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

TL;DR

This work investigates irreducible multipartite entanglement (IrME) in monitored random Clifford circuits by quantifying GHZ entanglement with a stabilizer-based index $g_n$. It reveals a robust GHZ$_3$ content in the volume-law phase, $\big\langle g_3 \big\rangle \approx 1.25$, largely independent of system size, measurement rate, and partitioning, up to a partitioning-induced phase transition (PIPT) near $N_B/N=1/2$, and shows dynamical phase transitions that govern the birth and death of GHZ$_3$ entanglement with critical times set by the bipartite entanglement speed $v_E$. In contrast, GHZ$_{n\ge4}$ entanglement is statistically significant only at the measurement-induced critical point, not in the bulk, highlighting a hierarchy between multipartite entanglement structures. The results connect dynamical entanglement formation to information-theoretic measures and operator growth concepts, suggesting broader implications for IrME in many-body quantum systems.

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 $|\text{GHZ}_3\rangle$ 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 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 $\text{GHZ}_{n\leq 3}$, $\text{GHZ}_{n\geq 4}$ 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.

Figures (11)

• Figure 1: (a) Setup: Random Clifford gates implemented upon neighboring qubits are arranged in a brickwork configuration. Each layer (counted by $t$) consists of two levels of random Clifford gates and after each level, projective Z-measurements are applied independently to every qubit with probability $p$. (b) Illustration of the three configurations of tripartitions. In each of them, we vary the relative size of the marked party while keeping the other two equal.
• Figure 2: Measurement- and partitioning-induced $\text{GHZ}_3$ transition of circuits in Config.(1). (a,b) $\langle g_3 \rangle$ as a function of $p$ with $N_B/N=1/3$ and $2/3$, respectively. The inset of (a) demonstrates the data collapse result, where the horizontal axis label is $x_p \equiv (p-p_c)N^{1/\nu}$. (c) $\langle g_3 \rangle$ as a function of $p$ and $N_B/N$ for $N=240$. (d) $\langle g_3 \rangle$ as a function of $N_B/N$ for $N=240$ and a series of $p$. (e,f) $\langle g_3 \rangle$ as a function of $N_B/N$ for $p=0$ in (e) and 0.08 in (f). The insets of (e,f) show the data collapse results, where the horizontal axis label is $x_B \equiv (n_B- n^{cr}_B)N^{1/\mu}$ with $n_B\equiv N_B/N$ and $n^{cr}_B=0.5$. The critical index $\mu$ equals 0.989 and 1.541 for $p=0$ and 0.08, respectively.
• Figure 3: Dynamics of $\text{GHZ}_3$ entanglement. (a,b) Examples of $\langle g_3 \rangle$ and the variances as a function of $\tau = t/N$ with $N=120, 180, 240$ for the equal partitions ($N_B/N=1/3$), and two different measurement probabilities $p=0, 0.02$. (c,d) Examples of $\langle g_3 \rangle$ as a function of circuit layer with $N=240$ for three partitions of $N_A/N>1/2$ without measurements. (e,f) Dynamical phase transitions of the birth and death of $\text{GHZ}_3$ in Configs. (2,3), respectively, with $p=0$.
• Figure 4: An illustration of gates that contribute to $\tilde{V}_k$ with $k=3$. Now BC is viewed as a single party so that the interaction occurs only at the boarder between A and B.
• Figure 5: The 3rd stage of the time-evolution of $\text{GHZ}_3$ entanglement behaves like a lower plateau when $p$ is 0.002 and $N=240$. The other parameters are same with Figs. \ref{['fig_transient']}(c) and (d).