Links and Quantum Entanglement

TL;DR

This paper explores the analogy between topological links and quantum entanglement, illustrating how different link structures correspond to inequivalent quantum entangled states, and discusses their unitary representations.

Contribution

It introduces a topological analogy for quantum entanglement using link systems and develops unitary representations of the Braid Group to model these quantum states.

Findings

The Borromean rings correspond to a specific quantum entangled state.

The NUS link represents an inequivalent entanglement structure.

Two quantum systems with different entanglement properties are locally unitarily equivalent.

Abstract

We discuss the analogy between topological entanglement and quantum entanglement, particularly for tripartite quantum systems. We illustrate our approach by first discussing two clearly (topologically) inequivalent systems of three-ring links: The Borromean rings, in which the removal of any one link leaves the remaining two non-linked (or, by analogy, non-entangled); and an inequivalent system (which we call the NUS link) for which the removal of any one link leaves the remaining two linked (or, entangled in our analogy). We introduce unitary representations for the appropriate Braid Group ($B_3$) which produce the related quantum entangled systems. We finally remark that these two quantum systems, which clearly possess inequivalent entanglement properties, are locally unitarily equivalent.