Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Quantum Information

THE REPRESENTATIONS OF TEMPERLEY-LIEB ALGEBRA AND ENTANGLEMENT IN A YANG-BAXTER SYSTEM

Chunfang Sun, Gangcheng Wang, Taotao Hu, CHENGCHENG ZHOU, Qingyong Wang, Kang Xue

Published Sep 1, 2009• 18 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749909005614

Related papers

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

Chunfang Sun, Gangcheng Wang, Hu, Taotao +3 more

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.

arXiv:0904.0090•Apr 1, 2009•3 citations

A 9 × 9 matrix representation of Temperley-Lieb algebra and corresponding entanglement

Lidan Gou, Ruihan Zhu

We present the matrix U . The 9 × 9 U matrix is a representation of specialized Temperley—Lieb algebra. Based on which, a unitary Ř matrix is generated via the Yang—Baxterization approach. The 9 × 9 Ř matrix, a solution of the Yang—Baxter equation, is universal for quantum entanglement.

Chinese Physics B•Feb 1, 2012•1 citation

METHOD OF CONSTRUCTING BRAID GROUP REPRESENTATION AND ENTANGLEMENT IN A 9 × 9 YANG–BAXTER SYSTEM

TAOTAO HU, GANGCHENG WANG, CHUNFANG SUN +3 more

In this paper, we present reducible representation of the n2 braid group representation which is constructed on the tensor product of n-dimensional spaces. Specifically, it is shown that via a combining method, we can construct more n2 dimensional braiding S-matrices which satisfy the braid relations. By Yang–Baxterization approach, we derive a 9 × 9 unitary [Formula: see text]-matrix according to a 9 × 9 braiding S-matrix we have constructed. The entanglement properties of [Formula: see text]-matrix is investigated, and the arbitrary degree of entanglement for two-qutrit entangled states can be generated via [Formula: see text]-matrix acting on the standard basis.

arXiv:0903.5230•Reviews in Mathematical Physics•Oct 1, 2009•2 citations

Quantum entanglement, unitary braid representation and Temperley-Lieb algebra

Ho, C.-L., A. I. Solomon, Oh, C.-H.

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.

•Nov 1, 2010•16 citations

Quantum entanglement, unitary braid representation and Temperley-Lieb algebra

C.-L. Ho, A. I. Solomon, C.-H. Oh

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.

arXiv:1011.6229•Europhysics Letters (EPL)•Nov 1, 2010•11 citations

YANG–BAXTER EQUATIONS IN QUANTUM INFORMATION

Mo‐Lin Ge, Kang Xue

The connection between Yang–Baxter system and quantum information has been discussed. Based on the topological basis for both Temperley–Lieb (TL) algebra and Birman–Wenzl (BW) algebra the representations of N by N braiding matrices associated with the corresponding N2 by N2 ones are obtained. Some of physical consequences of the braiding matrices connecting with quantum information are shown.

International Journal of Modern Physics B•Sep 18, 2012•13 citations
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.
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 $ \breve{R}(θ,φ_{1},φ_{2})$-matrix, a solution of the Yang-Baxter Equation. This 9×99\times99×9 Yang-Baxter matrix is universal for quantum computing.
In this paper, we present reducible representation of the n2 braid group representation which is constructed on the tensor product of n-dimensional spaces. Specifically, it is shown that via a combining method, we can construct more n2 dimensional braiding S-matrices which satisfy the braid relations. By Yang–Baxterization approach, we derive a 9 × 9 unitary [Formula: see text]-matrix according to a 9 × 9 braiding S-matrix we have constructed. The entanglement properties of [Formula: see text]-matrix is investigated, and the arbitrary degree of entanglement for two-qutrit entangled states can be generated via [Formula: see text]-matrix acting on the standard basis.
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.