Citation: Flammia, Steven T., et al. "Topological entanglement Rényi entropy and reduced density matrix structure." Physical review letters 103.26 (2009): 261601.
Web: https://arxiv.org/abs/0909.3305
Tags: Kitaev-quantum-double, Information-theory
This paper takes a look at the Renyi topological entanglement entropy of quantum double topological orders. Renyi entropy is a generalization of von Neumann entropy, chosen for its algebraic properties and not its physical relevance. Its algebraic properties in particular imply that the usual definition of TEE also gives a Renyi variant which is stable. The Renyi entropy is parameterized by a parameter \alpha, which varies from 0 to infinity. When \alpha=1 we recover von Neumann entropy.
The interesting result is that for doubled topological order, all Renyi topological entanglement entropies are equal to log(D). They give a direct proof first using the Levin-Wen Hamiltonian, and then the Kitaev quantum double Hamiltonian.
The authors then observe that all of the Renyi entropies being equal to log(D) implies a special structure on the reduced density matrix inside a disk, where the topological part of its spectrum is concentrated at a point, making it "flat". The philosophical argument they give for why this should happen is that TQFTs correspond to CFTs. The CFT for a doubled topological order can be chosen to be trivial since doubled topological orders admit gapped boundaries. Hence, this implies a special structure on the reduced density matrices.
An interesting point to observe is that Steve Simon in his book topological quantum present an extension of Levin-Wen's result:
> Simon, Steven H. Topological quantum. Oxford University Press, 2023.
He proves that the Renyi entropies of all topological phases at all integers \alpha are equal to log(D). This is suspicious because the philosophical argument of Levin-Wen does not apply in this case. Simon's argument is based on the manifold-surgery approach of Dong et al.:
> Dong, Shiying, et al. "Topological entanglement entropy in Chern-Simons theories and quantum Hall fluids." Journal of High Energy Physics 2008.05 (2008): 016.
It is unclear whether Simon's proof is correct - at the present moment I do not understand it. It would be interesting to either find a gap in his argument, find another argument supporting his conclusion, or find a counterexample.