Knots and entanglement

TL;DR

This paper extends the entanglement bootstrap approach to (3+1)-dimensions, studying knotted excitations and fusion processes in topological orders, with implications for quantum information storage.

Contribution

It introduces a generalized framework for analyzing knotted excitations and fusion in higher-dimensional topological orders, including new theorems and models.

Findings

Knot complement of a trefoil can store quantum information

Fusion spaces associated with knots are identified

Spiral maps elucidate consistency relations for torus knots

Abstract

We extend the entanglement bootstrap approach to (3+1)-dimensions. We study knotted excitations of (3+1)-dimensional liquid topological orders and exotic fusion processes of loops. As in previous work in (2+1)-dimensions, we define a variety of superselection sectors and fusion spaces from two axioms on the ground state entanglement entropy. In particular, we identify fusion spaces associated with knots. We generalize the information convex set to a new class of regions called immersed regions, promoting various theorems to this new context. Examples from solvable models are provided; for instance, a concrete calculation of knot multiplicity shows that the knot complement of a trefoil knot can store quantum information. We define spiral maps that allow us to understand consistency relations for torus knots as well as spiral fusions of fluxes.