Classifications,criteria And Properties Of Quantum Correlations And Their Applications In Quantum Many-body Systems

Quantum correlations play a central role in the study of quantum information and quantum foundation.In this work,we made a systematic study of the definition,classifi-cation,criteria and properties of quantum correlations.We first studied the classification of quantum correlations,including bipartite and multipartite quantum total correlation,quantum discord,quantum entanglement,quantum steering and Bell nonlocality.Then we discussed the criteria and properties of quantum correlations,finally,we studied entanglement area law of quantum neural network states and their applications.For bipartite case,we know that there is a hierarchical structure for the set of states which exhibit different quantum correlations,this can be proved by constructing some special one-parameter states.For multipartite case,we firstly pointed out that the gener-alization of bipartite correlation is genuine multipartite quantum correlations.We made a deep and thorough study of genuine multipartite quantum correlations.Then,by con-structing special quantum channels,we can map a bipartite quantum correlated state to a genuine multipartite correlated state.By this method,we proved that there is hierar-chical structure for genuine multipartite quantum correlations which is very similar as bipartite case.From the weakest one to the strongest one,we have:genuine multipar-tite total correlation,genuine multipartite discord,genuine multipartite entanglement,genuine multipartite steering,genuine multipartite Bell nonlocality.A thorough study of Bell nonlocality and quantum contextuality was presented in this work.Since quantum contextuality is a generalization of Bell nonlocality in one-particle system,we can study them in a unified framework.We developed three differ-ent kinds of framework to study them:(?)compatibility graph framework based on no-disturbance principle;(?)exclusivity graph framework based on exclusivity principle;and(?)entropic cone framework based on entropic no-disturbance principle.We stud-ied the Bell-Kochen-Specker theorem in these three different frameworks and analyzed their connections and differences.Based on these results,we took a systematic study of monogamy relations of Bell nonlocality and quantum contextuality.We proved that there is a monogamy relation between CHSH inequalities,between cycle-type quantum contextuality test inequalities,between CHSH inequality and KCBS inequality.We dis-cussed the physical origin of these monogamy relations,this shed new lights into the study of basics principles of quantum mechanics.We also discussed the applications of Bell nonlocality,we presented a method to to construct communication complexity problems from Bell inequalities based on compatibility graph.We also made a study of criteria of quantum entanglement and quantum steering based on uncertainty relations.We first discussed the universal uncertainty relation and analyzed the connections and difference of it with other type of uncertainty relations.Then we developed the criteria of entanglement and steering,and discussed the advan-tages and shortcomings of this kind of criteria.To remedy the shortcoming of the kind of criteria,we also developed the criteria based on fine-grained uncertainty relations.Finally,we studied the quantum neural network states and their entanglement prop-erties.We introduced the neural network representation of quantum sates and density operators,and compared and analyzed the states constructed from different neural net-works.We studied the entanglement area law of neural network states in the framework of entanglement-geometry duality.We proved that the neural network state with local connections obey the entanglement area law.The application of entanglement area law in image classification problem is also presented.