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EPJ Web of Conferences

Quantum rms error and Heisenberg’s error-disturbance relation

Paul Busch

Published Jan 1, 2014• 2 citations
Publisher

Abstract

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, a Heisenberg-type tradeoff relation for joint measurements of position and momentum has been formulated and proven in [Phys. Rev. Lett. 111, 160405 (2013)]. Here I show how the apparent conflict is resolved by a careful consideration of the quantum generalization of the notion of root-mean-square error. The claim of a violation of Heisenberg’s principle is untenable as it is based on a historically wrong attribution of an incorrect relation to Heisenberg, which is in fact trivially violated. We review a new general trade-off relation for the necessary errors in approximate joint measurements of incompatible qubit observables that is in the spirit of Heisenberg’s intuitions. The experiments mentioned may directly be used to test this new error inequality.

DOI
10.1051/epjconf/20147801002

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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, a Heisenberg-type tradeoff relation for joint measurements of position and momentum has been formulated and proven in [Phys. Rev. Lett. 111, 160405 (2013)]. Here I show how the apparent conflict is resolved by a careful consideration of the quantum generalization of the notion of root-mean-square error. The claim of a violation of Heisenberg’s principle is untenable as it is based on a historically wrong attribution of an incorrect relation to Heisenberg, which is in fact trivially violated. We review a new general trade-off relation for the necessary errors in approximate joint measurements of incompatible qubit observables that is in the spirit of Heisenberg’s intuitions. The experiments mentioned may directly be used to test this new error inequality.
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,BA,BA,B are tightly bounded from below by a quantity that measures the degree of incompatibility of AAA and BBB. 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.
While the slogan "no measurement without disturbance" has established itself under the name of the Heisenberg effect in the consciousness of the scientifically interested public, a precise statement of this fundamental feature of the quantum world has remained elusive, and serious attempts at rigorous formulations of it as a consequence of quantum theory have led to seemingly conflicting preliminary results. Here we show that despite recent claims to the contrary [L. Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)], Heisenberg-type inequalities can be proven that describe a tradeoff between the precision of a position measurement and the necessary resulting disturbance of momentum (and vice versa). More generally, these inequalities are instances of an uncertainty relation for the imprecisions of any joint measurement of position and momentum. Measures of error and disturbance are here defined as figures of merit characteristic of measuring devices. As such they are state independent, each giving worst-case estimates across all states, in contrast to previous work that is concerned with the relationship between error and disturbance in an individual state.
Recent years have witnessed a controversy over Heisenberg's famous error-disturbance relation. Here the conflict is resolved by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. Two approaches to adapting the classic notion of root-mean-square error to quantum measurements are discussed. One is based on the concept of a noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven.
Recently, Busch, Lahti, and Werner (arXiv:1306.1565v1 [quant-ph]) claimed that Heisenberg's error-disturbance relation can be proved in its original form with new formulations of error and disturbance, in contrast to the theory proposed by the present author and confirmed by recent experiments. Despite their claim, it is shown here that a class of solvable models of position measurement with explicit interaction Hamiltonians escape the Busch-Lahti-Werner relation. It is also made clear where their proof fails. Those models have unambiguously defined zero root-mean-square error and finite root-mean-square disturbance in every input state and are naturally considered to violate Heisenberg's error-disturbance relation in any conceivable formulation.
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.
Abstract Defining and measuring the error of a measurement is one of the most fundamental activities in experimental science. However, quantum theory shows a peculiar difficulty in extending the classical notion of root-mean-square (rms) error to quantum measurements. A straightforward generalization based on the noise-operator was used to reformulate Heisenberg’s uncertainty relation on the accuracy of simultaneous measurements to be universally valid and made the conventional formulation testable to observe its violation. Recently, its reliability was examined based on an anomaly that the error vanishes for some inaccurate measurements, in which the meter does not commute with the measured observable. Here, we propose an improved definition for a quantum generalization of the classical rms error, which is state-dependent, operationally definable, and perfectly characterizes accurate measurements. Moreover, it is shown that the new notion maintains the previously obtained universally valid uncertainty relations and their experimental confirmations without changing their forms and interpretations, in contrast to a prevailing view that a state-dependent formulation for measurement uncertainty relation is not tenable.