Measurement speed limit and Cramér-Rao bound for time-averaged estimators

Abstract

We consider a measured system $X$ containing a parameter $\theta$ of interest and the problem of estimating this parameter by observing a second, coupled measurement system $Y$. We show that, in thermodynamic equilibrium, the Fisher information of $Y$ associated with the parameter is bounded by that of $X$, implying that conservative interactions between $X$ and $Y$ reduce the precision of parameter estimation. Conversely, out of equilibrium, we show that the Fisher information of $Y$ can exceed that of $X$, so that non-conservative interactions can potentially increase the precision of parameter estimation. For idealized, back-action-free measurements, where the coupling to $Y$ does not perturb $X$, we derive a speed limit for the increase of Fisher information of $Y$. This speed limit only depends on the properties of the measured system $X$ and places a fundamental bound on the precision of any back-action-free measurement. As a corollary, we obtain a generalized Cramér-Rao bound (CRB) for time-averaged estimators. We illustrate our results for linear dynamics and suggest a scheme for disturbance-free non-equilibrium sensing with improved accuracy.

arXiv comment17 pages, 6 figures