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

Understand

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.