Braid group, Temperley-Lieb algebra, and quantum information and computation

Abstract

In this paper, we explore algebraic structures and low dimensional topology underlying quantum information and computation. We revisit quantum teleportation from the perspective of the braid group, the symmetric group and the virtual braid group, and propose the braid teleportation, the teleportation swapping and the virtual braid teleportation, respectively. Besides, we present a physical interpretation for the braid teleportation and explain it as a sort of crossed measurement. On the other hand, we propose the extended Temperley-Lieb diagrammatical approach to various topics including quantum teleportation, entanglement swapping, universal quantum computation, quantum information flow, and etc. The extended Temperley-Lieb diagrammatical rules are devised to present a diagrammatical representation for the extended Temperley-Lieb category which is the collection of all the Temperley-Lieb algebras with local unitary transformations. In this approach, various descriptions of quantum teleportation are unified in a diagrammatical sense, universal quantum computation is performed with the help of topological-like features, and quantum information flow is recast in a correct formulation. In other words, we propose the extended TemperleyLieb category as a mathematical framework to describe quantum information and computation involving maximally entangled states and local unitary transformations.

Dual bases in Temperley–Lieb algebras, quantum groups, and a question of Jones

“…The Temperley-Lieb algebras form a very important class of finite-dimensional algebras, arising in a remarkable variety of mathematical and physical contexts including lattice models [TL71], knot theory [KL94], subfactors and planar algebras [JS97], quantum groups [Ban96,Wor87b], and topological quantum computation [Abr08,Zha09,DRW16]. Given a complex number d ∈ C * and a natural number k ∈ N, the kth Temperley-Lieb algebra TL k (d) (with loop parameter d) is a unital finite-dimensional complex associative algebra given by a finite set of generators 1, u 1 , .…”

Quantum teleportation and Birman–Murakami–Wenzl algebra

“…Previous research has extensively studied the relationship between quantum teleportation, the Temperley-Lieb algebra and the Yang-Baxter equation [20,21]. The extended Temperley-Lieb diagrammatical approach [20,21] is devised to characterize topological features of quantum entanglement and quantum teleportation. However, the former research either concentrates on the topic of quantum teleportation using the Yang-Baxter gate or on the topic of quantum teleportation using the Temperley-Lieb projector.…”

“…Since a representation of the BMW algebra is generated by both the Yang-Baxter gate and the Temperley-Lieb projector, there remains a natural question to be answered: what about quantum teleportation using the BMW algebra? In this paper, we investigate this problem and expect to find something novel which is not presented in [20,21].…”

“…Furthermore, we realize that the tangle relations defining the BMW algebra involving both the Temperley-Lieb projector and the Yang-Baxter gate give rise to the teleportation protocol directly. Moreover, we are able to construct interesting representations of the BMW algebra in the extended Temperley-Lieb diagrammatical approach [20,21] different from the representation in [22,23].…”

“…Four appendices are added to complete the paper. Appendix A reviews a relation between the Brauer algebra and quantum teleportation [20], since the BMW algebra is a deformation of the Brauer algebra [26]. Appendix B collects the extended Temperley-Lieb configurations of the Yang-Baxter gate which are not in the main context of the paper.…”