Generation of quadripartite entanglement from a hybrid scheme with a four-wave mixing process and linear beam splitters

We theoretically propose a hybrid scheme with nonlinear four-wave mixing process and linear beam splitters to generate continuous-variable quadripartite entanglement. To expose the entanglement properties of the generated state, the coupling strength matrix is derived to reveal the interactions of the resulting modes at first. The generated state is verified to be a H-graph state with a non-full rank coupling strength matrix by using graphical calculus for Gaussian pure states. Then based on the eigenmodes for the generated state, we find that the introduction of beam splitters does not alter the deduced eigenmodes of the nonlinear four-wave mixing process, and only contributes to generate quadripartite entanglement. In the end, we quantitatively investigate the internal entanglement structure of this scheme. Our work paves the way to generate quadripartite entanglement from a hybrid scheme with nonlinear four-wave mixing process and linear beam splitters.