Quantum Nonlocality And Multipartite Entanglement

Quantum nonlocality theory and quantum entanglement theory are very important in quantum mechanics,and they are also the basic theories of quantum secure communication.Quantum nonlocality can be used for quantum data hiding and quantum secret sharing,while multipartite entanglement plays a central role in quantum key distribution,quantum teleportation,and quantum error-correcting codes.Therefore,the theoretical research on quantum nonlocality and multipartite entanglement not only contributes to the development of quantum mechanics,but also promotes the development of quantum secure communication.This dissertation specifically studies unextendible product bases and strong quantum nonlocality related to quantum nonlocality,as well as k-uniform states and quantum information masking related to multipartite entanglement,and specific research content and innovation points can be summarized into the following three parts:Firstly,we establish a relation between hypercube’s and unextendible product bases.Unextendible product bases have quantum nonlocality,which can guarantee information security.There are many studies on the minimum size of unextendible product bases currently,but the results on the large sizes of unextendible product bases,and explicit constructions are few.In this dissertation,we give the correspondence between tile structures and unextendible product bases in bipartite systems.By using this correspondence,we construct unextendible product bases with large sizes in bipartite systems.We also extend tile structures to multipartite systems,and establish a relation between hypercubes and unextendible product bases in multipartite systems.Based on the decomposition of hypercubes,we construct unextendible product bases with large sizes in three-partite and four-partite systems.Since unextendible product bases are locally indistinguishable,entanglement resources are necessary to distinguish them.So we also study the entanglement-assisted discrimination for unextendible product bases in bipartite systems.Secondly,we establish a relation between hypercubes and strong quantum nonlocality.Strong nonlocality can further improve information security,but there are only few strongly nonlocal orthogonal product sets in three-and four-partite systems currently.In this dissertation,by using the previously mentioned decomposition of hypercubes,we construct strongly nonlocal orthogonal product sets in three-,four-and fivepartite systems and strongly nonlocal orthogonal entangled sets in three-partite systems,and also prove that our unextendible product bases in three-partite and four-partite systems have strong quantum nonlocality.Furthermore,by using cyclic permutation group action,we construct strongly nonlocal orthogonal entangled sets in general N-partite homogeneous systems,and when N=3,4,we find strongly nonlocal orthogonal genuinely entangled sets.Finally,we establish a relation between quantum error-correcting codes and quantum information masking,and give the explicit constructions of 2,3-uniform states in heterogeneous systems.At present,there are few constructions of k-uniform states in heterogeneous systems,and quantum information masking in multipartite systems has a large security hole.In this dissertation,by using mixed orthogonal arrays,we construct a series of 2,3-uniform states in heterogeneous systems,and give two methods of generating(k-1)-uniform states from k-uniform states.By using shadow inequalities,we give some results on the nonexistence of absolutely maximally entangled states in heterogeneous systems.Furthermore,we propose the concept of k-uniform quantum information masking,which requires that collusion between any k parties can not reveal the encoded information.We establish a relation between quantum error-correcting codes in heterogeneous systems and k-uniform quantum information masking.Based on this relation,we show that the no-masking theorem is a special case of the quantum Singleton bound for quantum error-correcting codes in heterogeneous systems essentially,and give a more general no-masking theorem.We also give some methods for constructing new quantum error-correcting codes from old quantum error-correcting codes in heterogeneous systems,and these methods can be used to construct k-uniform states in heterogeneous systems.