Locally inequivalent four-qubit hypergraph states

Hypergraph states as real equally weighted pure states are important resources for quantum codes of non-local stabilizers. Using local Pauli equivalence and permutational symmetry, we reduce the 32 768 four-qubit real equally weighted pure states to 28 locally inequivalent hypergraph states and several graph states. The calculation of geometric entanglement supplemented with entanglement entropy confirms that further reduction is impossible for true hypergraph states.