| Quantum entanglement plays a key role in quantum information processing as an important resource.Characterization of quantum entangled states is a basic and impor-tant issue in the theory of quantum entanglement.Entanglement of two-qubit quantum states or some special quantum states has obtained good research results so far.However,the problem of entanglement characterization for mixed states of quantum systems with Hilbert space dimensions greater than 2?3,especially for multiparty situations,is still an open question.The monogamy and polygamy relations can provide a characterization of the entanglement distribution in multipartite systems,which is one of the most important ways to investigate the entanglement structures and properties of many-body quantum states.The main content of this thesis is to investigate the monogamy and polygamy relations of multiparty quantum entanglement.The thesis is organised as follows:In chapter 1,we review the main concepts of theory of entanglement related to the research contents,including quantum entanglement,entanglement measure,entanglement of assistance,monogamy and polygamy relations of entanglement and so on.In chapter 2,we investigate one class of entanglement measures based on distance,such as Bures measure of entanglement and geometric measure of entanglement.First,we study the maximally entangled states in terms of entanglement measures based on distance.Using the basic properties of the distance entanglement measures,we conclude that any entanglement measures based on distance must be maximal on pure states.Fur-thermore,we prove that theth power of Bures measure of entanglement and geometric measure of entanglement,as special cases of entanglement measures based on distance,obeys a class of general monogamy inequalities in multiqubit states forα≥1.We fur-ther establish a class of tight monogamy inequalities of multiqubit systems in terms of theth power of Bures measure of entanglement and geometric measure of entanglement forα≥1.In chapter 3,we present a new measure of entanglement without convexity—loga-rithmic convex roof extended negativity（LCREN）,which not only overcomes the short-comings of the logarithmic negativity failing to recognize positive partial transpose entan-gled states,but also satisfies many important properties of an entanglement measure.It can be seen that so far all entanglement measures found to fulfill a monogamy inequality turn out to be convex.We investigate the general monogamy inequality for logarithmic negativity and LCREN.To our surprise,we find that theth power of logarithmic nega-tivity and LCREN obey a class of general monogamy inequalities in multiqubit systems,2?2?3 systems and 2?2?2nsystems forα≥4 ln 2.Given that the logarithmic negativ-ity and LCREN are not convex,our results indicate that the lack of convexity of both the logarithmic negativity and LCREN is not sufficient enough to destroy the monogamous,which is indeed a fundamental property of any entanglement measure,but not a unique feature of a convex entanglement measure.By using the convex-roof extended negativity of assistance（CRENo A）,we also define a dual to LCREN,namely LCRENo A.We further prove that the βth power of LCRENo A,obeys a class of general polygamy inequalities in multiqubit systems for 0≤β≤2.Using the power of the logarithmic negativity and LCREN,we further establish a class of tight monogamy inequalities of multiqubit systems,2?2?3 systems and 2?2?2nsystems in terms of theth power of logarithmic negativity and LCREN forα≥4 ln 2.We also show that the βth power of LCRENo A obeys a class of tight polygamy inequalities of multiqubit systems for 0≤β≤2.In chapter 4,we investigate the tight monogamy and polygamy relations of multi-party entanglement in arbitrary-dimensional quantum systems.We first provide a class of tight monogamy relations of multiparty entanglement with larger lower bounds in com-parison to all known entanglement monogamy relations,in terms of theth power of the bipartite entanglement measure.We also give a class of tight polygamy relations of multiparty entanglement with smaller upper bounds than the existing ones,in terms of theth power of the entanglement of assistance.We provide examples in which our new monogamy and polygamy relations are tighter than the previous ones.Using these results we can provide a precise characterization of the entanglement distribution in multipartite systems. |
