Quantum divergence and barycenter associated with the spectral geometric mean

Abstract

We introduce new type of quantum divergence defined by the difference of trace values between the weighted arithmetic and spectral geometric means. Several interesting properties related to such quantum divergence such as the invariance property, the in-betweenness property, and the characterization of the (s-)geodesically convex set have been established. We also show the existence and uniqueness of barycenter, minimizing the weighted sum of such quantum divergences.