Relation between fundamental estimation limit and stability in linear quantum systems with imperfect measurement

TL;DR

This paper explores how the fundamental limits of estimating quantum observables are affected by measurement imperfections and system stability, revealing that instability increases the uncertainty bound while stability can achieve the Heisenberg limit.

Contribution

It establishes a modified uncertainty relation for continuously monitored linear quantum systems considering measurement efficiency and stability.

Findings

Uncertainty bound increases with measurement inefficiency in unstable systems.

Stable systems can reach the Heisenberg uncertainty limit.

Modified uncertainty relation depends on system stability and measurement efficiency.

Abstract

From the noncommutative nature of quantum mechanics, estimation of canonical observables $\hat{q}$ and $\hat{p}$ is essentially restricted in its performance by the Heisenberg uncertainty relation, $\mean{\Delta \hat{q}^2}\mean{\Delta \hat{p}^2}\geq \hbar^2/4$. This fundamental lower-bound may become bigger when taking the structure and quality of a specific measurement apparatus into account. In this paper, we consider a particle subjected to a linear dynamics that is continuously monitored with efficiency $\eta\in(0,1]$. It is then clarified that the above Heisenberg uncertainty relation is replaced by $\mean{\Delta \hat{q}^2}\mean{\Delta \hat{p}^2}\geq \hbar^2/4\eta$ if the monitored system is unstable, while there exists a stable quantum system for which the Heisenberg limit is reached.