Theoretical Study On The Geometric Phase And Quantum Entanglement In Quantum Systems

With the development of quantum information(QI) theory, quantum entangle-ment and geometric phase have been studied extensively because of their wideapplications in QI. In this thesis, the quantum geometric phase and quantumentanglement in some special quantum systems are discussed respectively. Thisdiscussion leads to some interesting results, which are helpful for the under-standing of the physical implications of quantum geometric phase and quantumentanglement.This paper consists of two parts. The first part is composed of Chaps. 2-5,which concentrate on the geometric phase in bipartite systems and many-bodysystems. The foundations for this part is presented in Chap. 2, which providesthe basic definition of geometric phase and its generalization. Especially we pro-vide a detailed investigation on the mixed-state geometric phase. In Chap. 3and 4 the diagonal and off-diagonal geometric phase respectively in bipartitesystems have been discussed and the effect of quantum entanglement has beenalso emphasized, which comes from one of the author's works. One can findthat because of quantum entanglement, geometric phase without the interactionbetween particles may be singular under some conditions, and at the other caseit is zero for Werner state or maximally entangled state. On the other hand, thebehavior of geometric phase can be explained properly by the entanglement whenintra-coupling is appearing. Especially it shows in Chap. 4 that quantum en-tanglement prohibits the appearance of singularity in geometric phase, for whichthe geometric phase is undefined and one has to define off-diagonal geometricphase, a novel connection have studied in Chap. 5 between the quantum phasetransition and geometric phase of the ground state in many-body systems, basedon the author's work. Given the study of geometric phase of ground state in1D spin-1/2 XY model and Lipkin-Meshkov-Glick model, it points out that the singularity in geometric phase around the critical point is originated from thequantum degeneracy. Furthermore the twist operator is introduced in two-bandmodel, and the general relation between geometric phase of ground state andcriticality is constructed. The possibility of the geometric phase of the groundstate as the universal order parameter is also discussedChap. 6 by itself consists of the second part of this thesis, which provides adetailed discussion of the so-called finite-time disentanglement in bipartite sys-tems first introduced by Yu and Eberly. The results show that this phenomenonis due to the energy transfer between the system and its surroundings or dueto the internal energy transfer. Meanwhile, one can also find the intimate cor-relation between the dynamics of entanglement and the energy transfer in thesystems. Finally the conclusions and discussions are presented.