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

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, $\mean{Δ\hat{q}^2}\mean{Δ\hat{p}^2}\geq \hbar^2/4$. 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 $η\in(0,1]$. It is then clarified that the above Heisenberg uncertainty relation is replaced by $\mean{Δ\hat{q}^2}\mean{Δ\hat{p}^2}\geq \hbar^2/4η$ if the monitored system is unstable, while there exists a stable quantum system for which the Heisenberg limit is reached.

4 pages