Citation: Zhang, Yi, et al. "Quasiparticle statistics and braiding from ground-state entanglement." Physical Review B—Condensed Matter and Materials Physics 85.23 (2012): 235151.
Web: https://arxiv.org/abs/1111.2342
Tags: Physical, Information-theory, Toric-code
This is a key paper in the development of of topological entanglement entropy as a diagnostic of topological order. The idea they have is to look at the entanglement entropy one obtains by cutting ground states of the torus into pairs of cylinders. The entropy one obtains is NOT independent of the choice of ground state. This means that the measured entropy can reveal features of the chosen ground state.
In particular, the authors show that ground states which maximize TEE correspond to states with well-defined anyonic content across the cut. The idea is that if there is not a well defined quasiparticle content, then the uncertainly will come from two places: uncertainty inherent in the ground state, and uncertainty about which quasiparticle is at the cut. Having a well-defined charge minimizes the uncertainty about what quasiparticle is at the cut, and hence maximizes TEE.
This main idea has several corollaries. One of this is an uncertainty principle. If a ground state has a well-defined anyonic content around one cut, it will be in a superposition of contents along the orthogonal content. It cannot have a well-defined content along both cuts. Hence, the sum of the TEE along both cuts will be bounded above! This bound is given by the usual TEE, and is dubbed the uncertainty principle.
Another corollary of the main idea is that it is easy to determine the S-matrix. By varying over coefficients in the ground states, it is easy to determine minimal entropy states. This means that one can numerically determine the anyon-basis for the ground states of the torus along the two orthogonal cuts. Then, computing the change of basis between these bases give the S-matrix. Hence, one can compute the S-matrix from entanglement! This approach is very suitable to variational Monte Carlo type simulations, and was demonstrated on two examples in the paper.
Another major takeaway from this paper is that well-defined charges are intimately tied to TEE-extremal points. This idea was later used to fantastic effect in the information convex:
> Shi, Bowen, and Yuan-Ming Lu. "Characterizing topological order by the information convex." Physical Review B 99.3 (2019): 035112.