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International Journal of Modern Physics B

YANG–BAXTER EQUATIONS IN QUANTUM INFORMATION

Mo‐Lin Ge, Kang Xue

Published Sep 18, 2012• 13 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979212430072

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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.
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.
The solutions of Yang–Baxter equation (YBE) associated with quantum information have been reviewed with new progress. The additivity rule for two particles is shown to obey Lorentzian type other than the Gallilean. Acting the braiding operations on the topological basis, it turns out that the [Formula: see text]-dimensional representations obeying YBE are nothing but the Wigner’s [Formula: see text]-functions. The connection between the extremization of [Formula: see text]-norm as well as of von Neumann entropy and the reduction of YBE to braid relation is also discussed.
The topological basis associated with Birman-Wenzl-Murakami algebra (BWMA) is constructed and the three dimensional forms of braiding matrices S have been found for both S+=SS^+=SS+=S and S+=S−1S^+=S^{-1}S+=S−1. A familiar spin-1 model related to braiding matrix associated with BWMA is discussed. The extreme points (θ=±π/2(θ=\pmπ/2(θ=±π/2 and ±π)\pmπ)±π) of L1-norm and von Neumann entropy are shown to be connected to each other. Through the general discussion and examples we then point out that the L1-norm describes quantum entanglement.
Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.
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.
Recent study suggests that there are natural connections between quantum information theory and the Yang--Baxter equation. In this paper, in terms of the generalized almost-complex structure and with the help of its algebra, we define the generalized Bell matrix to yield all the GHZ states from the product base, prove it to form a unitary braid representation and present a new type of solution of the quantum Yang--Baxter equation. We also study Yang-Baxterization, Hamiltonian, projectors, diagonalization, noncommutative geometry, quantum algebra and FRT dual algebra associated with this generalized Bell matrix.