The Bures metric: from positive linear functionals to completely positive maps

Abstract

This thesis aims to explore the evolution of the notion of the Bures metric in Operator Algebras, from its introduction by D. Bures in 1969 for normal states of von Neumann algebras to its extension to completely positive maps of C∗-algebras by D. Kretschmann, D. Schlingemann, and R. Werner in 2008. While Bures’ work is rooted in von Neumann algebras, our primary focus will be on unital C∗-algebras. We will explore the definitions and main properties of the Bures metric for positive linear functionals, providing bounds and additional insights into fidelity — a measure closely related to the Bures metric. The thesis extends these concepts to completely bounded maps and completely positive maps, introducing the Bures metric for such mappings and characterizing them in terms of their Stinespring representations. We show that the Bures metric forms a true metric space on the set of positive linear functionals, as well as on the set of completely positive maps that take values on injective C∗-algebras. This offers both theoretical bounds and practical methods to compute distances in these sets.