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We construct an analogue of the classical $L^p$-Wasserstein distance for thestate space of a $C^*$-algebra. Given an abstract Lipschitz gauge on a$C^*$-algebra $\mathcal{A}$ in the sense of Rieffel, one can define theclassical $L^p$-Wasserstein distance on the state space of each commutative$C^*$-subalgebra of $\mathcal{A}$. We consider a projective limit of thesemetric spaces, which appears to be the space of all quasi-linear states,equipped with a distance function. We call this distance the projective$L^p$-Wasserstein distance. It is easy to show, that the state space of a$C^*$-algebra is naturally embedded in the space of its quasi-linear states,hence, the introduced distance is defined on the state space as well. We showthat this distance is reasonable and well-behaved. We also formulate asufficient condition for a Lipschitz gauge, such that the correspondingprojective $L^p$-Wasserstein distance metricizes the weak$^*$-topology on thestate space.

$L^p$-Wasserstein distances on state and quasi-state spaces of $C^*$-algebras