Towards a Monge–Kantorovich metric in noncommutative geometry

We investigate whether the identification between Connes' spectral distance in noncommutative geometry and the Monge–Kantorovich distance of order 1 in the theory of optimal transport – that has been pointed out by Rieffel in the commutative case – still makes sense in a noncommutative framework. To this aim, given a spectral triple $(\mathcal A,\mathcal H, D)$ with noncommutative $\mathcal A$, we introduce a “Monge–Kantorovich”-like distance $W_D$ on the space of states of $\mathcal A$, taking as a cost function the spectral distance $d_D$ between pure states. We show in full generality that $d_D\leq W_D$, and exhibit several examples where the equality actually holds true, in particular on the unit two-ball viewed as the state space of $M_2(\mathbb C)$. We also discuss $W_D$ in a two-sheet model (product of a manifold by $\mathbb C^2$), pointing towards a possible interpretation of the Higgs field as a cost function that does not vanish on the diagonal.