Proof of Heisenberg's error-disturbance principle

TL;DR

This paper rigorously proves Heisenberg's error-disturbance principle, clarifying the roles of measurement resolution and disturbance in quantum measurements, and establishing the relation's validity under specific measurement definitions.

Contribution

The paper provides a formal proof that Heisenberg's error-disturbance relation holds when the post-measurement observable is defined to minimize measurement resolution.

Findings

Heisenberg's relation is valid when using the post-measurement resolution.

The disturbance magnitude is independent of the measurement value definition.

The proof clarifies the interpretation of error and disturbance in quantum measurements.

Abstract

In Heisenberg's error-disturbance relation for electron position measurement, the measurement error must be the one that determines the uncertainty in the electron position just after the measurement. It is the resolution $\epsilon(x_t)$ i.e., the measurement error of the post-measurement observable $\hat{x}_t$, not the precision $\epsilon(x_0)$ i.e., that of the pre-measurement observable $\hat{x}_0$ that determines the uncertainty of the observable $\hat{x}_t$. Therefore, Heisenberg's relation must be interpreted as one between the resolution $\epsilon(x_t)$ and the disturbance $\eta(y_0)$. The magnitude of the disturbance $\eta(y_0)$ is independent of the definition of the measurement value of the observable $\hat{x}_t$. Heisenberg's error-disturbance relation is proven to hold true in general, when the measurement value of $\hat{x}_t$ is defined in order to minimize the resolution $\epsilon(x_t)$.