  We analyze a quantum version of the Monge--Kantorovich optimal transport
problem. The quantum transport cost related to a Hermitian cost matrix $C$ is
minimized over the set of all bipartite coupling states $\rho^{AB}$ with fixed
reduced density matrices $\rho^A$ and $\rho^B$ of size $m$ and $n$. The minimum
quantum optimal transport cost $\rT^Q_{C}(\rho^A,\rho^B)$ can be efficiently
computed using semidefinite programming. In the case $m=n$ the cost $\rT^Q_{C}$
gives a semidistance if and only if $C$ is positive semidefinite and vanishes
exactly on the subspace of symmetric matrices. Furthermore, if $C$ satisfies
the above conditions, then $\sqrt{\rT^Q_{C}}$ induces a quantum analogue of the
Wasserstein-2 distance. Taking the quantum cost matrix $C^Q$ to be the
projector on the antisymmetric subspace, we provide a semi-analytic expression
for $\rT^Q_{C^Q}$ for any pair of single-qubit states and show that its square
root yields a transport distance on the Bloch ball. Numerical simulations
suggest that this property holds also in higher dimensions. Assuming that the
cost matrix suffers decoherence and that the density matrices become diagonal,
we study the quantum-to-classical transition of the Earth mover's distance,
propose a continuous family of interpolating distances, and demonstrate that
the quantum transport is cheaper than the classical one. Furthermore, we
introduce a related quantity -- the SWAP-fidelity -- and compare its properties
with the standard Uhlmann--Jozsa fidelity. We also discuss the quantum optimal
transport for general $d$-partite systems.
