# Jones index, secret sharing and total quantum dimension

August 08, 2016

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.

Keywords:

*none*