Proof of the Standard Quantum Limit for Monitoring Free-Mass Position

TL;DR

This paper proves that the standard quantum limit (SQL) for monitoring a free mass's position during a time interval cannot be surpassed, establishing a fundamental quantum constraint on measurement precision.

Contribution

The paper provides a rigorous proof that the SQL always holds for a specific class of position measurements involving a probe interaction.

Findings

The SQL sets a lower bound on measurement uncertainty for free mass position monitoring.

The proof applies to measurements where a probe is prepared in a known state before interaction.

The result confirms the fundamental quantum limit in continuous position measurements.

Abstract

The measurement result of the moved distance for a free mass m during the time t between two position measurements cannot be predicted with uncertainty smaller than sqrt{hbar t/2m}. This is formulated as a standard quantum limit (SQL) and it has been proven to always hold for the following position measurement: a probe is set in a prescribed position before the measurement. Just after the interaction of the mass with the probe, the probe position is measured, and using this value, the measurement results of the pre-measurement and post-measurement positions are estimated.