  A quantum version of the Monge-Kantorovich optimal transport problem is
analyzed. The transport cost is minimized over the set of all bipartite
coupling states $\rho^{AB}$, such that both of its reduced density matrices
$\rho^A$ and $\rho^B$ of size $N$ are fixed. We show that, selecting the
quantum cost matrix to be proportional to the projector on the antisymmetric
subspace, the minimal transport cost leads to a semidistance between $\rho^A$
and $\rho^B$, which can be bounded from below by the rescaled Bures distance
and from above by the root infidelity. In the single qubit case we provide a
semi-analytic expression for the optimal transport cost between any two states,
and prove that its square root satisfies the triangle inequality and yields an
analogue of the Wasserstein distance of order two on the set of density
matrices. Assuming that the cost matrix suffers decoherence, we study the
quantum-to-classical transition of the Earth Mover's distance, propose a
continuous family of interpolating distances, and demonstrate in the case of
diagonal mixed states that the quantum transport is cheaper than the classical
one.
