Quantum entanglement, unitary braid representation and Temperley-Lieb algebra

¹ Department of Physics, and Center for Quantum Technologies, National University of Singapore 117543, Singapore
² Department of Physics, Tamkang University - Tamsui 251, Taiwan, ROC a
³ Department of Physics and Astronomy, The Open University - Walton Hall, Milton Keynes MK7 6AA, UK
⁴ LPTMC, Université Pierre et Marie Curie - 75252 Paris Cedex 05, France

Received: 3 June 2010
Accepted: 14 October 2010

Abstract

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.