Algebraic Structures Underlying Quantum Information Protocols

In this paper, we describe the teleportation from the viewpoints of the braid group and Temperley–Lieb algebra. We propose the virtual braid teleportation which exploits the teleportation swapping and identifies unitary braid representations with universal quantum gates, and suggest the braid teleportation which is explained in terms of the crossed measurement and the state model of knot theory. In view of the diagrammatic representation for the Temperley– Lieb algebra, we devise diagrammatic rules for an algebraic expression and apply them to various topics around the teleportation: transfer operator and acausality problem; teleportation and measurement; all tight teleportation and dense coding schemes; the Temperley–Lieb algebra and maximally entangled states; entanglement swapping; teleportation and topological quantum computing; teleportation and the Brauer algebra; multipartite entanglements. All examples clearly suggest the Temperley–Lieb algebra under local unitary transformations to be the algebraic structure underlying the teleportation. We show the teleportation configuration to be a fundamental element in the diagrammatic representation for the Brauer algebra and propose a new equivalent approach to the teleportation in terms of the swap gate and Bell measurement. To explain our diagrammatic rules to be a natural diagrammatic language for the teleportation, we compare it with the other two known approaches to the quantum information flow: the teleportation topology and strongly compact closed category, and make essential differences among them as clear as possible.