How to embed any anyon symmetry into simple quantum systems

Anyons are exotic quasiparticles whose exchange statistics define fusion categories—algebraic structures encoding quantum symmetries. The authors prove that any such symmetry can be realized on a tensor product of infinite-dimensional Hilbert spaces (like harmonic oscillators), and that all realizations of the same symmetry are equivalent up to local rearrangement. This settles a structural question about topological order and provides a unified framework for understanding when different physical systems encode identical quantum information.