Generalized-mean Cramer-Rao bounds for multiparameter quantum metrology

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 combinations. By defining the weighted f-mean of estimation error and quantum Fisher information, we derive various quantum Cramer-Rao bounds on mean error in a very general form and give their refined versions with complex quantum Fisher information matrices. We show that the geometric- and harmonic-mean quantum Cramer-Rao bounds can help to reveal a larger forbidden region of estimation error for a complex signal in coherent light accompanied by a 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 and is useful in quantifying asymmetry and coherence as quantum resources.