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arXiv:0904.0090

The representations of Temperley-Lieb algebras and entanglement in a Yang-Baxter system

Chunfang Sun, Gangcheng Wang, Hu, Taotao, Zhou, Chengcheng, Qingyong Wang, Kang Xue

Published Apr 1, 2009• 3 citations
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Abstract

A method of constructing Temperley-Lieb algebras(TLA) representations has been introduced in [Xue \emph{et.al} arXiv:0903.3711]. Using this method, we can obtain another series of $n^{2}\times n^{2}$ matrices $U$ which satisfy the TLA with the single loop $d=\sqrt{n}$. Specifically, we present a $9\times9$ matrix $U$ with $d=\sqrt{3}$. Via Yang-Baxterization approach, we obtain a unitary $ \breve{R}(θ,φ_{1},φ_{2})$-matrix, a solution of the Yang-Baxter Equation. This $9\times9$ Yang-Baxter matrix is universal for quantum computing.

DOI
10.48550/arxiv.0904.0090

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A method of constructing Temperley-Lieb algebras(TLA) representations has been introduced in [Xue \emph{et.al} arXiv:0903.3711]. Using this method, we can obtain another series of n2×n2n^{2}\times n^{2}n2×n2 matrices UUU which satisfy the TLA with the single loop d=nd=\sqrt{n}d=n​. Specifically, we present a 9×99\times99×9 matrix UUU with d=3d=\sqrt{3}d=3​. Via Yang-Baxterization approach, we obtain a unitary R˘(θ,φ1,φ2) \breve{R}(θ,φ_{1},φ_{2})R˘(θ,φ1​,φ2​)-matrix, a solution of the Yang-Baxter Equation. This 9×99\times99×9 Yang-Baxter matrix is universal for quantum computing.
A method of constructing Temperley-Lieb algebra (TLA) representations has been introduced in [Xue et al. arXiv:0903.3711]. Using this method, we can obtain another series of n2 × n2 matrices U which satisfy the TLA with the single loop [Formula: see text]. Specifically, we present a 9 × 9 matrix U with [Formula: see text]. Via Yang-Baxterization approach, we obtain a unitary [Formula: see text]-matrix, a solution of the Yang-Baxter equation. This 9 × 9 Yang-Baxter matrix is universal for quantum computing.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly , from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly from separable basis states important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
This work concerns studies of boundary integrability of the vertex models from representations of the Temperley–Lieb algebra associated with the quantum group 𝒰 q [ X n ] for the affine Lie algebras X n = , , and . A systematic computation method is used to construct solutions of the boundary Yang–Baxter equations. We find a 2 n 2 + 1 free parameter solution for spin-( n − 1/2) and vertex models. It turns out that for spin- n , and vertex models, the solution has 2 n 2 + 2 n + 1 free parameters.
This work concerns to the studies of boundary integrability of the vertex models from representations of the Temperley-Lieb algebra associated with the quantum group Uq[Xn]{\cal U}_{q}[X_{n}]Uq​[Xn​] for the affine Lie algebras XnX_{n}Xn​ = A1(1)A_{1}^{(1)}A1(1)​, Bn(1)B_{n}^{(1)}Bn(1)​, Cn(1)C_{n}^{(1)}Cn(1)​ and Dn(1)D_{n}^{(1)}Dn(1)​. A systematic computation method is used to constructed solutions of the boundary Yang-Baxter equations. We find a 2n2+12n^{2}+12n2+1 free parameter solution for $A_{1}^{(1)} $ spin-(n−1/2)(n-1/2)(n−1/2) and $ C_{n}^{(1)}$ vertex models. It turns that for $A_{1}^{(1)} $ spin-nnn, $ B_{n}^{(1)}$ and Dn(1)D_{n}^{(1)}Dn(1)​ vertex models, the solution has 2n2+2n+12n^{2}+2n+12n2+2n+1 free parameters.