Combinatorial and Geometrical Structures of Multipartite Quantum Systems, pp. 589-604
Authors: (Hoshang Heydari, Quantum and Field Theory Group, Dept. of Physics, Stockholm Univ., AlbaNova Univ. Center, Stockholm, Sweden)
Abstract: In recent years we have witnessed an increase of interests in geometrical and topological structures of bipartite and multipartite states based on Hopf fibration, Segre variety, hyperdeterminant, etc. We have also discussed geometrical and topological structure of multipartite states based on the Segre variety, Grassmann variety, and Hopf fibration. In this paper we investigate the combinatorial and geometrical structures of multipartite states based on the construction of toric varieties. In particular, we describe pure quantum systems in terms of algebraic toric varieties and projective embedding of these varieties in suitable 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. We discuss in details construction of two-, three-, and multi- qubits states. The construction is a generalization of the complex multi-projective Segre variety and it is also a systematic way of looking at structure of any quantum system such as a single qubit state or a complex multi-qubit system with many party involved. These geometrical results can contribute to our better understanding of complex multipartite quantum systems.