Two states

D. Bures defined a metric $β$ on states of a $C^*$-algebra and this concept has been generalized to unital completely positive maps $ϕ: \mathcal A \to \mathcal B$, where $\mathcal B$ is either an injective $C^*$-algebra or a von Neumann algebra. We introduce a new distance $γ$ 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 $γ$ looks like Wasserstein metric on probability measures. Surprisingly, when the range algebra $\mathcal B$ is injective, $γ$ and $β$ are related by the following explicit formula: $β^2= 2-\sqrt{4- γ^2} .$ A deep result of Choi and Li on constrained dilation is the main tool in proving this formula.

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