Notes on Bell states and quantum teleportation

TL;DR

This paper explores the algebraic and topological properties of Bell states and quantum teleportation, providing new insights into their structure and topological features in quantum information science.

Contribution

It introduces a basis theorem, basis group, and twist operator, and employs algebraic and topological tools to analyze generalized Bell states and teleportation.

Findings

Extension of Bell basis remains orthonormal

Topological diagrammatic description of teleportation

Topological nature of quantum entanglement

Abstract

Bell states and quantum teleportation play important roles in the study of quantum information and computation. But a comprehensive theoretical research on both of them remains to be performed. This work aims to investigate important algebraic properties of generalized Bell states as well as explore topological features of quantum teleportation. First, the basis theorem and basis group are introduced to explain that the extension of a generalized Bell basis by a unitary matrix is still an orthonormal basis. Then a twist operator is defined to make a connection between a generalized multiple qubit Bell state and a tensor product of two qubit Bell state. Besides them, the Temperley--Lieb algebra, the braid group relation and the Yang--Baxter equation are used to provide a topological diagrammatic description of generalized Bell states and quantum teleportation. It turns out that our approach is able to present a clear illustration of relevant quantum information protocols and exhibit a topological nature of quantum entanglement and quantum teleportation.