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TMF, 2014 Volume 181, Number 1, Pages 19–38 (Mi tmf8723)

This article is cited in 2 papers

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

Mo-Lin Gea, Li-Wei Yua, Kang Xueb, Qing Zhaoc

a Chern Institute of Mathematics, Nankai University, Tianjin, China
b Department of Physics, Northeast Normal University, Changchun, China
c Physics College, Beijing Institute of Technology, Beijing, China

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 $\theta$ 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.

Keywords: Yang–Baxter equation, quantum entanglement, topological quantum field theory, Wigner function $D$, Kitaev's toy model.

Received: 01.06.2014

DOI: 10.4213/tmf8723


 English version:
Theoretical and Mathematical Physics, 2014, 181:1, 1145–1163

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© Steklov Math. Inst. of RAS, 2025