Multipartite Greenberger-Horne-Zeilinger entanglement in monitored random Clifford circuits

Interactions in many-body systems are typically short-range and few-body. We investigate how such local interactions build up long-range and intrinsically multipartite entanglement by studying the $n$-partite Greenberger-Horne-Zeilinger $({\text{GHZ}}_{n})$ entanglement in monitored random Clifford circuits, which is well-known for a measurement-induced transition between phases of volume-law and area-law (bipartite) entanglement. We obtain a series of results: (1) About 1.25 $|{\text{GHZ}}_{3}\ensuremath{\rangle}$ can be extracted from states in the volume-law phase. This value is remarkably universal, independent of both the measurement rate and partitioning details, until a phase transition (either measurement-induced or a newly identified partitioning-induced transition) is approached. (2) Dynamically, the creation (sometimes also the annihilation) of ${\text{GHZ}}_{3}$ entanglement occurs suddenly via dynamical phase transitions (DPTs). The critical points of these DPTs are governed by the entanglement speed $({v}_{E})$ of biaprtite entanglement. (3) In stark contrast to ${\text{GHZ}}_{n\ensuremath{\le}3}$, ${\text{GHZ}}_{n\ensuremath{\ge}4}$ entanglement is statistically significant only at the measurement-induced critical point, not in the bulk of the volume-law phase. Our results uncover a rich hierarchy of multipartite entanglement structures.