Geometry of distributive multiparty entanglement in 4 − qubit hypergraph states

A detailed investigation of the multiparty entanglement present in the 4 − qubit quantum hypergraph states is presented, following a measurement-based geometrical approach. Considering a classification of the 4 − party quantum system represented by a mathematical hypergraph based on the connections between its vertices, the genuine 4 − party entanglement present in each bi-partition of the states have been measured. A strong correlation between the connectivity of the vertices of the hypergraphs and the genuine 4 − party entanglement has been found. The equivalence of the genuine 4 − party entanglement present in each bi-partition is shown considering similar connectivity of the vertices. This explicates the cyclic permutation symmetry of the multiparty entanglement present in the 4 − qubit hypergraph states. Physically, one may expect the quantum systems with superposition of many states to behave in this symmetric manner while mapped into a network-type picture, which the authors have quantified, as well as classified in this work.