Bures Distance For Completely Positive Maps

TL;DR

This paper extends Bures distance, originally defined for states, to completely positive maps between C*-algebras using Hilbert C*-modules, providing new metrics, examples, and a rigidity theorem.

Contribution

It introduces a Hilbert C*-module framework for Bures distance on completely positive maps and establishes conditions for the metric to be valid.

Findings

The Bures distance forms a metric for maps into von Neumann algebras.

Several examples and counterexamples illustrate the theory.

A rigidity theorem shows maps close to identity contain the original algebra.

Abstract

D. Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between $C^*$-algebras by D. Kretschmann, D. Schlingemann and R. F. Werner. We present a Hilbert $C^*$-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close to the identity map contain a copy of the original algebra.