On a parametric family of distance measures that includes the Hellinger and the Bures distances

Abstract

In this paper we define a parametric family of certain two-variable maps on positive cones of C⁎-algebras. The square roots of the values of those maps under a faithful tracial positive linear functional (in the cases where the square roots are well defined, i.e., those values are non-negative real numbers on the whole cones) as two-variable numerical functions can be considered as a family of potential distance measures which includes the well known Hellinger and Bures metrics. We study that family from various points of view. The main questions concern the mentioned problem of well-definedness and, whenever we have an affirmative answer to that question, the problem whether those distance measures are true metrics. Besides, we obtain some related trace characterizations. Our study is not complete, we formulate a few probably quite difficult open questions.