A Realization Of Birman-Wenzl-Murakami Algebra And Corresponding Berry Phase

Very recently, Yang-Baxter equation and braid group theory have been introduced to the field of quantum information and quantum computation. These make the development of Yang-Baxter equation and also provide a novel way to study the quantum information and quantum computation.It is well known that the Yang-Baxter equation plays a fundamental role in the theory （1+1）- or 2- dimensional quantum integrable systems. Then how is the Yang-Baxter equation resolved directly? Because the asymptotic behavior of the （?） -matrix is the braid group representation, a solution （?） -matrix of Yang-Baxter equation can be obtained from a given braid group representation by introducing a spectral parameter. That is Yang-Baxterization.The phase factor of a wave function is the source of all interference phenomena and one of most fundamental concepts in quantum physics. When a Hamiltonian undergoes a cyclic evolution, the quantum states pick up a geometric phase in addition to the dynamical phase. This adiabatic geometric phase is called Berry phase. Anandan and Aharonov generalized Berry’s result to nonadiabatic cyclic evolutions. The last two decades witnessed many fascinating applications of geometric phases in many areas of physics. In recent years, new perspectives on geometric phases were opened in considering their possible applications in quantum computing, motivated by their insensitivity to dynamical detail.In this paper, we study the theory and applications of Yang-Baxter equation, mostly working over Birman-Wenzl-Murakami algebra, Temper-Lieb algebra, braid group representation and Berry phase. This paper is organized as follows:Chapter 1 introduces the theory about Yang-Baxter equation. In Chapter 2, the background of our study and the importance of the investigation are introduced, the general situation of quantification of quantum information and geometric phase.In Chapter 3, we solve the braid group relations, and obtain a 9×9 S -matrix, a solution of the braid relation. A matrix representation of specialized Birman-Wenzl-Murakami algebra is obtained.In Chapter 4, for a given braid group representation S -matrix, we obtain the trigonometric solution of Yang-Baxter equation. A unitary matrix （ ）（?） x ,φ₁ ,φ₂ is generated via the Yang-Baxterization approach. We construct a Yang-Baxter Hamiltonian through the unitary matrix （ ）（?） x ,φ₁ ,φ₂. Berry phase of this Yang-Baxter system is investigated in detail.In Chapter 5, according to E -matrix satisfying the Temper-Lieb algebra, we derive a unitary matrix （?） （ u）, a rational solution of Yang-Baxter equation. Similarly the Hamiltonian can be constructed relating to （?） （ u）-matrix, and Berry phase of this system is analyzed.