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arXiv:1504.03779

Proof of Heisenberg's error-disturbance principle

Seiji Kosugi

Published Apr 15, 2015• 0 citations
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Abstract

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 $ε(x_t)$.

DOI
10.48550/arxiv.1504.03779

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Comment on "Proof of Heisenberg's error-disturbance principle"

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Recently, Kosugi [arXiv:1504.03779v2 [quant-ph]] argued that Heisenberg's error-disturbance relation (EDR) must be interpreted as being between the resolution, the preparational error for the post-measurement observable, and the disturbance. He further claimed that Heisenberg's EDR can be proven to hold true in general, when the meter observable is modified as one of its functions. Here, some comments are given to suggest that the above claims are not supported.

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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​).
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.
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.
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​.
Heisenberg's uncertainty principle is one of the main tenets of quantum theory. Nevertheless, and despite its fundamental importance for our understanding of quantum foundations, there has been some confusion in its interpretation: Although Heisenberg's first argument was that the measurement of one observable on a quantum state necessarily disturbs another incompatible observable, standard uncertainty relations typically bound the indeterminacy of the outcomes when either one or the other observable is measured. In this paper, we quantify precisely Heisenberg's intuition. Even if two incompatible observables cannot be measured together, one can still approximate their joint measurement, at the price of introducing some errors with respect to the ideal measurement of each of them. We present a tight relation characterizing the optimal tradeoff between the error on one observable vs. the error on the other. As a particular case, our approach allows us to characterize the disturbance of an observable induced by the approximate measurement of another one; we also derive a stronger error-disturbance relation for this scenario.
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.