Jones index, secret sharing and total quantum dimension

TL;DR

This paper explores how the total quantum dimension in topologically ordered systems quantifies the capacity for secret sharing, using advanced mathematical tools like subfactor index theory to provide a rigorous framework.

Contribution

It introduces a rigorous mathematical approach to relate total quantum dimension with secret sharing capabilities in topological phases, extending previous heuristic methods.

Findings

Total quantum dimension quantifies secret sharing capacity.

Mathematical rigor achieved via subfactor index theory.

Relation to topological entanglement entropy discussed.

Abstract

We study the total quantum dimension in the thermodynamic limit of topologically ordered systems. In particular, using the anyons (or superselection sectors) of such models, we define a secret sharing scheme, storing information invisible to a malicious party, and argue that the total quantum dimension quantifies how well we can perform this task. We then argue that this can be made mathematically rigorous using the index theory of subfactors, originally due to Jones and later extended by Kosaki and Longo. This theory provides us with a "relative entropy" of two von Neumann algebras and a quantum channel, and we argue how these can be used to quantify how much classical information two parties can hide form an adversary. We also review the total quantum dimension in finite systems, in particular how it relates to topological entanglement entropy. It is known that the latter also has an interpretation in terms of secret sharing schemes, although this is shown by completely different methods from ours. Our work provides a different and independent take on this, which at the same time is completely mathematically rigorous. This complementary point of view might be beneficial, for example, when studying the stability of the total quantum dimension when the system is perturbed.