Solutions of the Yang-Baxter equation associated with a topological basis and applications in quantum information

Abstract

We discuss a new type of solutions of the Yang-Baxter equation, called type-II solutions. They are related to quantum entanglements. The action of the corresponding braiding operator on the topological basis associated with a topological quantum field theory generates a (2J+1)-dimensional matrix form of the R-matrix for spin J, i.e., the Wigner function D with the spectral parameter θ denoting the entanglement degree. We present concrete examples for J = 1/2 and J = 1 in an explicit form. We show that the Hamiltonian related to the type-II R-matrix is Kitaev’s toy model.