Symmetry Protected Topological phases under Decoherence

We study symmetry protected topological (SPT) phases under various types of decoherence, which drives a pure SPT state into a mixed state. We demonstrate that the system can still retain the nontrivial topological information from the SPT ground state even under decoherence, which can arise from noise or weak measurement. The main quantity that we investigate is the ``strange correlator" proposed previously as a diagnosis for the SPT ground states, and in this work we generalize the notion of strange correlator to the mixed state density matrices. Using both exact calculations of the stabilizer Hamiltonians, as well as field theory evaluation, we demonstrate that under decoherence the characteristic behaviors of the SPT states can persist in the ``type-II" strange correlators. Furthermore, in some cases, the ``type-I'' strange correlators exhibit transition behaviors depending on the strength of decoherence, which alerts the presence of information-theoretic transition. We show that in the regime where type-I strange correlators are non-trivial, topological information of the decohered SPT state can be efficiently identified from experiments.