After a brief guide to the formalism of quantum mechanics, we introduce a quantum version of the Wasserstein distance of order 1, a well-known notion from the theory of optimal mass transport. Then, we apply it to measure the local distinguishability of quantum states. In particular, we focus on finite quantum spin systems using the approach based on optimal transport to derive concentration inequalities for local observables and to study the equivalence of the thermodynamical ensembles.