Construction of genuine multipartite entangled states

Genuine multipartite entanglement is of great importance in quantum information, especially from the experimental point of view. Nevertheless, it is difficult to characterize genuine multipartite entangled states, because the genuine multipartite entanglement is unruly. There are scattered results on the criteria for genuine multipartite entangled states. We propose another product based on the Kronecker product in this paper. The Kronecker product is a common product in quantum information with good physical interpretation. We mainly investigate whether the proposed product of two genuine multipartite entangled states is still a genuine entangled one. We understand the entanglement of the proposed product better by characterizing the entanglement of the Kronecker product. Then we show the proposed product is a genuine multipartite entangled state in two cases. The results develop nontrivial criteria for a class of genuine multipartite entangled states. One can use such criteria to systematically construct genuine multipartite entangled states of more parties.