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The description of the complex separability structure of quantum states interms of partially ordered sets has been recently put forward. In this work, weaddress the question of how to efficiently determine these structures forunknown states. We propose an experimentally accessible and scalable iterativemethodology that identifies, on solid statistical grounds, sufficientconditions for non-separability with respect to certain partitions. Inaddition, we propose an algorithm to determine the minimal partitions (thosethat do not admit further splitting) consistent with the experimentalobservations. We test our methodology experimentally on a 20-qubit IBM quantumcomputer by inferring the structure of the 4-qubit Smolin and an 8-qubit Wstates. In the first case, our results reveal that, while the fidelity of thestate is low, it nevertheless exhibits the partitioning structure expected fromthe theory. In the case of the W state, we obtain very disparate results indifferent runs on the device, which range from non-separable states to veryfragmented minimal partitions with little entanglement in the system.Furthermore, our work demonstrates the applicability of informationallycomplete POVM measurements for practical purposes on current NISQ devices.

Experimentally accessible non-separability criteria for multipartite entanglement structure detection