Two states

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.