Two states

Abstract

D. Bures defined a metric $\beta$ on states of a $C^{\ast }$-algebra and this concept has been generalized to unital completely positive maps $\varphi :\mathcal{A}\to \mathcal{B}$, where $\mathcal{B}$ is either an injective $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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.

arXiv comment28 pages, The abstract and some statements revised based on a referee report. To appear in the Houston Journal of Mathematics