Uncertainty Relations for Accurate Measurements

TL;DR

This paper critiques previous neutron spin experiments, clarifies the correct definition of measurement-value operators for accurate measurements, and derives new uncertainty relations for such measurements, challenging existing interpretations of Heisenberg's principle.

Contribution

It introduces a proper definition of measurement-value operators for accurate measurements and derives new uncertainty relations based on this framework.

Findings

Previous experiments misinterpreted due to incorrect measurement operators.

New uncertainty relations are established for accurate pre- and post-measurement observables.

Heisenberg's original relation is clarified as a relation between measurement error and disturbance.

Abstract

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 $\hat{x}_0$. In addition, we have presented an uncertainty relation for accurate measurements of a post-measurement observable $\hat{x}_t$. 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 $\hat{x}_t$ and the disturbance of an object observable $\hat{y}_0$.