Research On Several Classes Of Entanglement Measures And Correlation Measures Of Quantum States For Multipartite Composite Systems

As an extremely important physical resource,quantum correlation has been widely applied to many fields of quantum information and quantum computation.At present,there are many quantum correlations,including quantum entanglement,quantum discord,measurement-induced nonlocality,quantum steering,Bell nonlocality and so on.Among them,entanglement is one of the special physical resources,which is of great theoretical value to identify and quantify the entanglement and strength of quantum states.Due to its separability concept is relatively complex,quantifying entanglement strength of quantum states for multipartite composite system is relatively difficult.Therefore,it is of certain significance to further study entanglement of quantum states for multipartite composite system,especially to give a reasonable entanglement measure.On the other hand,it is found that there are many forms of quantum correlation in separable states,and all of these quantum correlations can be used as important resources in various aspects of quantum computation.At present,many quantum correlations have been proposed in bipartite and multipartite composite systems.However,the internal structure of the multipartite quantum state is very complex,and quantifying the correlation strength contained in the multipartite quantum states is still an unsolved problem.Hence,further research on the correlation measure of the multipartite quantum states is another valuable research problem.This thesis mainly studies two aspects.Firstly,three kinds of entanglement measures of quantum states for bipartite and multipartite composite systems are given.Secondly,we introduce two class quantum correlation measures of the quantum state with respect to the k-partition for multipartite composite systems.The specific research contents are as follows:(1)Firstly,we introduce a class of preserving Hermitian linear maps,based on these maps we define an entanglement measure for bipartite quantum states,and prove that it satisfies the necessary conditions of entanglement measures.Secondly,in(Phys.Rev.A,2011,84(1):01230),the authors proposed the mirror entanglement measure for the bipartite pure states.We extend this measure to the case of the multipartite quantum states with respect to the k-partition,and obtain the mirror entanglement measure of the multipartite k-nonseparable states.We also prove that the measure is well defined.Finally,an entanglement measure based on Hellinger distance is introduced for bipartite composite systems,and the necessary properties of satisfying the entanglement measure are proved,a relation between the given entanglement measure and the coherence measure based on Hellinger distance is obtained.(2)We study two kinds of correlation measures of quantum states for multipartite composite systems.Firstly,the equivalent condition of k-product state is obtained,furthermore,a class of correlation measure of k-product state based on trace distance is introduced,and we prove that it satisfies some necessary physical conditions of correlation measure.Secondly,the definition of k-product eigenstates is introduced,and the quantum relation entropy correlation measure of non-k-product eigenstates for multipartite composite system is given,and its properties are studied.