Notes on Bell states and quantum teleportation

TL;DR

This work develops a unified algebraic and topological framework for Bell states and quantum teleportation, introducing the basis theorem and basis group to characterize when an extended Bell basis remains orthonormal under local actions, and defining the twist operator to connect two-qudit Bell states with multipartite Bell states. It leverages the Temperley–Lieb algebra, braid group relations, and Yang–Baxter equations to provide a diagrammatic/topological description of teleportation and entanglement, including explicit teleportation equations for single and multi-qudit/multi-qubit scenarios and a multipartite concurrence measure. The paper demonstrates that generalized Bell states admit complete sets of observables, clear circuit representations, and topological descriptions that illuminate the entanglement structure and measurement-based teleportation, with potential implications for quantum networks, integrable quantum computation, and topological quantum information processing. Overall, it establishes a deep link between quantum information protocols and low-dimensional topology, highlighting how SWAP, twist, and Yang–Baxter gates generate rich families of entangled states and teleportation processes that can be analyzed diagrammatically. The results offer a foundation for further exploration of generalized Pauli bases, multipartite entanglement measures, and diagrammatic methods in quantum computation and communication.

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.

Figures (21)

• Figure 1: Quantum circuit to create a generalized $2n$-qubit Bell state $|\mathcal{B}_{2n}(\underline{\alpha\beta})\rangle$.
• Figure 2: Rules to draw operators, state vectors and inner products.
• Figure 3: Rules to draw the transfer operator.
• Figure 4: Rules to draw generalized two qudit Bell states.
• Figure 5: Various combinations of cup and cap states.