Research On Measures Of Quantum Correlations Of Multipartite Quantum States With Respect To K-partitions

In quantum information theory,quantum correlation contained in quantum state is an extremely important physical resource,among which quantum entanglement is particularly important in many quantum correlations.In the bipartite composite quantum system,there are many valuable results on entanglement detection and entanglement quantification of quantum states.However,there are many separability concepts in multipartite composite systems,such as complete separability,semiseparability,strong k-separability and k-separability,so it is difficult to detect its separability and quantify its strength of entanglement.How to detect its separability and quantify its strength of entanglement is still a hot research topic.In addition,quantum entanglement is not the only quantum correlation,and it is found that quantum correlation exists in separable states,and many methods have been proposed to describe the nonclassical correlation,including quantum discord,geometric quantum discord,measurement induced nonlocality and so on.The quantum correlations play a key role in quantum information theory,so it is of great theoretical value to study the quantum correlations of bipartite or multipartite quantum states.Now,there are many valuable results on quantifying the correlation strength of quantum states in bipartite composite system,but there are relatively few results about this field in multipartite composite systems,especially in the case of k-partitions.This paper mainly studies two aspects:one is to introduce two kinds of entanglement measures of quantum states with respect to k-partitions in multipartite composite systems,and we research the properties of these entanglement measures;the other is to study the quantum correlation measures of multipartite quantum states with respect to k-partitions,and further to discuss their properties.The concrete research contents are as follows:(1)we propose two kinds of entanglement measures of multipartite quantum states with respect to k-partitions.Firstly,we study the Schmidt number entanglement measure of multipartite k-nonseparable states,and prove that this measure satisfies invariance under local unitary transformation,convexity and monotonicity under LOCC and so on.We give a super bound and lower bound of k-MESN for multipartite pure states.Secondly,we introduce the entanglement measure of k-nonseparable states based on fidelity,and also show that this measure satisfies invariance under local unitary transformation,convexity,monotonicity under LOCC and invariance when an uncorrelated ancilla is appended to the system and so on.The measure can quantify entanglement strength of multipartite k-nonseparable states.(2)we research some quantum correlation measures of quantum states in multipartite composite systems.Firstly,we propose multipartite quantum correlation measures induced by local projection measurements based on unified(q,s)-entropies,and these measures can quantify the correlation strength of multipartite quantun states.Secondly,the concept of the density matrice having a k-product eigenbasis is introduced,then two different methods are given to quantify the correlation strength of the density matrice having no a k-product eigenbasis.Finally,we introduce the k-global quantum discord.The measure can also quantify the correlation strength of multipartite quantun states.