Symmetry protected topological phases under decoherence

We study a class of symmetry protected topological (SPT) phases under various types of decoherence, which can drive 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. 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 the strange correlator to mixed-state density matrices. Using both exact calculations of the stabilizer Hamiltonians and field theory evaluations, we demonstrate that under decoherence the nontrivial features of the SPT state can persist in the two types of strange correlators: type-I and type-II. We show that the nontrivial type-I strange correlator corresponds to the presence of the SPT information that can be efficiently identified from experiments. The nontrivial type-II strange correlator corresponds to the presence of the original SPT information in the density matrix, which in principle can be identified to distinguish decohered mixed states of an SPT and trivial states. Therefore, our work provides a unified framework to understand decohered SPT phases from the information-theoretic viewpoint.