Topological order subject to local errors II: diagnostics of error-induced phases

Topologically ordered ground states can encode information nonlocally and protect it from local errors up to a finite error threshold. The capability of encoding information defines distinct error-induced phases. In this talk, we propose three complementary information-theoretic diagnostics of such phases intrinsic to the corrupted mixed state: (1) quantum relative entropy between an error-corrupted ground state and excited state; (2) coherent quantum information; (3) topological entanglement negativity. In the example of 2D Toric code with local incoherent errors, three diagnostics simultaneously undergo the transition and consistently probe the error-induced phases. We analytically establish this result by mapping the three diagnostics to observables in a series of 2D classical statistical mechanics models. These observables all detect the ferromagnetic transitions in the classical models and thus exhibit singular behaviors at the same error threshold. We numerically verify our results using Monte-Carlo simulations.