Measurement speed limit and Cramér-Rao bound for time-averaged estimators
Abstract
We consider a measured system X containing a parameter θ 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.
Related
- Work fluctuation speed limit in boundary conformal field theories
- Speed Limit for Information Acquisition in Stochastic Learning Dynamics
- Similarities between the speed limit in relativity and the sound barrier in inviscid fluid flows
- Probability bounds on trajectory-level exceedances of quantum speed limits under continuous-measurement feedback
- Quantum Fisher Information as the Speed Limit for Multipartite Entanglement
BibTeX
@misc{odajima2026measurement,
title = {{Measurement speed limit and Cramér-Rao bound for time-averaged estimators}},
author = {Hiroyuki Odajima and Andreas Dechant},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32252},
eprint = {2610.32252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32252},
}
RIS
TY - UNPB
TI - Measurement speed limit and Cramér-Rao bound for time-averaged estimators
AU - Hiroyuki Odajima
AU - Andreas Dechant
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32252
UR - https://arxiv.org/abs/2610.32252
AB - We consider a measured system X containing a parameter θ 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.
ER -