Ethntanglement Of The Symmetric States

With the rapid development of quantum information technology,the qualitative and quan-titative study of quantum entangled states has become an urgent problem to be solved.This area involves a wide range of research.They have been integrated and promoted each other,both theoretical and experimental problems,both physical and mathematical aspects.They have be-come an important research direction in contemporary quantum theory.However,so far,there are still many theoretical problems that have not been solved in quantum entanglement state,which hinders the process of experimental work.Many of these theoretical problems are direct mathematical problems.For example,the quantum entanglement classification and the operable criterion,the mathematical characterization of the maximum entangled state and the quantita-tive description of the quantum entanglement state.Developing corresponding mathematical theories to characterize quantum entangled states,especially putting forward the mathemati-cal discriminant method that can be operated in experiments,has become a top priority for the development of quantum information technology.In this paper,the entanglement criteria of diagonal symmetric states and the entanglement classification of C3(?)C3 symmetric states are mainly studied.First,we discuss the entangle-ment classification of C3(?)C3 symmetric state in detail.We find that any C3(?)C3 symmetric entangled state is equivalent to the two class of states.According to the definition of the bipar-tite diagonal symmetric state and the vectors in the C3(?)C3 Dicke state basis,we present the general form of density matrix of C3(?)C3 diagonal symmetric state.Through the reintegration of C3(?)C3(?)C3 Dicke state basis,we obtain that C3(?)C3(?)C3 is isomorphic to C3(?)C6 and redefine the density matrix of C3(?)C6 diagonal symmetric states,which is extended to Cd(?)Cd(?)cd symmetric quantum system,so the density matrix of Cd(?)Cd(d+1)/2 diagonal symmetric states is also redefined.If the partial transposition of the bipartite quantum states is negative,the quantum state is entangled.According to the Peres-Horodecki criterion,the suffi-cient conditions for the entanglement of C3(?)C3 in diagonal symmetric states are obtained.For C3(?)C3 diagonal symmetric state,p satisfies the PPT(Positive Partial Transposed)condition if and only if p is separable.So we get the separable criterion of C3(?)C3 diagonal symmetric state.Then,according to k-separability criteria for n-partite quantum states,we study the entan-glement criterion of C3(?)C6 and Cd(?)Cd(d+1)/2 diagonal symmetric state respectively.Finally,we also use algebraic method to study the PPT condition of C3(?)C6 diagonal symmetric state and get the conclusion about the rank of density matrix and the dimension of the value domain of the subsystem.