Quantum entanglement, unitary braid representation and Temperley-Lieb algebra

TL;DR

This paper introduces a new class of braiding operators derived from the Temperley-Lieb algebra that generalize the Bell matrix, enabling direct generation of complex entangled states in multi-qubit systems for topological quantum computing.

Contribution

It presents a novel set of braiding operators from the Temperley-Lieb algebra that unify and extend previous operators, allowing direct creation of multi-qubit entangled states.

Findings

Operators generate important entangled states directly from separable states

Unification of Hadamard and Bell matrices within a single framework

Potential applications in fault-tolerant topological quantum computation

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 {\it 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.