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

We investigate the fragility of quantum information in a topologically ordered state, namely, the ground state of a weakly Zeeman perturbed honeycomb Kitaev model under unitary (Hamiltonian) evolution interrupted by random local projective measurements of the spins, using the tripartite mutual information $\ensuremath{\gamma}$---equivalent to the topological entanglement entropy for topologically ordered states---as a measure. Our findings show that in the steady state, $\ensuremath{\gamma}$ remains significant up to high measurement rates per site $r$ comparable to the Kitaev coupling scale when the initial state has Ising topological order (ITO), beyond which it degrades with increasing $r.$ We contrast this with the Abelian ${Z}_{2}$ topologically ordered ground state in the toric code regime of the Kitaev model with the same energy per site, which has garnered greater attention in the literature of fault-tolerant quantum computation. We observe that $\ensuremath{\gamma}$ in the toric code regime of the Kitaev model collapses rapidly beyond very small values of $r.$ Further, we also find that at high measurement rates, while topological order is lost, certain observables leave distinct signatures of the initial quantum state, thus providing a diagnostic of the topological order.