If there are N distinct fusion channels in the presence of a pair of particles, the system exhibits N-fold
degeneracy spanned by these states. We refer to this non-local space shared by the non-Abelian
anyons, regardless of where they are located, as the fusion space. Under the assumption that all
microscopics of the system giving rise to the anyons are decoupled from the low-energy physics, the
states in the fusion space are perfectly degenerate. As it is a collective non-local property of the
anyons, no local perturbation can act on it and it is hence a decoherence-free subspace. As such it is
an ideal place to non-locally encode quantum information. We stress that the fusion space arises from
the distinct ways anyons can be fused over how they are fused. If two anyons are actually fused and the
outcome of the fusion is detected, this would correspond to performing a projective measurement in the
"Introduction to Topological Quantum Computation"
Leveraging the stories you can tell about the particles in your system as a computational substrate built out of nothing but mathematics and a delicately engineered suspension of both belief and disbelief is something people are working towards building primitive prototypes of, right now, as I type this.
What an absurdly beautiful universe.