  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.

Functional Analysis