BURES DISTANCE FOR $\alpha$-COMPLETELY POSITIVE MAPS AND TRANSITION PROBABILITY BETWEEN $P$-FUNCTIONALS

Abstract

In this paper we discuss the Bures distance between $\alpha$-CP maps on a $C^*$-algebra and the transition probability between $P$-functionals on a *-algebra. We first review the notion of $\alpha$-CP maps and the representation theorem associated to $\alpha$-CP maps. Using the Krein space representation and the set of intertwiners between Krein space representations, we study the Bures distance between $\alpha$-CP maps. We prove that the transition probability between $P$-functionals can be estimated by some functionals using $J$-representations on Krein spaces.