This paper explores the phenomenon of spontaneous strong-to-weak symmetry breaking (SSSB) in mixed states, using the purification perspective. The authors analyze how SSSB in mixed states can be mapped to symmetry-protected topological (SPT) order in the purified state. They show that mixed-state long-range order and SSSB can be characterized by the Rényi-2 correlator, which mirrors the long-range order in the purified SPT state. The study also establishes a correspondence between fidelity correlators in the mixed state and strange correlators in the purification, which signify SPT order. The purification perspective is extended to explore intrinsic mixed-state topological order and decoherent SPT phases. The paper begins by introducing the general formalism of mixed states, quantum channels, and purifications. It then discusses weak and strong symmetries, and how they are defined in the context of mixed states. The authors analyze how SSSB can be driven by finite-depth quantum channels and how it manifests in mixed states. They demonstrate that SSSB in a mixed state implies that its purification exhibits a non-vanishing strange correlator, a hallmark of an SPT wavefunction. They link these strange correlators to previously discussed observables, including the fidelity correlator and type-2 strange correlator. The paper then explores SSSB in 1D and 2D systems, showing how it can be mapped to SPT order in the purified state. They analyze the averaged strange correlator in the purified state and its relation to various SSSB observables in the mixed state. The authors also broaden the scope of SSSB to include a wider array of symmetry groups, such as higher-form symmetry, subsystem symmetry, and continuous groups. Finally, they explore mixed-state topological orders and SPT phases from a purification perspective, highlighting the implications of SSSB for mixed-state topological order and decoherence-induced transitions. The paper concludes with a discussion of the physical interpretation of the Rényi-2 correlator and its relation to concepts of pure-state order.This paper explores the phenomenon of spontaneous strong-to-weak symmetry breaking (SSSB) in mixed states, using the purification perspective. The authors analyze how SSSB in mixed states can be mapped to symmetry-protected topological (SPT) order in the purified state. They show that mixed-state long-range order and SSSB can be characterized by the Rényi-2 correlator, which mirrors the long-range order in the purified SPT state. The study also establishes a correspondence between fidelity correlators in the mixed state and strange correlators in the purification, which signify SPT order. The purification perspective is extended to explore intrinsic mixed-state topological order and decoherent SPT phases. The paper begins by introducing the general formalism of mixed states, quantum channels, and purifications. It then discusses weak and strong symmetries, and how they are defined in the context of mixed states. The authors analyze how SSSB can be driven by finite-depth quantum channels and how it manifests in mixed states. They demonstrate that SSSB in a mixed state implies that its purification exhibits a non-vanishing strange correlator, a hallmark of an SPT wavefunction. They link these strange correlators to previously discussed observables, including the fidelity correlator and type-2 strange correlator. The paper then explores SSSB in 1D and 2D systems, showing how it can be mapped to SPT order in the purified state. They analyze the averaged strange correlator in the purified state and its relation to various SSSB observables in the mixed state. The authors also broaden the scope of SSSB to include a wider array of symmetry groups, such as higher-form symmetry, subsystem symmetry, and continuous groups. Finally, they explore mixed-state topological orders and SPT phases from a purification perspective, highlighting the implications of SSSB for mixed-state topological order and decoherence-induced transitions. The paper concludes with a discussion of the physical interpretation of the Rényi-2 correlator and its relation to concepts of pure-state order.