Using formal definitions for the measurement precision and the disturbance (measurement back-action) η, Ozawa (2003 Phys. Rev. A 67 042105) has shown that Heisenberg's claimed relation between these quantities is false in general. Here, we show that the quantities introduced by Ozawa can be determined experimentally, using no prior knowledge of the measurement under investigation—both quantities correspond to the root-mean-squared difference given by a weak-valued probability distribution. We propose a simple three-qubit experiment that can illustrate the failure of Heisenberg's measurement–disturbance relation and the validity of an alternative relation proposed by Ozawa.
Using formal definitions for the measurement precision and the disturbance (measurement back-action) η, Ozawa (2003 Phys. Rev. A 67 042105) has shown that Heisenberg's claimed relation between these quantities is false in general. Here, we show that the quantities introduced by Ozawa can be determined experimentally, using no prior knowledge of the measurement under investigation—both quantities correspond to the root-mean-squared difference given by a weak-valued probability distribution. We propose a simple three-qubit experiment that can illustrate the failure of Heisenberg's measurement–disturbance relation and the validity of an alternative relation proposed by Ozawa.
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.
In this comment on the paper by F. Kaneda, S.-Y. Baek, M. Ozawa and K. Edamatsu [Phys. Rev. Lett. 112, 020402, 2014, arXiv:1308.5868], we point out that the claim of having refuted Heisenberg's error-disturbance relation is unfounded since it is based on the choice of unsuitable and operationally problematical quantifications of measurement error and disturbance. As we have shown elsewhere [PRL 111, 160405, 2013], for appropriate choices of operational measures of error and disturbance, Heisenberg's heuristic relation can be turned into a precise inequality which is a rigorous consequence of quantum mechanics.
In [1], Busch et al. showed that it is possible to construct an error-disturbance relation having the same form as Heisenberg's original heuristic definition[2], in contrast to the theory proposed by Ozawa[3] which we and others recently confirmed experimentally[4,5]. With Ozawa's definitions of measurement error and disturbance, a relation of Heisenberg's form is not in general valid, and a new error-disturbance relationship can be derived. Here we explain the different physical significance of the two definitions, and suggest that Ozawa's definition better corresponds to the usual understanding of the disturbance that Heisenberg discussed.
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.
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-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.
Reports on experiments recently performed in Vienna [Erhard et al., Nature Phys. 8, 185 (2012)] and Toronto [Rozema et al., Phys. Rev. Lett. 109, 100404 (2012)] include claims of a violation of Heisenberg's error-disturbance relation. In contrast, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013)]. To resolve the apparent conflict, we formulate here a general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined errors in an approximate joint measurement of a pair of $\ifmmode\pm\else\textpm\fi{}1$-valued observables A,B are tightly bounded from below by a quantity that measures the degree of incompatibility of A and B. The claim of a violation of Heisenberg is shown to fail because it is based on unsuitable measures of error and disturbance. Finally we show how the experiments mentioned may directly be used to test our error inequality.