Quantum entanglement, unitary braid representation and Temperley-Lieb algebra

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. Copyright c © EPLA, 2010 Introduction. – Recent developments in fault-tolerant quantum computation using the braiding of anyons [1], have stimulated interest in applying the theory of braid groups to the fields of quantum information and quantum computation. In this respect, an interesting result is the realization that a specific braiding operator is a universal gate for quantum computing in the presence of local unitary transformations [2]. This operator involves a unitary matrix R that generates the four maximally entangled Bell states from the standard basis of separable states. This has led to further investigation on the possibility of generating other entangled states by appropriate braiding operators [3–5]. In [4], unitary braiding operators were used to realize entanglement swapping and generate the Greenberger-Horne-Zeilinger (GHZ) state [6], as well as the linear cluster states [7]. Further generalizations of the braiding operators to bipartite quantum systems with states of arbitrary dimension, i.e., qudits, were obtained by the approach of Yang-Baxterization [8,9]. (a)Permanent address. The GHZ state was not directly generated by the braiding operator in [4]. The resulting state was transformed, by use of a local unitary transformation, to the GHZ state. We argue here that this state does not, in fact, possess the same entanglement properties as the GHZ state. In this note we show how the Bell states, the generalized GHZ states and some cluster-like states may be generated directly from a braiding operator. We adopt a different approach, based on the Temperley-Lieb algebra (TLA) [10], to obtain a class of unitary representations of the braid group, and with it the required braiding operator. We first obtain an explicit representation of the TLA, and then find the braid group representation via the Jones representation [11]. Braid group and quantum entanglement. – The m-stranded braid group Bm is generated by a set of elements {b1, b2, . . . , bm−1} with defining relations: bibj = bjbi, |i− j|> 1; bibi+1bi = bi+1bibi+1, 1 i <m. (1) Quantum computing requires that quantum gates be represented by unitary operators. Thus, for applications of