Coarse-Grained Cramér-Rao Bound in Singular Parameter Regimes of Quantum Metrology

In quantum metrology, the Cramér-Rao Lower Bound (CRLB) sets the ultimate precision limit for estimating parameters that define a family of probability distributions arising from measurement outcomes. However, when the Fisher Information Matrix (FIM) becomes singular due to specific input states or parameter configurations, the CRLB becomes undefined, preventing meaningful error bounds. We introduce a coarse-graining approach that regularizes such singularities by evaluating the CRLB over a small neighbourhood in parameter space. Our method involves a Taylor expansion of the FIM around the parameter point of interest and computes the average CRLB within this reduced region, yielding a well-defined, unbiased precision bound. We demonstrate this technique for a two-parameter estimation task in SU(2) quantum interferometry, where both the beam-splitter ratio and path-length difference are unknown. For input states that lead to a singular FIM, our method provides a viable alternative to biased estimators, extending the utility of the CRLB in regimes previously considered ill-posed. This framework enhances robustness and interpretability in precision bounds across singular regions of quantum metrological models.