The Aplication Of The Yang-Baxter Equation In The Quantum Information Theory

Along with quantum information and quantum computation fast development, the quantum information theory has attracted more and more attention as an important physical resource. While Yang-Baxter Equation (YBE) and the related mathematics physical property is one of the front branches in the area of theoretical physics and mathematics physics research in the last few years, It contains quite rich physical content, in recent years, YBE as well as braid group have already been applied into the domain of quantum information and quantum computation successfully. In this thesis, the quantum entanglement, the geometry, the quantum transition and the quantum teleportation in the Yang-Baxter system have been discussed respectively. These discussions lead YBE and braid group into more widespread domain of quantum information theory, so it has great significance to deeperly explore the function and value in the quantum information theory. The dissertation consists of seven chapters, and the main contents of our work are given from Chapters 3 to 6.In Chapter 1 and Chapter 2, the background of our study and the importance of the investigation are introduced, the general advancement and development of the qu- antum information theory, and the situation of the research of YBE in the quantum in- formation theory are briefly described. The form of the YBE, its algebra form: braid group algebra, Tempely-Lieb algebra and two-groups algebra are described in detail. In Chapter 3, we investigated the quantum entanglement and the geometry in the Yang-Baxter system. First, we construct a new8×8M′matrix from the4×4M matrix, where M′/Mare the image of the braid group representation. The8×8M matrix and the4×4M both satisfy extraspecial 2-group algebra relations. By Yang–Baxteration approach, we can derive a unitaryR( matrix from the matrixM′. Three-qubit entangled states can be generated by using theR( matrix. A Hamiltonian for three qubits is cons- tructed from the unitaryR( matrix. We then study the entanglement and Berry phase of the Yang–Baxter system. Next, we present the reducible representation of the n 2braid group representation which is constructed on the tensor product of n-dimensional spa- ces. Specifically, it is shown that via a combining method, we can construct more n 2 dimensional braiding S-matrices which satisfy the braid relations. we derive a 9×9 unitaryR( matrix according to a 9×9braiding S-matrix we have constructed by Yang- Baxterization approach,. The entanglement property ofR( matrix is investigated, and the arbitrary degree of entanglement for two-qutrit entangled states can be generated via R( matrix acting on the standard basis.In Chapter 4, we investigated the sudden death of entanglement in Yang-Baxter systems. First, some Hamiltonians for two qubits are constructed from the unitary R( (θ,φ)matrices, whereθandφare time-independent parameters. We show that the entanglement sudden death (ESD) can happen in these closed Yang–Baxter systems. It is found that the ESD is not only sensitive to the initial condition, but also has a great connection with different Yang–Baxter systems. Especially, we find that the meaningful parameterφhas a great influence on ESD. Next, we derive the unitary Yang- Baxter R( (θ,φ)matrices from the8×8M′matrix and the4×4M matrix by Yang- Baxteration approach. In Yang-Baxter systems, we explore the evolution of tripartite negativity for three qubits Greenberger-Horne-Zeilinger (GHZ)-type states and W-type states and investigate the evolution of the bipartite concurrence for two qubits Bell-type states. We show that tripartite entanglement sudden death (ESD) and bipartite ESD all can happen in Yang-Baxter systems and find that ESD all are sens- itive to the initial condition. We find that in the Yang-Baxter system, GHZ-type states can have bipartite entanglement and bipartite ESD, and find in some initial conditions, W-type states have tripartite ESD while they have no bipartite Entanglement. It is wo- rth noting that the meaningful parameter ’ has great influence on bipartite ESD for two qubits Bell-type states in the Yang-Baxter system.In Chapter 5, two-body interacting Hamiltonians are constructed from the unitary (φ)matrices and we get Hamiltonians of 4 qubits by combination. Through stu- dying the second derivative of the ground-state (GS) energy and the energy gap betw- een the ground state and the first excited state we can see that quantum phase transiti- ons can happen in these closed Yang-Baxter systems. For the ground state of one of the Yang-Baxter systems, we also investigate the critical behavior of nearest neighb- oring entanglement and Berry phase in parameters space. It shows that GS nearest ne- ighboring entanglement and Berry phase are good indicators to QPTs in the combined Yang-Baxter system.In Chapter 6, by means of Temperley-Lieb Algebra(TLA) and Topological basis, we make a new realization of topological basis, and get sixteen orthogonal complete topological basis which are all maximally entangled states for four quasi-particles.Then we present an explicit protocol for teleporting an arbitrary two-qubit state via atopological basis entanglement channel.Finally, the conclusions and discussions are presented.