The Research On Separability Of States In Quantum Information

In order to study the separability of quantum states in multipartite quantum systems, weconstruct Bell operators with Hermitian observables. By discussing the upper of the average, wegive out all the classification of multipartite quantum separable states. Firstly, we consider theoperator constructed by Pauli matrices. Using Cauchy-Schwarz inequality and linear program-ming, we obtained a set of Bell inequalities, which can well distinguish all possible k-separablestates in22222systems, including fully separable, bi-separable, tri-separable andfour-separable states. As the generators of SU(3) are Hermitian, we reconstruct the operator andpresent another set of Bell inequalities to characterize full separable and bi-separable quantumstates in333systems. Then we construct an operator based on the generalized Pauli ma-trices to consider the classification. It can get lower bound and is easy to realize, we can alsogeneralize it to N N N systems. Therefore, we classify the states in tri-partite systems.At last, the Laplacian matrices of graphs are studied as density matrices in quantum mechanics.We get a criterion for entanglement by matrix realignment, and generalize the entanglementproperties of mixed density matrices to higher dimension. We prove that the density matrix ofthe tensor product of four graphs is four-separable, but if a density matrix is four-separable, itcan not necessarily be written as the tensor of some graphs. We study the entanglement of twospecial graphs (the star graph and the nearest point graph). We derive that star graph ρ(K1, n1)is four-partite entangled for n=n1n2n3n4≥16, and also give a sufficient and necessary con-dition for the separability of the nearest point graph.