Proof of Heisenberg’s Error-Disturbance Relation

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.

Figure 1

Scenario of preparation uncertainty. ${\Delta }_{\rho }$ is the root of the variance of the distribution obtained for the indicated observable in the state $\rho$. In this pair of experiments no particle is subject to both a position and a momentum measurement.

Figure 2

Scenario of measurement uncertainty for successive measurements, as discussed by Heisenberg (middle row). An approximate position measurement $Q^{\prime }$ is followed by an ideal momentum measurement, effectively given a measurement $P^{\prime }$ on the initial state. The accuracy $\Delta (Q,Q^{\prime })$ quantifies the difference between the output distributions of $Q^{\prime }$ and an ideal position measurement $Q$ (first row). Similarly, the momentum disturbance $\Delta (P,P^{\prime })$ quantifies the difference between the distributions obtained by $P^{\prime }$ and by an ideal momentum measurement $P$ (last row). The definitions for these $\Delta$ quantities (see text) can be applied, more generally, to an arbitrary joint measurement $M$ (dashed box). This can be any device producing, in every shot, a $q$ value and a $p$ value. $Q^{\prime }$ and $P^{\prime }$ are then defined as the marginals of $M$, obtained by ignoring the other output.