Symmetry protected topological phases under decoherence

Abstract

We investigate mixed states exhibiting nontrivial topological features, focusing on symmetry-protected topological (SPT) phases under various types of decoherence. Our findings demonstrate that these systems can retain topological information from the SPT ground state despite decoherence. In the ''doubled Hilbert space,'' we define symmetry-protected topological ensembles (SPT ensembles) and examine boundary anomalies in this space. We generalize the concept of the strange correlator, initially used to diagnose SPT ground states, to identify anomalies in mixed-state density matrices. Through exact calculations of stabilizer Hamiltonians and field theory evaluations, we show that nontrivial features of SPT states persist in two types of strange correlators: type-I and type-II. The type-I strange correlator reveals SPT information that can be efficiently detected and used experimentally, such as in preparing long-range entangled states. The type-II strange correlator encodes the full topological response of the decohered mixed state, reflecting the SPT state's pre-decoherence presence. Our work offers a unified framework for understanding decohered SPT phases from an information-theoretic perspective.