Quantum Entanglement And Its Applications In Strongly Correlated Systems

Quantum entanglement plays a very important role in the quantum informa-tion theory. Many quantum processing cannot occur without entanglement. In a sense, quantum entanglement is a kind of resource. In this dissertation, the properties of entanglement and its application in condensed matter physics are studied.In chapter 1, a brief review of the quantum information theory is given. We discuss the concepts and some properties of pure state and mixed state respec-tively. In particular, we discuss how to detect and measure the entanglement. We give a brief introduction to the entanglement of formation, entanglement of distillation and the relative entropy of entanglement. Then we discuss some kinds feasible measures, including von Neumann entropy, linear entropy, Neg-ativity and so on. The quantum channel is then studied. In the last section of the chapter, we discuss the some applications for quantum entanglement, including dense coding, teleportation and cryptography.In chapter 2, we discuss the entanglement in strongly correlated systems. The pairwise entanglement of XXZ model is studied. Much attention is devoted to study the properties in the vicinity of quantum phase transition points. From the viewpoint of conformal field theory, we discuss the relationship between entanglement entropy and central charges, the length of rings. We then discuss the information loss along the renormalization group trajectory.In chapter 3, basing on the Lie algebra, we extend the notion of concurrence introduced by Hill and Wootters to high dimension Hilbert space. We call this generalized quantity "concurrence vector". For pure state, we can use the norm of concurrence vector to measure the entanglement of the state. We also try to extend the concurrence vector to study mixed state. In the chapter, we calculate the entanglement of 3×3 bipartite system as an example. We introduce the concept of "concurrence edge" and discuss the existence condition. In chapter 4, we study a composite bipartite system which contains spin-1 and spin-1/2. We discuss the ground-state entanglement when the system is degenerate. We introduced one notion "average entanglement" to measure the degenerate ground state.In chapter 5, we study the entanglement in Bose-Einstein Condensate, espe-cially the Hydrogen atom and Lithium cold atom gases. We calculate the en-tanglement of the system at zero temperature and finite temperature. At very low temperature, one can manipulate the entanglement of the atom through adjusting the external fields.In chapter 6, we construct a system which carries out SU(3) representation. The system is interacted with external fields. We give the entanglement as some parameters are varied. We discuss the physical significance when the entanglement approaches its most maximally entangled. We also discuss the three-level Lipkin-Meshkov-Glick Model and spin-1 correlated spin.