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

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 $\theta$ 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.