The quantum coupling of two given quantum states denotes the set of bipartite states whose marginal states are these given two states. In this paper, we provide tight inequalities to describe the structure of quantum coupling. These inequalities directly imply that the trace distance between two quantum states cannot be determined by the quantum analog of the earth mover’s distance, thus ruling out the equality version of the quantum Kantorovich–Rubinstein theorem for trace distance even in the finite-dimensional case. In addition, we provide an inequality that can be regarded as a quantum generalization of the Kantorovich–Rubinstein theorem. Then, we generalize our inequalities and apply them to the three tripartite quantum marginal problems. Numerical tests with a three-qubit system show that our criteria are much stronger than the known criteria: the strong subadditivity of entropy and the monogamy of entanglement.
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