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

From the noncommutative nature of quantum mechanics, estimation of canonical observables $\widehat{q}$ and $\widehat{p}$ 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.