The Pairwise Concurrence In 3-qubit And 4-qubit Symmetric Quantum States

Entanglement is a valuable resource,and occurs in a complex system consisting of two or more quantum subsystems.So the study of entanglement measure is a key problem in the field of quantum information.Concurrence is widely accepted as a entanglement measure.And as the entanglement measure of 2-qubit quantum states,its specific calcu-lation formula has been well known.k-ME concurrence,as a generalized entanglement concurrence,is used to measure entanglement of multipartite quantum states.It satisfies many important properties of entanglement measure and can detect all k-inseparability states in any dimension explicitly.The topic of this thesis is the pairwise concurrence in 3-qubit and 4-qubit symmetric quantum states.We investigate the maximum entanglement in the 2-qubit reduced states of 3-qubit and 4-qubit symmetric quantum states.Then we obtain the conditions for 3-qubit,4-qubit symmetric quantum states when their pairwise concurrence values are maximum.In chapter 1,we introduce some basic concepts related to the content of this the-sis,including quantum bit,density matrix and reduced density matrix,entanglement measure,pairwise concurrence,symmetric state and X-state,etc.In chapter 2,we introduce the Hilbert subspaces VH stabled by permutation groups H on ternary and quaternion sets,and calculate the pairwise concurrence of 3-qubit and 4-qubit even X type symmetric quantum states and the non-even X type symmetric quantum states in these symmetric spaces,then we can obtain the conditions for the maximum entanglement of the 2-qubit reduced states with 3-qubit and 4-qubit symmetric quantum states.In chapter 3,we briefly introduce k-ME concurrence of even X type symmetric quantum states,and simplify the formulas for calculating k-ME concurrence of 3-qubit and 4-qubit even X type symmetric quantum states.In chapter 4,in 3-qubit and 4-qubit quantum systems,we introduce GHZ states and W states mixed with white noise respectively,calculate the values of pairwise concurrence of the mixed quantum states.In chapter 5,we introduce even X type mixed symmetric quantum states in a special form.In different dimensional subspaces,we calculate the pairwise concurrence values of the mixed states in the same form.