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The quantum marginal problem is concerned with characterizing whichcollections of quantum states on different subsystems are compatible in thesense that they are the marginals of some multipartite quantum state. Presentedhere is a countable family of inequalities, each of which is necessarilysatisfied by any compatible collection of quantum states. Additionally, thisfamily of inequalities is shown to be sufficient: every incompatible collectionof quantum states will violate at least one inequality belonging to the family.

A sufficient family of necessary inequalities for the compatibility of quantum marginals