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 $\gamma$—equivalent to the topological entanglement entropy for topologically ordered states—as a measure. Our findings show that in the steady state, $\gamma$ 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 $\gamma$ 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.

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• Dynamical phase transitions
• Quantum entanglement
• Topological order
• Strongly correlated systems
• Kitaev model