  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.
