The quantum marginals problem (QMP) aims to understand how the various marginals of a joint quantum state are related to one another by deciding whether or not a given collection of marginals is compatible with some joint quantum state. Although existing techniques for the QMP are well developed for the special case of disjoint marginals, the same is not true for the generic case of overlapping marginals. The leading technique for the generic QMP, published by Yu et. al. (2021), resorts to evaluating a hierarchy of semidefinite programs.

In this talk, I will introduce a slightly different approach to the QMP by demonstrating how to construct a simple hierarchy of operator inequality constraints each of which are necessarily satisfied by any collection of marginals of a joint quantum state. Then, using state-estimation techniques and large deviation theory, I will sketch the proof that the satisfaction of these inequalities is additionally sufficient for a collection of marginals to be compatible with some joint quantum state.