# Kitaev's quantum double model as an error correcting code

## TL;DR

Kitaev's quantum double models are proven to function as quantum error-correcting codes for any finite group, with detailed analysis of their local properties and topological entanglement entropy.

## Contribution

The paper provides an explicit proof that Kitaev's quantum double models serve as quantum error-correcting codes for arbitrary finite groups and clarifies the nature of their local properties and entanglement entropy.

## Key findings

- Any two zero-energy density states in a contractible region have identical reduced states.
- Local properties are determined by trivial holonomies, not all fluxes.
- Topological entanglement entropy is independent of the inclusion of the 'log dim R' term.

## Abstract

Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes -- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the "log dim R" term is included in the definition of entanglement entropy.