  We study ensembles described by density matrices with potentially nontrivial
topological features. In particular, 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. In the "doubled Hilbert space", we provide a general
definition for symmetry protected topological ensemble (SPT ensemble), and the
main quantity that we investigate is various types of (boundary) anomalies in
the doubled Hilbert space. We show that the notion of the strange correlator,
previously proposed to as a diagnosis for the SPT ground states, can be
generalized to capture these anomalies in 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 and
utilized from experiments, such as for the purpose of preparing for long-range
entangled states. The nontrivial type-II strange correlator encodes the full
topological response of the decohered mixed state density matrix, i.e., the
information about the presence of the SPT state before decoherence. Therefore,
our work provides a unified framework to understand decohered SPT phases from
the information-theoretic viewpoint.
