Abstract We study operator means more general than the Kubo-Ando means. They are given as minima of geodesically convex functions and allow uniquely defined extensions to any number of variables. Multivariate hyper-means is a class of means of this type bounded from below by the arithmetic mean. We extend the definition of a hyper-man in the bivariate case and discover bivariate means that are not restrictions of the multivariate means studied earlier. We introduce the notion of reduced relative quantum entropy and prove that it is convex. The result is used to give a simplified proof of a theorem of Lieb and Seiringer.
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 Cram\'er-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 Cram\'er-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.