as is widely accepted.
In its original formulation, Heisenberg's uncertainty principle dealt with the relationship between the error of a quantum measurement and the thereby induced disturbance on the measured object. Meanwhile, Heisenberg's heuristic arguments have turned out to be correct only for special cases. An alternative universally valid relation was derived by Ozawa in 2003. Here, we demonstrate that Ozawa's predictions hold for projective neutron-spin measurements. The experimental inaccessibility of error and disturbance claimed elsewhere has been overcome using a tomographic method. By a systematic variation of experimental parameters in the entire configuration space, the physical behavior of error and disturbance for projective spin-12\frac{1}{2}21​ measurements is illustrated comprehensively. The violation of Heisenberg's original relation, as well as the validity of Ozawa's relation become manifest. In addition, our results conclude that the widespread assumption of a reciprocal relation between error and disturbance is not valid in general.
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 x^t\hat{x}_tx^t​, rather than a prior observable x^0\hat{x}_0x^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 XXX of a posterior probe observable X^t\hat{X}_tX^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 εX(x^t)ε_{X}(\hat{x}_t)εX​(x^t​) and the disturbance ηX(p^0)η_{X}(\hat{p}_0)ηX​(p^​0​) holds strictly at the level of individual measurements, characterized by the specific readout XXX. 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.
Heisenberg’s uncertainty principle in a formulation of uncertainties, intrinsic to any quantum system, is rigorously proven and demonstrated in various quantum systems. Nevertheless, Heisenberg’s original formulation of the uncertainty principle was given in terms of a reciprocal relation between the error of a position measurement and the thereby induced disturbance on a subsequent momentum measurement. However, a naive generalization of a Heisenberg-type error-disturbance relation for arbitrary observables is not valid. An alternative universally valid relation was derived by Ozawa in 2003. Though universally valid, Ozawa’s relation is not optimal. Recently, Branciard has derived a tight error-disturbance uncertainty relation (EDUR), describing the optimal trade-off between error and disturbance under certain conditions. Here, we report a neutron-optical experiment that records the error of a spin-component measurement, as well as the disturbance caused on another spin-component to test EDURs. We demonstrate that Heisenberg’s original EDUR is violated, and Ozawa’s and Branciard’s EDURs are valid in a wide range of experimental parameters, as well as the tightness of Branciard’s relation.