Citation: Kim, Isaac H., et al. "Universal lower bound on topological entanglement entropy." Physical Review Letters 131.16 (2023): 166601.
Web: https://arxiv.org/abs/2302.00689
Tags: Information-theory
In this paper the authors prove that even though spurious TEE is possible, it will always be positive. Hence, cuts with no spurious TEE can be identified as the cuts which minimize TEE. In particular, TEE can be defined as the lower bound of the usual definition of TEE over all perturbations by local unitaries, as the radius gets large.
Clearly, this new definition of TEE is invariant under perturbations - it is defined as a minimum over perturbations, so one can always perturb back! This universal quantity is equal to (log D) whenever that makes sense.
The main idea of the proof is to look at "reference states". These are states for which TEE is well behaved - it gives the same value on any choice of annulus. It is then shown that these reference states minimize the TEE over all states. This is done by first showing that the TEE for any state can be lower bounded in terms of a difference of entropies between a maximally and minimally entangled state on the annulus. This first part of the proof is done by first restricting to unitaries on a simply connected subset of the annulus, and then extending to the full annulus in a clever way. The difference of entropies between maximally and minimally entangled states is then related to TEE by the same entropy-maximization arguments as in
> Shi, Bowen, and Yuan-Ming Lu. "Characterizing topological order by the information convex." Physical Review B 99.3 (2019): 035112.
The full proof is actually quite technical. It involves properly developing the theory of quantum Markov chains and proving a large number of technical lemmas.