Barycenter of the arithmetic-harmonic quantum divergence

A notion of divergence is a very important and useful tool to measure the difference between probability distributions or between data (information). We consider a quantum divergence constructed by the difference of two-variable weighted arithmetic and harmonic means on the open convex cone of positive definite Hermitian matrices, called the arithmetic-harmonic quantum divergence. We see its invariance properties and study the barycenter minimizing the weighted sum of arithmetic-harmonic quantum divergences to given variables. We provide the lower bound for the barycenter of the arithmetic-harmonic quantum divergence in terms of Loewner order and its upper bound in terms of operator norm.