The paper explores the decoherence effect in mixed-state ensembles from a purification perspective, focusing on spontaneous strong-to-weak symmetry breaking (SSSB) triggered by local quantum channels. The authors map the decoherence process to unitary operations in the purified state within an extended Hilbert space, showing that mixed-state long-range order and SSSB can be mapped to symmetry-protected topological (SPT) order in the purified state. They establish a correspondence between fidelity correlators in the mixed state, which measure SSSB, and strange correlators in the purification, which signify SPT order. The paper also discusses the implications for mixed-state topological order and decoherent symmetry-protected topological phases, providing insights into the nature of SSSB and its relation to SPT states.The paper explores the decoherence effect in mixed-state ensembles from a purification perspective, focusing on spontaneous strong-to-weak symmetry breaking (SSSB) triggered by local quantum channels. The authors map the decoherence process to unitary operations in the purified state within an extended Hilbert space, showing that mixed-state long-range order and SSSB can be mapped to symmetry-protected topological (SPT) order in the purified state. They establish a correspondence between fidelity correlators in the mixed state, which measure SSSB, and strange correlators in the purification, which signify SPT order. The paper also discusses the implications for mixed-state topological order and decoherent symmetry-protected topological phases, providing insights into the nature of SSSB and its relation to SPT states.