Is the Heisenberg uncertainty relation really violated?

TL;DR

This paper revisits the Heisenberg uncertainty principle, proposing revised definitions of measurement error and disturbance that restore its validity in continuous linear measurement interactions.

Contribution

It introduces new definitions of error and disturbance considering measurement gain, confirming the Heisenberg relation's validity in specific measurement scenarios.

Findings

Heisenberg relation holds for continuous linear measurements with revised definitions.

Revised definitions account for measurement gain effects.

Clarifies physical meaning of large errors and disturbances.

Abstract

It has been pointed out that for some types of measurement the Heisenberg uncertainty relation seems to be violated. In order to save the situation a new uncertainty relation was proposed by Ozawa. Here we introduce revised definitions of error and disturbance taking into account the gain associated with generalized measurement interactions. With these new definitions, the validity of the Heisenberg inequality is recovered for continuous linear measurement interactions. We also examine the changes in distribution functions caused by the general measurement interaction and clarify the physical meanings of infinitely large errors and disturbances.