Researchers have demonstrated that decoherence, traditionally viewed as a destructive process for quantum states, can actually induce and organize a series of nontrivial mixed quantum states. Specifically, they have studied the effect of subsystem decoherence on an extended cluster state, which is a symmetry-protected topological (SPT) phase. This model incorporates multiple subsystem $Z_2$ symmetries, revealing unexpected behavior where decoherence not only degrades the system but transforms it into new phases with distinct quantum properties.

The study reveals that subsystem decoherence induces local charge fluctuations, leading to the formation of a mixed SPT state in the unaffected subsystems. Starting from an extended cluster state, a hierarchy of mixed-state SPT phases emerges as step-by-step subsystem decoherence progresses. These mixed-state SPT phases retain the strong symmetries that protected the initial cluster SPT. Furthermore, these SPT phases can be characterized by Rényi-2 string orders, a metric that quantifies long-range quantum correlation.

As subsystems are progressively decohered, this hierarchy of mixed-state SPT phases terminates in a $Z_2$ strong-to-weak spontaneous symmetry breaking (SWSSB) state on the final remaining subsystem. At this point, a long-range entangled state emerges, specifically a glassy Greenberger-Horne-Zeilinger (GHZ) state. This work underscores that decoherence is not merely a detrimental factor, but can be a mechanism to induce and organize a series of nontrivial quantum states, offering a systematic route from mixed-state SPT order to SWSSB with glassy GHZ-type long-range entanglement.