Unified strategy for non-invertible Fisher information matrix in quantum metrology

Abstract

In quantum multi-parameter estimation, the precision of estimating unknown parameters is bounded by the Cramér-Rao bound (CRB), defined via the inverse of the Fisher information matrix (FIM). However, in certain scenarios–such as distributed quantum sensing–the FIM becomes non-invertible due to parameter redundancy, which depends on the probe state and measurement. This issue is often handled using a weaker form of the CRB, potentially overestimating the uncertainty and underrepresenting achievable precision. Here, we propose an alternative approach by introducing equality constraints to remove redundancy and define the CRB via the Moore-Penrose pseudoinverse of the FIM. This framework enables systematic treatment of both simultaneous estimation and distributed sensing cases. We demonstrate its utility by reanalyzing several known examples within this unified perspective, highlighting improved interpretability and practical relevance. Our results offer a concrete guideline for addressing non-invertible FIMs and enhancing the precision of quantum multi-parameter estimation in realistic scenarios.