The Wasserstein distance associated with Monge optimal-transport problem consists in minimising a cost. Kantorovich has shown there exists a dual formulation, consisting in maximising a profit. The distance in noncommutative geometry is defined by Conner as the search for a supremum, and may be seen as a non-commutative generalisation of Kantorovich distance. We will see that there is no obvious noncommutative equivalent to the Wassertein distance, although there do exist a dual formulation for Connes distance (as an infimum rather than a supremum).
Thomas Strobl