Toric varieties: Simple combinatorial and geometrical structure of multipartite quantum systems

We investigate the geometrical structure of multipartite states based on the construction of toric varieties. We show that the toric variety represents the space of general pure states and projective toric variety defines the space of separable set of multi-qubits states. We also discuss in details the construction of single-, two-, three-, and multi- qubits states. This construction gives a very simple and elegant visual representation of the geometrical structure of multipartite quantum systems.