We consider the entanglement marginal problem, which consists of deciding whether a number of reduced density matrices are compatible with an overall separable quantum state. To tackle this problem, we propose hierarchies of semidefinite programming relaxations of the set of quantum state marginals admitting a fully separable extension. We connect the completeness of each hierarchy to the resolution of an analog classical marginal problem and thus identify relevant experimental situations where the hierarchies are complete. For finitely many parties on a star configuration or a chain, we find that we can achieve an arbitrarily good approximation to the set of nearest-neighbour marginals of separable states with a time (space) complexity polynomial (linear) on the system size. Our results even extend to infinite systems, such as translation-invariant systems in 1D, as well as higher spatial dimensions with extra symmetries.
Determining if the state of a multi-partite quantum system is separable or entangled is a difficult task. First, acquiring a full state description of an n-partite system requires a number of experiments exponential in n. Second, even if we had such a description, the computational resources required to run general algorithms for entanglement detection scale terribly with the system size.
In this paper, we provide general methods to decide whether the near-neighbor statistics of a many-body quantum state, which can be estimated through a small number of experiments, are compatible with the existence of an overall separable state. Our methods provide an efficient characterization of quantum entanglement in physical systems where the parts or sites are arranged in 1D and tree-like geometries. We also show how to detect entanglement in higher dimensional scenarios subject to global symmetries.
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