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

Understand

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.