Theory And Application Of Quantum Entanglement Measures

Quantum information,as a rising interdisciplinary field which combines quantum mechanics,informatics and mathematics,becomes a new frontier science for information processing.It has widely attracted the focus of international academia,as well as been one of the core scientific problems of China 's basic science in the twenty-first century.It is not only widely believed that quantum information processing offers the secure and high rate information transmission,fast computational solution of certain important problems,which are at the heart of the modern information technology.But also,it provides new angles,tools and methods which help in understanding other fields of science.The key concept of quantum information is the quantum entanglement and one of the most fun-damental parts of quantum entanglement is the entanglement measure.Any quantitative study on both quantum communication and capability of quantum computation will rely on the entanglement measure.Without a proper entanglement measure,no further research will be carried out.In the case of multipartite states,given that there does not exist a single measure that can successfully account for all possible entanglement characteristics and applications,each measure usually performs better for a specific purpose and is always needed to choose the one that better fits our needs Therefore,the characterization of the entanglement measures for multipartite states is a long-term task.Quantum entanglement is not shareable at liberty when distributed among three or more parties.This property is called entanglement monogamy,which is of paramount importance in many protocols of quantum information and quantum communication.This thesis focuses on the measure of multipartite entanglement,entanglement monogamy and their applications.In the third chapter,by the reduced density matrices corresponding to all possi-ble partitions of the entire system,a bounded entanglement measure is constructed for arbitrary-dimensional multipartite quantum states.Not only a physical interpretation is provided,but also an upper bound is obtained,by which a necessary condition for the maximally entangled states is given to positively answer a conjecture about judging the maximally entangled states[Phys.Lett.A 2000,273,213].In particular,for three-qubit quantum systems,we prove that our entanglement measure satisfies the relation of monogamy.These results has been published in an international SCI publication[Quantum Inf.Process 2016,15(6),2406-2424].In the fourth chapter,we further prove that our entanglement measure still satisfies the monogamous relation in the n-qubit system.This implies that the satisfaction of the entanglement monogamy is related to the number of particles in the quantum system.In addition,we prove that our measure also satisfies the monogamous relation for three-qutrit system.This result helps us to acquire a condition for the separability of a class of two-qutrit mixed states.Our result,for the first time,finds the existence of entangle-ment monogamy beyond the two-dimensional system.We conjecture that entanglement monogamy exists in every many-body system and this needs further study.These results will be published online in Scientific Reports.The fifth chapter on the entanglement measure of mixed states in three-dimensional system,through a characterization for the extreme maps in the set of positive linear maps on the three-dimensional matrix space,gives a necessary and sufficient condition for the separability of two-qutrit mixed states,based on which an entanglement measure for two-qutrit mixed states is proposed.