  We investigate the geometrical structures of multipartite states based on
construction of toric varieties. In particular, we describe pure quantum
systems in terms of affine toric varieties and projective embedding of these
varieties in complex projective spaces. We show that a quantum system can be
corresponds to a toric variety of a fan which is constructed by gluing together
affine toric varieties of polytopes. Moreover, we show that the projective
toric varieties are the spaces of separable multipartite quantum states. The
construction is a generalization of the complex multi-projective Segre variety.
Our construction suggests a systematic way of looking at the structures of
multipartite quantum systems.
