We study the effect of subsystem decoherence on an extended cluster state, which is a symmetry-protected topological (SPT) phase. The model includes many subsystem $Z_2$ symmetries. We report that subsystem decoherence induces local charge fluctuations, leading to a mixed SPT state in the unaffected subsystems. If we start from the extended cluster state, hierarchical mixed-state SPT phases emerge in response to step-by-step subsystem decoherences. These mixed-state SPT phases keep strong symmetries, the symmetry of which protects symmetries for the initial cluster SPT. Moreover, these SPT states can be characterized by Rényi-2 string orders. Then, as the subsystems are progressively decohered, the hierarchy of mixed-state SPT phases terminates in a $Z_2$ strong-to-weak spontaneous symmetry breaking (SWSSB) state on the final remaining subsystem, where a long-range entangled state appears, namely, a glassy Greenberger-Horne-Zeilinger (GHZ) state. Our work demonstrates that decoherence is not merely a destructive process, but can induce and organize a series of nontrivial mixed states. This reveals a systematic route from mixed-state SPT order to SWSSB order with the glassy GHZ-type long-range entanglement.

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• Density matrix renormalization group
• Open quantum systems & decoherence
• Spontaneous symmetry breaking
• Symmetry protected topological states