Yang-Baxter Equation, Quantum Entanglement And Majorana Fermions

Quantum entanglement plays an important role in quantum information and computation, and there are various descriptions about quantum entanglement. Since2004, Kauffman and Lomonaco first pointed out that braiding operators are universal quantum gates, then they presented the relationship between topologi-cal entanglement and quantum entanglement. The4-D representation of braiding operator is closely related to2-qubit Bell basis that is the maximal entangled2-qubit pure state. The generalization of braiding relation-Yang-Baxter equa-tion’s relation to quantum entanglement also attracts much attention. Based on one YBE solution R12(θ,φ), which is known as2body S-matrix, some issues have been discussed, such as generating2-qubit pure entangled state, the rela-tionship among the Hamiltonian, Berry phase and entanglement, the application of Temperley-Lieb algebra(relates to YBE) in generating N-qubit GHZ states, and the application of l1-Norm’s extreme value in quantum information and so on.In this thesis, with respect to3body S-matrix constrained by YBE, we focus on two parts. The first part is about the relation between3body S-matrix and3-qubit entanglement, the second part is the discussion about the Hamiltonian derived from S-matrix. In Chapter1, we review the braiding operator and2-qubit entanglement’s relation. In Chapter2, we give the explicit form of3body S-matrix R123(η,β,φ) and act it on3-qubit natural basis. In Chapter3, based on2-body S-matrix represented by Spin-1/2lattice, the Hamiltonian and Berry phase derived from the3-body S-matrix is presented and discussed. In Chapter4, we construct the chain model and compare it with1D Kitaev toy model. In Chapter5, we give examples to show the parameter-dependence of chain model in entanglement transfer. In Chapter6, the solution to YBE, which is represented by Majorana fermions that satisfy Clifford algebra, is presented. We find that the corresponding Hamiltonian possess topological phase, which is exactly ID Kitaev model; Based on Clifford algebra, we construct the4n-D(n odd) matrix representation of solution to YBE. Acting Rin(θ=π/4) on2n-qubit natural basis, we generate2n-qubit GHZ state. This result can be regarded as the generalization of the Bell state in Kauffman’s paper, Conclusion and discussion is made in the last chapter.