Quantum Limits of Measurements and Uncertainty Principle

TL;DR

This paper rigorously analyzes quantum measurement limits using the Robertson uncertainty relation, establishing bounds on measurement errors, conditions for the standard quantum limit, and models that can bypass these limits.

Contribution

It provides a rigorous mathematical framework for quantum measurement limits, including bounds on errors and conditions for the standard quantum limit.

Findings

Established a lower bound for measurement error products.

Provided conditions under which the standard quantum limit holds.

Constructed models that can circumvent the standard quantum limit.

Abstract

In this paper, we show how the Robertson uncertainty relation gives certain intrinsic quantum limits of measurements in the most general and rigorous mathematical treatment. A general lower bound for the product of the root-mean-square measurement errors arising in joint measurements of noncommuting observables is established. We give a rigorous condition for holding of the standard quantum limit (SQL) for repeated measurements, and prove that if a measuring instrument has no larger root-mean-square preparational error than the root-mean-square measurement errors then it obeys the SQL. As shown previously, we can even construct many linear models of position measurement which circumvent this condition for the SQL.