| The Yang-Baxter equation (or star-triangle relation, YBE) was introduced tophysics by C.N.Yang (1967) and R.J.Baxter (1971) independently. In1978-79, thequantum inverse scattering method (QISM) was introduced by Faddeev, Sklyanin andTakhajan. In their theory the basic commutation relation of operators is described by asolution of YBE. In the beginning of1980s, the study of YBE has been activelyperformed by Faddeev and Leningrad Scholars. These works led to the idea ofintroducing quantum algebra and Yangian. It turned out that YBE plays a crucial rolein statistical model, many-body problems, quantum integrable model and knot theory,etc.In recent years, the researchers found that Yang-Baxter equation and quantuminformation theory, topological quantum computing are closely related. By means ofunitary solution of Yang-Baxter equation to study quantum entanglement, quantumteleportation, and topological quantum computing has become a hot research topic.These researches greatly enriched the branch field theory of the Yang-Baxter equation.In this dissertation, based on the unitary solution of YBE, braid group representationand T-L algebra, we investigated the representation of T-L algebra, quantumentangled state, quantum gate, and Berry’s phase of Yang-Baxter system.The structure of this dissertation is arranged as follows:Chapter one gives the research background of this dissertation. Firstly, weintroduced the development of quantum computation theory, and some conceptsassociated with quantum computation theory, such as quantum entanglement,quantum gate and Berry’s phase, etc. Secondly, we reviewed the development of YBE,and also reviewed the relation between YBE and quantum computation theory.In Chapter two, by virtue of the Yang-Baxterization approach, we investigatedquantum entanglement properties of the matrix which acts on the tensor productspace V(?) V, such set Yang-Baxter (?)-matrix can be constructed from the T-L algebra.First, we investigated the construction of high dimensional T-L matrix. By means ofour method, a set of high dimensional T-L matrices with topological parameter d=(?) can be constructed. It was to mention that this set of T-L matrices can be viewed as the generalization of "eight-vertex" T-L matrix with topological parameter d=(?). Based on Yang-Baxterization method, we investigated the entanglement properties of the spectrum parameter dependent solution of YBE. The results showed that such unitary solutions of YBE can generate entangled states possess arbitrary entanglement degree, and when we changing the spectral parameter θ, we can obtain a set of entangled states which can be viewed as generalized Bell basis in tensor product space HN(?)HN.In chapter three, we presented a method to construct "X" form unitary solution of YBE, which act on the tensor product space Vj1(?)Vj2(here j1and j2are half-integers). By means of Yang-Baxter transformation, a set of entangled states for (2j1+1)×(2j2+1) dimensional system can be constructed. When j1=3/2and j2=1/2, a few body Hamiltonian with Yangian symmetry was constructed and Yangian generators as shift operators for this Yang-Baxter system are investigated in detail.Chapter four contains two parts of contents. First, we discussed the universality for an "O" form braid matrix in quantum computation. The result shows that such braid matrix local equivalent to the (Double controlled-NOT) DCNOT gate. Second, by means of time-dependent Yang-Baxter transformation, we investigated the Berry’s phase in high dimensional Yang-Baxter system, and the result shows that, the Berry’s phase for high dimensional Yang-Baxter system can be explained under the framework of su(2) theory.In chapter five, based on T-L algebra, we investigated some properties of topological basis associated spin singlet pair. Then we discussed the quantum Zeno effect and quantum tunneling effect of the "eight vertex" model, and we compared this model with the quantum double well model.Finally, we gave the conclusion and prospect for the dissertation. |
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