Quantitative threshold for sufficient Fisher information

Determine, from the principal Fisher information operator eigenvalue \(\lambda_1\), whether the associated principal eigenvector provides sufficient information about a hidden parameter for accurate retrieval, and derive a theoretical prediction of the relevant threshold across different scattering systems and parameters.

Background

The paper demonstrates experimentally that the principal eigenvector of the Fisher information operator can estimate small changes in a hidden parameter and can focus energy onto the parameter’s physical location. However, the authors report that they cannot quantitatively infer from the value of the principal eigenvalue alone whether the available information is sufficient for reliable parameter retrieval. Although empirical results suggest that $\lambda_1$ must exceed a threshold, the threshold varies with the system and parameter and lacks a theoretical characterization. The authors identify a future statistical investigation of Fisher eigenvalues as a possible route, considering factors such as the number of channels, symmetries, absorption, and port coupling.