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

TL;DR

This work addresses the challenge of non-invertible Fisher information matrices in quantum multiparameter estimation due to parameter redundancy. It introduces a unified strategy that imposes equality constraints to remove redundant parameters and uses the Moore-Penrose pseudoinverse $F_x^{+}$ to define the Cramér-Rao bound on the reduced parameter space, applicable to both simultaneous estimation and distributed sensing. The approach extends naturally to the quantum Fisher information matrix, clarifying attainability and guiding measurement design when generators commute or do not. Through analyses of various probe states, including GHZ-like, NOON-like, and cyclic phase states, and extensions to many-body systems, the framework provides practical guidelines to avoid underestimating uncertainty and to achieve the bound on the estimable parameter set. Together, these insights offer a robust, broadly applicable methodology for precise quantum metrology in realistic, potentially redundancy-rich 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.

Figures (1)

• Figure 1: Flow chart for guiding how to use the CRI in multi-parameter estimation. SE and DQS denote simultaneous estimation and distributed quantum sensing, respectively.