Proof of Heisenberg’s Error-Disturbance Relation

Abstract

While the slogan "no measurement without disturbance" has established itself under the name 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 [Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)], Heisenberg-type inequalities can be proven that describe a trade-off 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. Heisenberg's 1927 paper [7] introducing the uncertainty relations is one of the key contributions to early quantum mechanics. It is part of virtually every quantum mechanics course, almost always in the version forwarded by Kennard [9], Weyl [10] and Robertson [11]. What is often overlooked, however, is that this popular version is only one way of making the idea of uncertainty precise. The original paper begins with a famous discussion of the resolution of microscopes, in which the accuracy (resolution) of an approximate position measurement is related to the disturbance of the particle's momentum.This situation is no way covered by the standard relations, since in an experiment concerning the KennardWeyl-Robertson inequality no particle meets with both a position and a momentum measurement. Heisenberg's semiclassical discussion has no immediate translation into the modern quantum formalism, particularly since the momentum disturbance prima facie involves the comparison of two (generally) non-commuting quantities, the momentum before and after the measurement. Such a translation does require some careful conceptual work, and one can arrive at different results. This is shown by the example of Ozawa [4], who defines a relation he claims to be a rigorous version of Heisenberg's ideas, and shows that it fails to hold in general. A suggested modification of the false relation has recently been verified experimentally [5,6]. This has been widely publicized as a refutation of Heisenberg's ideas, in apparent contradiction to our main result. However, there is no contradiction, and the disagreement only shows that there is a grain of rigorously explicable truth in Heisenberg, provided one looks in the right place for it. While Ozawa aims to describe the interplay between error and disturbance for an individual state, our approach gives a state-independent characterization...

Optimized entropic uncertainty for successive projective measurements

“…It was also proved experimentally that the Heisenberg's relation (1) is violated in spin measurements, while the improved relationship (3) remains valid [12]. Later, the errordisturbance relation was improved in a stronger form [6,14]. The EDUR reduces to the Robertson uncertainty relations (2) when there is no error in the first measurement (X) = 0 and the disturbance is replaced by the statistical deviation of the measurement Y as η(Y ) = δ(Y ).…”

“…Through extensive investigations, it has been known that various distinctive properties of quantum mechanics can be derived from the principle [2]. However, its precise underlying meaning has so far eluded many attempts to explain its diverse features [3][4][5][6][7]. Heisenberg proposed the uncertainty relation in 1927 after postulating the kinematics of quantum canonical variables that do not commute [1].…”

Quantum rms error and Heisenberg’s error-disturbance relation

“…As shown in [11] and [12], the following measurement error relation holds for approximate joint measurements of position Q and momentum P in terms of observable C, D:…”

“…The present contribution is a concise report on joint work with P. Lahti and R.F. Werner [10][11][12][13].…”

Heisenberg's uncertain legacy

“…Only in recent years has it been challenged by the suggestion that the precision of measurement can do better 2 -an idea prompted originally by attempts to estimate the feasibility of gravitational-wave detection. Opposed points of view, and experimental measurements of the revised error-disturbance relation, have been put forward [3][4][5][6] . The upshot is that whether or not you think Heisenberg's relation was correct depends on what you think he meant by it: whether he was talking about one-shot measurements or averages.…”