Dynamical one-from-many quantum metrology: Sum rule and matrix-free precision bound

How can practical quantum single-parameter estimation be performed accurately when nuisance parameters and environmental noises are inevitably present, especially given that traditional multi-parameter methods break down due to singular quantum Fisher information matrices?

The authors investigate dynamical one-from-many parameter estimation by analyzing the scaling invariance of quantum Lindblad master equations. By scaling time and model parameters, they derive an exact sum rule connecting the quantum Fisher information of all model parameters to the quantum Fisher information with respect to time. Recognizing that this forces the augmented quantum Fisher information matrix to become singular, they bypass matrix inversion entirely and derive a novel, matrix-free scalar precision bound that is valid for both optimal and suboptimal system observables.

The primary theoretical breakthrough is the discovery of a sum rule that necessitates treating time on an equal footing with all Hamiltonian and noise parameters in an augmented parameter set. Because the resulting augmented quantum Fisher information matrix is inherently non-invertible, conventional matrix-based Cramér-Rao bounds fail. The newly derived matrix-free quantum precision bound successfully circumvents this singularity, reproducing standard results in optimal limits while providing significantly tighter bounds out of equilibrium for unitary models and quantum thermometry scenarios.

This framework resolves a major bottleneck in quantum metrology by offering a systematic operational strategy for single-parameter estimation plagued by nuisance parameters. By replacing complex matrix inversions with versatile scalar response functions, the approach enables accurate precision tracking in complex open quantum systems and noisy environments where traditional multi-parameter methods fail.