Topological Aspects of Quantum Entanglement

Abstract

Kauffman and Lomonaco explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In the work of G. Alagic, M. Jarret, and S. Jordan it is shown that entanglement is a necessary condition for forming invariants of knots from braid closures via solutions to the Yang-Baxter Equation. We show that the arguments used by these authors generalize to essentially the same results for quantum invariant state summation models of knots. We also given an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing.The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement, and the ER = EPR hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space, that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.