  The earth mover's distance is a measure of the distance between two
probabilistic measures. It plays a fundamental role in mathematics and computer
science. The Kantorovich-Rubinstein theorem provides a formula for the earth
mover's distance on the space of regular probability Borel measures on a
compact metric space. In this paper, we investigate the quantum earth mover's
distance. We show a no-go Kantorovich-Rubinstein theorem in the quantum
setting. More precisely, we show that the trace distance between two quantum
states can not be determined by their earth mover's distance. The technique
here is to track the bipartite quantum marginal problem. Then we provide
inequality to describe the structure of quantum coupling, which can be regarded
as quantum generalization of Kantorovich-Rubinstein theorem. After that, we
generalize it to obtain into the tripartite version, and build a new class of
necessary criteria for the tripartite marginal problem.
