Citation: Kato, Kohtaro, Fabian Furrer, and Mio Murao. "Information-theoretical analysis of topological entanglement entropy and multipartite correlations." Physical Review A 93.2 (2016): 022317.
Web: https://arxiv.org/abs/1505.01917
Tags: Information-theory, MTC-reconstruction
This paper introduced the merging lemma, which is a key tool in the entanglement bootstrap program. They introduced this technique to prove a longstanding conjecture, which we state now. Let \rho_{ABC} be the reduced density matrix of the topologically ordered state on a rotationally symmetric three-way-cut circle. Let \tilde{\rho}_{ABC} be the state with maximum entropy which has the same reduced density matrices as \rho_{ABC} on AB, BC, AC. The entropy difference S(\tilde{\rho}_{ABC})-S(\rho_{ABC}) is equal to the TEE. The mergining lemma is useful here because it allows the authors to merge the density matrices on AB, BC, and AC, and a general fact about the merging lemma is that it results in entropy-maximized states. Hence, the authors can explicitly construct \tilde{\rho} and thus compute its entropy.
The authors of this paper also compare TEE to a secret sharing protocol - they show that the TEE can be interpreted as the amount of information that can be stored in \rho_{ABC} which cannot be deduced from AB, BC, or AC. I'll give some details because I found this confusing at first. The idea is that you start with some big (finite) pool of topologically ordered states. You communicate with a partner by sending them these topologically ordered states. The idea is that you want to communicate in such a way that even if a third party had access to all of your communicators on AB, BC, or AC they would not be able to figure out what you were saying. This limits the amount of information you are able to send. In particular, the amount of information which can be stored per state in your pool is equal to the TEE of that state!
Both of these main results were previously known conjectures, so this shows that the merging lemma really is a conjecture-busting result. A reference for these conjectures is
> Zhou, D. L., and L. You. "Characterizing the complete hierarchy of correlations in an $ n $-party system." arXiv preprint quant-ph/0701029 (2007).