  We construct an analogue of the classical $L^p$-Wasserstein distance for the
state space of a $C^*$-algebra. Given an abstract Lipschitz gauge on a
$C^*$-algebra $\mathcal{A}$ in the sense of Rieffel, one can define the
classical $L^p$-Wasserstein distance on the state space of each commutative
$C^*$-subalgebra of $\mathcal{A}$. We consider a projective limit of these
metric 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 show
that this distance is reasonable and well-behaved. We also formulate a
sufficient condition for a Lipschitz gauge, such that the corresponding
projective $L^p$-Wasserstein distance metricizes the weak$^*$-topology on the
state space.
