Generalized-mean Cram\'er-Rao Bounds for Multiparameter Quantum Metrology

TL;DR

This paper introduces generalized quantum Cramér-Rao bounds using various mean functions, revealing new insights into quantum estimation errors and resource quantification in multiparameter quantum metrology.

Contribution

It develops a unified framework for quantum Cramér-Rao bounds based on weighted $f$-means, extending the analysis beyond the traditional arithmetic mean and linking to quantum resources.

Findings

Geometric- and harmonic-mean bounds reveal more forbidden error regions.

$f$-mean quantum Fisher information quantifies quantum resources like coherence.

Refined bounds incorporate complex quantum Fisher information matrices.

Abstract

In multiparameter quantum metrology, the weighted-arithmetic-mean error of estimation is often used as a scalar cost function to be minimized during design optimization. However, other types of mean error can reveal different facets of permissible error combination. By introducing the weighted $f$-mean of estimation error and quantum Fisher information, we derive various quantum Cram\'er-Rao bounds on mean error in a very general form and also give their refined versions with complex quantum Fisher information matrices. We show that the geometric- and harmonic-mean quantum Cram\'er-Rao bounds can help to reveal more forbidden region of estimation error for a complex signal in coherent light accompanied by thermal background than just using the ordinary arithmetic-mean version. Moreover, we show that the $f$-mean quantum Fisher information can be considered as information-theoretic quantities to quantify asymmetry and coherence as quantum resources.