Teleportation, Braid Group and Temperley--Lieb Algebra

Abstract

We explore algebraic and topological structures underlying the quantum teleportation phenomena by applying the braid group and Temperley--Lieb algebra. We realize the braid teleportation configuration, teleportation swapping and virtual braid representation in the standard description of the teleportation. We devise diagrammatic rules for quantum circuits involving maximally entangled states and apply them to three sorts of descriptions of the teleportation: the transfer operator, quantum measurements and characteristic equations, and further propose the Temperley--Lieb algebra under local unitary transformations to be a mathematical structure underlying the teleportation. We compare our diagrammatical approach with two known recipes to the quantum information flow: the teleportation topology and strongly compact closed category, in order to explain our diagrammatic rules to be a natural diagrammatic language for the teleportation.

33 pages, 19 figures, latex. The present article is a short version of the preprint, quant-ph/0601050, which includes details of calculation, more topics such as topological diagrammatical operations and entanglement swapping, and calls the Temperley--Lieb category for the collection of all the Temperley--Lieb algebra with physical operations like local unitary transformations

References in corpus (22)

• Quantum Mechanics helps in searching for a needle in a haystack
• Quantum cryptography: Public key distribution and coin tossing
• Quantum Teleportation is a Universal Computational Primitive
• Teleportation of Quantum States
• All Teleportation and Dense Coding Schemes
• Braiding Operators are Universal Quantum Gates
• Comparing Quantum Entanglement and Topological Entanglement
• Instantaneous measurement of non-local variables
• Abstract Physical Traces
• Teleportation Topology
• Introducing categories to the practicing physicist
• Quantum Knots
• Entanglement Criteria - Quantum and Topological
• The logic of entanglement
• Atemporal diagrams for quantum circuits
• Quantum information-flow, concretely, and axiomatically
• Unitary Solutions to the Yang-Baxter Equation in Dimension Four
• Physical Traces: Quantum vs. Classical Information Processing
• Virtual Extension of Temperley--Lieb Algebra
• Teleportation from a Projection Operator Point of View
• Permutation and Its Partial Transpose
• Extraspecial 2-groups and images of braid group representations

Cited by in corpus (11)

• Temperley-Lieb Algebra, Yang-Baxterization and Universal Gate
• Quasitriangular Hopf algebras, braid groups and quantum entanglement
• Tensor Universality, Quantum Information Flow, Coecke's Theorem, and Generalizations
• Quantum teleportation in the commuting operator framework
• Birman-Wenzl-Murakami Algebra and the Topological Basis
• Geometric representations of braid and Yang-Baxter gates
• Combinatorial properties of Temperley Lieb algebras
• A general formulation based on algebraic spinors for the quantum computation
• GHZ transform (I): Bell transform and quantum teleportation
• A study on the relations between the topological parameter and entanglement
• Quantum Algebras Associated With Bell States