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

Abstract: We investigate $n$-partite Greenberger-Horne-Zeilinger ($\text{GHZ}n$) entanglement in monitored random Clifford circuits and uncover a series of novel phase transitions and dynamical features. Random Clifford circuits are well-known for a measurement-induced transition between phases of volume-law and area-law (bipartite) entanglement. We numerically find that the volume-law phase is characterized by, approximately, a constant amount of $\text{GHZ}_3$ entanglement, despite the variations of system size and the rate of measurement. The area-law phase does not harbor such tripartite entanglement. A measurement-induced phase transition is also seen from $\text{GHZ}_3$ entanglement. Moreover, the amount of $\text{GHZ}_3$ entanglement generally does not change with the partitioning of the system, but abruptly drops to zero when a subsystem has more than half of the qubits, giving rise to a partitioning-induced phase transition. Dynamically, $\text{GHZ}_3$ entanglement does not grow gradually, as is typical for bipartite entanglement, but instead emerges suddenly through dynamical phase transitions (DPTs). Furthermore, in some situations we observe transient $\text{GHZ}_3$ entanglement that will suddenly disappear through another DPT. We also studied multipartite $\text{GHZ}{n\geq 4}$ entanglement and find they emerge exclusively at the measurement-induced criticality, providing new insights into the long-range correlations of critical states.