ON THE BURES METRIC, $C^*$-NORM AND QUANTUM METRIC

Abstract Given a unital $C^*$ -algebra and a faithful trace, we prove that the topology on the associated density space induced by the $C^*$ -norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide a metric on the density space induced by a quantum metric in the sense of Rieffel and prove that the induced topology is the same as the topology induced by the Bures metric and $C^*$ -norm when the $C^*$ -algebra is assumed to be finite dimensional. Finally, we provide an example where the Bures metric and induced quantum metric are not metric equivalent. Thus, we provide a bridge between these aspects of quantum information theory and noncommutative metric geometry.