Citation: Shi, Bowen, and Yuan-Ming Lu. "Characterizing topological order by the information convex." Physical Review B 99.3 (2019): 035112.
Web: https://arxiv.org/abs/1801.01519
Tags: Information-theory
This article takes a deep look at topological order through the lens of the information convex, defined to be the set of reduced density matrices on some region which minimize all nearby terms of the Hamiltonian. The key observation is that anyons can be identified as extremal points in the information convex an an annulus.
The information convex associated to any region forms a convex set. This is because a convex combination of density matrices is again a density matrix, and a convex combination of density matrices which satisfy all nearby terms of the Hamitlonian will still satisfy all of the nearby terms. There is a well-defined notion of extremal point in a convex set - these are the points which are not in the interior of any line segment.
The information convex of a simply connected region is shown to be a point. This is a restatement of the indistinguishably of local ground states. The extremal points in the information convex of an annulus correspond to anyons, and the algebra of the corresponding density matrices is well-behaved.
There is a general principle that specifies how to maximize the entropy of a linear combination of density matrices. This principle can be applied to the annulus, to find the entropy-maximizing density matrix. Seeing as the entropies of the extremal points can be computed in terms of quantum dimension, this entropy maximizing density matrix has coefficients related to the quantum dimension. There is a quick argument which related this entropy maximizing state to the Levin-Wen prescription of TEE, and thus TEE can be interpreted in terms of the information convex. This maximization principle is one of the reasons that von Neumann entropy seems like the right choice in these settings, and not Renyi entropy.
This paper has two excellent features. The first is that all of its results generalize to gapped boundaries and boundary defects, and the paper spends a lot of time making this clear. Secondly, all of its results are verified explicitly in the case of the Kitaev quantum double model. This takes up a large amount of space, but motivates that the answers presented in this paper really are correct and should generalize to other doubled phases.