Quantum Entanglement Under Different Quantum Network Structures

Quantum entanglement is a fundamentally important field in quantum mechanics,which plays a vital role in quantum information processing.A lot of researchers have paid much more attention to quantum entanglement theory.It is of great significance to study the quantum entanglement in different network structures.In this paper,we mainly study the maximally entangled states of multiple particle system and the properties of the quantum entangled states in the network structure of multiple nodes.In chapter 1,we mainly review the knowledge of the quantum theory that will be used in the subsequent section of this thesis,including quantum entanglement,quantum density matrix,tensor product,monogamy,entropy and rank.In chapter 2,we firstly review the concept and the properties of the absolutely maximally entangled state and the planar maximally entangled state.Secondly,by comparing the conditions that these two kinds of maximally entangled states satisfied,we propose another kind of maximally entangled state.This maximally entangled state is called the planar two-regionfour-part maximally entangled state(PKME(4)).Because the particles in a plane can be divide into two region with four parts.Obviously,the condition of the planar two-region-four-part maximally entangled state is less than the condition of absolutely maximally entangled state and more than the condition of planar maximally entangled state.Thus,there are more this kind of the maximally entangled states than the absolutely maximally entangled states,and the number of the planar maximally entangled states is more than the number of this kind of states.Thirdly,we prove that there are this kind of maximally entangled states and more complex maximally entangled states in the even-particle systems.And we also demonstrate that there are this kind of maximally entangled states and more complex maximally entangled states in the special odd-particle systems.In chapter 3,we firstly generalize the planar two-region-four-part maximally entangled state to the planar two-region-28)-part maximally entangled state,which is denoted as PKME(28)).Secondly,we find some PKME(28))in the special particle systems.Because there are not the absolutely maximally entangled states in four-particle and seven-particle systems,so one question is whether there are PKME(28))states in four-particle and seven-particle systems.We show that there are the planar two-region-four-part maximally entangled state in four-particle system,and there are the planar two-region-six-part maximally entangled state in seven-particle system.In the end,the specific applications of these two kinds of planar maximally entangled states are discussed.In chapter 4,we firstly review the constraints of the various quantum entangled states which can be prepared in the three-node network structures.Secondly,two kinds of quantum entanglement constrains on the quantum entangled states prepared in the three-note network structures are investigated.Then we define the quantum mutual information for four parties.Finally,the constraints of the quantum entangled states prepared in the special situations in the four-node network structures are studied.