Proof of Heisenberg's error-disturbance relation

TL;DR

This paper rigorously proves Heisenberg-type inequalities that quantify the fundamental trade-off between measurement precision and disturbance in quantum position-momentum measurements, resolving previous conflicting claims.

Contribution

It provides a rigorous, state-independent formulation of the error-disturbance relation for position and momentum, clarifying the fundamental quantum measurement limits.

Findings

Heisenberg-type inequalities are proven for position and momentum measurements.

The inequalities are state independent, providing worst-case estimates.

The results resolve previous conflicting claims about the 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.