Entanglement Criteria And Entanglement Measurements For States In Multipartite Composite Quantum Systems

Quantum information and quantum computation is a newly emerging and fast developing cross subject in recent decade years, since it has tremendous application value and great scientific significance, this subject has attracted more and more noticed by physicists、computer scientists、mathematicians and many experts and scholars of other fields. Quantum entanglement is an important physical resource of quantum in-formation theory, how to detect entanglement and quantize entanglement amount for states in bipartite and multipartite quantum systems is a very important subject. In finite-dimensional cases, there are many valuable results about detecting entanglement and quantizing entanglement amount. But it is very difficult to detect separability and quantize entanglement amount for states in infinite-dimensional composite quantum systems, there is only a few results about these fields. In addition, the partial separa-bility for finite-dimensional or infinite-dimensional multipartite quantum states is just beginning to be developed, in particular, there is only a few results about k-separability and entanglement measurement relative to k-partition for multipartite quantum states.This doctoral thesis mainly study entanglement criteria and entanglement mea-surements for states in bipartite or multipartite quantum systems. The main research contents are as follows:1. We give a trace inequality criterion for states in infinite-dimensional bipartite and multipartite quantum systems, and make a comparison with reduction criterion; We give two class reduction criteria for states in infinite-dimensional multipartite quantum systems; For the case of2(?)∞and N(?)∞, we obtain two entanglement criteria, which generalized the results of finite-dimensional systems; We obtain some necessary conditions for fully separable multipartite quantum states based on local orthogonal observables.2. We study the k-separability for multipartite quantum states, and obtain some equivalent conditions for k-separable pure states in finite-or infinite-dimensional multi-partite quantum systems. Furthermore, some necessary conditions are presented about k-separable multipartite quantum states in finite-dimensional quantum systems.3. We introduce two new entanglement measurements,k-ME EoF and k-ME Neg-ativity with respect to k-partition in finite-dimensional multipartite quantum systems, and prove these measurements satisfy some necessary properties, such as, its value is zero for k-separable states, and it is invariant under local unitary transformations, and its entanglement amount cannot increase under local operations and classical commu-nication. In addition, we obtain a lower bound about k-ME EoF and k-ME Negativity, respectively.4. We generalized the geometry entanglement measure to infinite dimensional multipartite quantum systems. And then, we study the k-ME Concurrence measure, give a upper bound about k-ME Concurrence measure in finite-dimensional multipartite quantum systems.5.For multipartite quantum states, we introduce||·||γ(k), and define an entanglement measure for multipartite quantum states with respect to k-partition, and prove that it satisfies some basic properties of entanglement measure.