Spatial structure of multipartite entanglement at measurement induced phase transitions

TL;DR

This work analyzes long-range genuine multipartite entanglement at measurement-induced phase transitions in local quantum circuits. By developing an entanglement-cluster picture and entanglement-weighted graphs, it derives general exponent relations and tests them across 1+1D and 2+1D measurement-only circuits. The authors obtain exact k-party entanglement exponents for the 1+1D case through mapping to percolation and conformal field theory, finding $\alpha_k = 2k$, and extend to 2+1D where exponents are about $\alpha_k \approx 3$–$3.5k$, supported by numerics. The results provide a concrete, scalable framework for understanding multipartite entanglement in non-unitary quantum circuit ensembles and suggest a broader applicability of cluster-based methods and percolation mappings. Overall, the paper advances a principled approach to quantify and bound GME in measurement-driven quantum dynamics with potential impact on quantum information and many-body physics.

Abstract

We study multiparty entanglement near measurement induced phase transitions (MIPTs), which arise in ensembles of local quantum circuits built with unitaries and measurements. In contrast to equilibrium quantum critical transitions, where entanglement is short-ranged, MIPTs possess long-range k-party genuine multiparty entanglement (GME) characterized by an infinite hierarchy of entanglement exponents for k>=2. First, we represent the average spread of entanglement with "entanglement clusters," and use them to conjecture general exponent relations: 1) classical dominance, 2) monotonicity, 3) subadditivity. We then introduce measure-weighted graphs to construct such clusters in general circuits. Second, we obtain the exact entanglement exponents for a 1d MIPT in a measurement-only circuit that maps to percolation by exploiting non-unitary conformal field theory. The exponents, which we numerically verify, obey the inequalities. We also extend the construction to a 2d MIPT that maps to classical 3d percolation, and numerically find the first entanglement exponents. Our results provide a firm ground to understand the multiparty entanglement of MIPTs, and more general ensembles of quantum circuits.

Figures (15)

• Figure 1: A pictoral representation of $2$-party entanglement between subregions $A_1, A_2$. Time flows upwards. In order for there to be large $k$-party entanglement through a cluster $\gamma$, there should be high entanglement regions (blue) through the bulk of $\gamma$ and low entanglement regions (red) between $\gamma$ and the complement of $A=A_1A_2$.
• Figure 2: Combining 2-party clusters $\gamma_A, \gamma_B, \gamma_C$ into a 4-party cluster $\gamma_{ABC}$(right). In the process, the top parts of $\gamma_C$ are replaced by $\gamma_A, \gamma_B$, while the region between $\gamma_A$ and $\gamma_B$ is replaced by $\gamma_C$ - these both correspond to a relaxing of conditions.
• Figure 3: Obtaining the loop and percolation models of a realization of a 1+1D measurement-only stabilizer circuit. (a) A particular realization $C$ of the ensemble, consisting of alternating layers of X and ZZ measurements. (b) The percolation model $P_{S,d}(C)$ of the circuit realization, in red, overlayed on top of the original circuit. From this model we can see that sites $\{1,2\}$ form a cat state with 2-party entanglement, and sites $\{3,5,6\}$ form a cat state with 3-party entanglement, while site 4 is unentangled with any neighbor.
• Figure 4: (a) $k$-party entanglement up to 5 parties over the generalized anharmonic ratio $\eta$. Subregions are 2-16 sites long, with separations up to 48 sites, and darker-colored points correspond to larger width subregions. (b) $k$-party mutual information over $\eta$, for width 8 subregions.
• Figure 5: Entanglement-weighted $\overline{W}[E_k]$, given entanglement between $k$ subregions (indicated by the boxes at the top of each figure) of width $w=4$ and different spacings $\Delta$. The weights $W_\varepsilon$ correspond to the average bond openness in the equivalent percolation model (i.e. the density of $ZZ$ measurements and/or the sparsity of $X$ measurements), convolved over a $ZZ$ and $X$ measurement layer.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Claim 7