# Quantum Earth mover's distance, No-go Quantum Kantorovich-Rubinstein theorem, and Quantum Marginal Problem. (arXiv:1803.02673v2 [quant-ph] UPDATED)

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.