  The description of the complex separability structure of quantum states in
terms of partially ordered sets has been recently put forward. In this work, we
address the question of how to efficiently determine these structures for
unknown states. We propose an experimentally accessible and scalable iterative
methodology that identifies, on solid statistical grounds, sufficient
conditions for non-separability with respect to certain partitions. In
addition, we propose an algorithm to determine the minimal partitions (those
that do not admit further splitting) consistent with the experimental
observations. We test our methodology experimentally on a 20-qubit IBM quantum
computer by inferring the structure of the 4-qubit Smolin and an 8-qubit W
states. In the first case, our results reveal that, while the fidelity of the
state is low, it nevertheless exhibits the partitioning structure expected from
the theory. In the case of the W state, we obtain very disparate results in
different runs on the device, which range from non-separable states to very
fragmented minimal partitions with little entanglement in the system.
Furthermore, our work demonstrates the applicability of informationally
complete POVM measurements for practical purposes on current NISQ devices.
