Multipartite Greenberger-Horne-Zeilinger Entanglement in Monitored Random Clifford Circuits

Abstract

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 $|{\text{GHZ}}_3⟩$ 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 occur 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 ${\text{GHZ}}_{n\le 3}$, ${\text{GHZ}}_{n\ge 4}$ 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 and previously overlooked hierarchy of multipartite entanglement structures.

arXiv commentLatest update: speed of GHZ spread. New results included: Dynamical phase transitions of the birth and death of GHZ entanglement; Multipartite GHZ emerges at the measurement-induced criticality. 6 pages, 4 figures, 5 pages supplemental material

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