Robustness of a state with Ising topological order against local projective measurements

We investigate the fragility of a topologically ordered state, namely, the ground state of a weakly Zeeman perturbed honeycomb Kitaev model to environment induced decoherence effects mimicked by random local projective measurements. Our findings show the nonabelian Ising topological order, as quantified by a tripartite mutual information (the topological entanglement entropy $\gamma$,) is resilient to such disturbances. Further, $\gamma$ is found to evolve smoothly from a topologically ordered state to a distribution of trivial states as a function of rate of measurement (temperature). We assess our model by contrasting it with the Toric Code limit of the Kitaev model, whose ground state has abelian $Z_2$ topological order, and which has garnered greater attention in the literature of fault-tolerant quantum computation. The findings reveal the topological order in the Toric Code limit collapses rapidly as opposed to our model where it can withstand higher measurement rates.