Two states

TL;DR

This paper introduces a new metric  for unital completely positive maps between C*-algebras, relating it to the existing Bures metric and Wasserstein metric, with applications to dilation theory.

Contribution

It defines a novel distance  for unital completely positive maps and establishes a precise relationship with the Bures metric using dilation theory.

Findings

 can be expressed via completely positive maps on free products

When  is injective,  and  are related by a specific formula

The relationship involves the Choi-Li constrained dilation theorem

Abstract

D. Bures defined a metric $\beta $ on states of a $C^*$-algebra and this concept has been generalized to unital completely positive maps $\phi : \mathcal A \to \mathcal B$, where $\mathcal B$ is either an injective $C^*$-algebra or a von Neumann algebra. We introduce a new distance $\gamma $ for the same classes of unital completely positive maps. We use in our definition the distance between representations on the same Hilbert $C^*$-module in contrast to the Bures metric which uses one representation and distinct vectors. This metric can be expressed in terms of a class of completely positive maps on free products of $C^*$-algebras and in this setting $\gamma $ looks like Wasserstein metric on probability measures. Surprisingly, when the range algebra $\mathcal B$ is injective, $\gamma $ and $\beta $ are related by the following explicit formula: $\beta ^2= 2-\sqrt{4- \gamma ^2} .$ A deep result of Choi and Li on constrained dilation is the main tool in proving this formula.