# A New Class of Criteria for Tripartite Marginal Problem. (arXiv:1803.02673v1 [quant-ph])

The question of whether given density operators for subsystems of a

multipartite quantum system are compatible to one common total density operator

is known as the quantum marginal problem. In this paper, we focus on its

tripartite version that of determining whether there exists tripartite quantum

state with bipartite reduced density matrices equal to the given bipartite

states. We first study the bipartite marginal problem and find out there is a

"duality" relation: The distance between the marginal states is at most the

fidelity between the probability distributions generated by measuring the

bipartite state with measurements onto the symmetric (anti-symmetric) space and

antisymmetric (symmetric) space; the fidelity between the marginal states is at

least the distance between the probability distributions by measuring the

bipartite state with measurements onto the symmetric (anti-symmetric) space and

antisymmetric (symmetric) space. Then we generalize it into the tripartite

version, and build a new class of necessary criteria for the tripartite

marginal problem.