Fundamental bounds for parameter estimation with few measurements

Bounding the optimal precision in parameter estimation tasks is of central importance for technological applications. In the regime of a small number of measurements, or low signal-to-noise ratios, the meaning of common frequentist bounds, such as the Cramér-Rao bound (CRB), becomes questionable. Here, we discuss a different (Barankin-like) approach to derive the conditions for the existence of optimal estimators and calculate their variance, for any number of measurement repetitions. We elaborate on connections to the shot-noise limit and the Kitaev phase estimation algorithm and derive an extended Cramér-Rao bound that is compatible with a finite variance in situations where the Barankin bound (BaB) is undefined. Finally, we show an exemplary numerical confrontation between frequentist and Bayesian approaches to parameter estimation. Published by the American Physical Society 2024