On the Bures metric, C*-norm, and the quantum metric

TL;DR

This paper explores the relationships between the Bures metric, C*-norm topology, and quantum metrics on density spaces, establishing conditions for their topological equivalence and differences within noncommutative geometry.

Contribution

It demonstrates the topological relationships between the Bures metric, C*-norm, and quantum metrics, providing examples of strict containment and non-equivalence.

Findings

The C*-norm topology is finer than the Bures metric topology.

In finite dimensions, the quantum metric induces the same topology as the Bures metric.

Examples show that these metrics are not always topologically equivalent.

Abstract

We prove that the topology on the density space with respect to a unital C*-algebra and a faithful 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 of when 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.