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

TL;DR

This paper introduces a unified method using equality constraints and the Moore-Penrose pseudoinverse to handle non-invertible Fisher information matrices in quantum multi-parameter estimation, improving precision analysis in distributed sensing.

Contribution

It proposes a systematic framework to address non-invertible FIMs in quantum metrology using constraints and pseudoinverse, unifying estimation approaches.

Findings

Reanalysis of known examples demonstrates improved interpretability.

Framework effectively handles parameter redundancy in distributed sensing.

Guidelines provided for practical quantum estimation scenarios.

Abstract

In quantum multi-parameter estimation, the precision of estimating unknown parameters is bounded by the Cramer-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.