  We explore the landscape of the decoherence effect in mixed-state ensembles
from a purification perspective. We analyze the spontaneous strong-to-weak
symmetry breaking (SSSB) in mixed states triggered by local quantum channels by
mapping this decoherence process to unitary operations in the purified state
within an extended Hilbert space. Our key finding is that mixed-state
long-range order and SSSB can be mapped into symmetry-protected topological
(SPT) order in the purified state. Notably, the measurement-induced long-range
order in the purified SPT state mirrors the long-range order in the mixed state
due to SSSB, characterized by the Renyi-2 correlator. We establish a
correspondence between fidelity correlators in the mixed state, which serve as
a measure of SSSB, and strange correlators in the purification, which signify
the SPT order. This purification perspective is further extended to explore
intrinsic mixed-state topological order and decoherent symmetry-protected
topological phases.
