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arXiv:2505.22221 · Physical Review Research

Tight qubit uncertainty relations studied through weak values in neutron interferometry

Andreas Dvorak, Ismaele V. Masiello, Yuji Hasegawa, Hartmut Lemmel, Holger F. Hofmann, Stephan Sponar

Published Nov 17, 2025• 1 citation
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Abstract

In its original formulation, Heisenberg's uncertainty principle describes a trade-off relation between the error of a quantum measurement and the thereby induced disturbance on the measured object. However, this relation is not valid in general. An alternative universally valid relation was derived by Ozawa in 2003, defining error and disturbance in a general concept, experimentally accessible via a tomographic method. Later, it was shown by Hall that these errors correspond to the statistical deviation between a physical property and its estimate. Recently, it was discovered that these errors can be observed experimentally when weak values are determined through a procedure named “feedback compensation.” Here, we apply this procedure for the complete experimental characterization of the error-disturbance relation between a which-way observable in an interferometer and another observable associated with the output of the interferometer, confirming the theoretically predicted relation. As expected for pure states, the uncertainty is tightly fulfilled.

DOI
10.1103/pthn-81pm

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In its original formulation, Heisenberg's uncertainty principle describes a trade-off relation between the error of a quantum measurement and the thereby induced disturbance on the measured object. However, this relation is not valid in general. An alternative universally valid relation was derived by Ozawa in 2003, defining error and disturbance in a general concept, experimentally accessible via a tomographic method. Later, it was shown by Hall that these errors correspond to the statistical deviation between a physical property and its estimate. Recently, it was discovered that these errors can be observed experimentally when weak values are determined through a procedure named “feedback compensation.” Here, we apply this procedure for the complete experimental characterization of the error-disturbance relation between a which-way observable in an interferometer and another observable associated with the output of the interferometer, confirming the theoretically predicted relation. As expected for pure states, the uncertainty is tightly fulfilled.
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’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.
Heisenberg's uncertainty principle is probably the most famous statement of quantum physics and its essential aspects are well described by a formulations in terms of standard deviations. However, a naive Heisenberg-type error-disturbance relation 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. 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 the Ozawa's and Branciard's EDURs are valid in a wide range of experimental parameters, applying a new measurement procedure referred to as two-state method.
Over the past few years, Heisenberg's error-disturbance uncertainty relation has experienced increased attention since several experimental publications verified the theoretical findings of Ozawa predicting the violation and thus necessary reformulation of Heisenberg's relation. However, soon after their appearance, an alternative theory was presented by Busch and co-workers, which proclaimed the validity of Heisenberg's relation and thus gave rise to heated debates. Here, we present an experimental comparison of the competing approaches by applying them to the same neutron optical measurement apparatus. Empirical results for the different definitions of error and disturbance are presented for special input states and configurations of the apparatus to illustrate the opposing approaches. The inequalities restricting errors and disturbances are particularly emphasized. Despite the strong controversy, in the case of projectively measured qubit observables, both approaches lead to equal outcomes.
Recently there have been claims of experimental violations of Heisenberg's error-disturbance relation [Rozema et al, PRL 109, 200404 (2012), Erhard et al, Nature Phys. 8, 185 (2012)]. These claims were contradicted by a proof of a form of such relations [Busch et al, PRL 111, 160405 (2013)], showing that the validity of any formalization of Heisenberg's principle depends on the choice of appropriate measures of error and disturbance. While the error quantity underlying the experiments was shown to be conceptually problematic in general, it turns out to coincide, in the qubit experiments in question, with the alternative error measure used by Busch et al. Here we analyze this coincidence and discover a technique, hitherto deemed unavailable, for determining the disturbance measure. We show that this procedure reflects directly the operational meaning of the disturbance measure, in contrast to the weak-value method used in the experiments of Rozema et al and several other groups. The latter is thus not fundamental for an understanding of the quantum mechanical error-disturbance trade-off. The strong scheme is also shown to be applicable to a wider class of joint measurement scenarios.
The uncertainty principle formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable such that their product should be no less than the limit set by Planck's constant. However, Ozawa in 1988 showed a model of position measurement that breaks Heisenberg's relation and in 2003 revealed an alternative relation for error and disturbance to be proven universally valid. Here, we report an experimental test of Ozawa's relation for a single-photon polarization qubit, exploiting a more general class of quantum measurements than the class of projective measurements. The test is carried out by linear optical devices and realizes an indirect measurement model that breaks Heisenberg's relation throughout the range of our experimental parameter and yet validates Ozawa's relation.