Separability Criteria And Entanglement Criteria Of Quantum State

In quantum theory,one basic problem is to determine whether a given bipartite state is entangled or separable.We first introduce the development process of quantum theories and some basic definitions,properties,assumptions,and so on.Then we give the main criterion for the study.In this paper,we discuss two methods of judgement.The first method,by using the Bloch representation of quantum states in terms of the Heisenberg-Weyl observable basis,we present a new entanglement criterion and separability criterion for bipartite quantum systems.The main discussion is to improve the criteria under the different bases and compare with the SU(N)basis.Above all,we give the construction of the Heisenberg-Weyl observable basis,then we give a general form of density matrix which based on the new basis.Next,constructed by using its various parameters and coefficients,we get a new matrix.Then we get the upper bound of the trace norm about the new matrix according to the separable condition,so that we know when the quantum state will be entangled.Finally,we generalize the criteria to the case of multipartite states.The second way,we construct a 3 × 3 parametric matrix map ?,which is positive but not completely positive,to get a lower bound of the concurrence.Under the circumstance of a specific state,we analyse it by choosing appropriate parameter values to get a new different map,so that the different lower bound of concurrence can be obtained.As for these two ways,we combine with the actual state and show the advantages of the criteria in this paper by comparing our new criteria with the previous criteria.Finally,we make a summary of the construction methods,theoretical basis,results and so on.