Taming multiparty entanglement at measurement-induced phase transitions

TL;DR

This paper tackles the problem of characterizing multiparty entanglement at measurement-induced phase transitions by simulating a trapped-ion native MIPT circuit that approaches Haar non-unitary CFT universality. The authors combine generalized multiparty mutual information and genuine multiparty negativity, computed via semidefinite programming, with conformal finite-size scaling around a critical point $p_c$ to reveal robust algebraic decay of correlations for $k=2,3,4$ parties, and to propose a conjecture $\alpha_k^{\rm MI}=k+2$. They demonstrate that mutual information exponents satisfy a monogamy-derived lower bound $\alpha_k^{\rm MI}\ge k$ and present strong evidence for monotonicity and subadditivity across party numbers, while GMN exponents show larger finite-size drifts and imply long-range quantum resources with entanglement halos in spacetime. Overall, the work advances understanding of non-unitary critical dynamics and entanglement organization, with implications for trapped-ion experiments and connections to holographic ideas.

Abstract

Measurement-induced phase transitions (MIPT) give rise to novel dynamical states of quantum matter realized by balancing unitary evolution and measurements. We present large-scale numerical simulations of a trapped-ion native MIPT, argued to belong to the universality class described by the Haar non-unitary conformal field theory. First, through a finite-size analysis we obtained the critical measurement rate, and correlation length exponent, which falls close to the percolation value. Second, by leveraging a monotone computable via semi-definite programming, we uncover robust algebraic decay of genuine multiparty entanglement (GME) versus separation for 2, 3, and 4 parties. The corresponding critical exponents are lower-bounded by those of the multiparty mutual information, which we determine up to 4 parties, and conjecture to be (k+2) for k parties. Additionally, we derive lower bounds for both GME and multiparty mutual information.

Figures (11)

• Figure 1: Monitored circuit model and conformal geometry. (a) Space-time diagram of the hybrid circuit dynamics. The system evolves via layers of unitary gates (blocks) applied in a brickwork pattern with periodic boundary conditions, interspersed with single-site projective measurements (yellow circles) performed with probability $p$. We probe the spatial structure of the critical state by computing multiparty entanglement for a subsystem of qubits (highlighted in purple) separated by a distance $x$. (b) Geometric representation of the circuit with periodic boundary conditions. To extract universal scaling exponents, Euclidean distances $x$ are replaced by chord lengths $l_x$ defined by (\ref{['eq:chord_length_definition']}).
• Figure 2: Critical point identification via tripartite mutual information (TMI).(a) TMI plotted as a function of measurement probability $p$ for system sizes $N \in \{12, 16, 20, 24\}$. Inset: The intersection of the curves for the largest system sizes, $N=20$ and $N=24$ shows that the critical measurement rate $p_c \lesssim 0.16$. (b) Finite-size scaling collapse of the TMI data. By rescaling the horizontal axis as $(p-p_c)L^{1/\nu}$, the data for $N=20$ and $24$ collapse onto a single universal curve. Inset: A collapse-diagnostic based on the difference between these two rescaled curves for different $\nu$. Optimizing the collapse over $\nu$ and $p_c$ yields a correlation length critical exponent $\nu = 1.48(4)$ at $p_c = 0.160(1)$ for the specific pair of curves at $N=20, 24$.
• Figure 3: Large-N limits of the critical measurement rate and correlation length exponent. a) Measurement probability $p_c$ determined by crossing of curves, as well as curve collapse, at $N_1$ and $N_2$ sites respectively. The extrapolation is a linear fit to the $p_c$ crossing data. b) Correlation length exponent $\nu$ obtained by fitting two curves of $N_1$ and $N_2$ sites respectively, over $(N_1 N_2)^{-1}$. Confidence intervals are set to the $O = 1.3O^*$ points. All linear fits assign the points equal weight, and omit the rightmost $N_1 = 12, N_2 =16$ point from the fitting.
• Figure 4: Spatial scaling of multiparty correlations.(a) Multiparty Mutual Information $(-1)^k\overline{I_k}$ and (b) Genuine Multiparty Negativity $\overline{\mathcal{N}_k}$ as a function of the effective distance scale $d$ at $p=0.17$, near the critical point. The number of parties $k$ is distinguished by symbol and color: 2-party (red circles), 3-party (blue triangles), and 4-party (purple squares). Color intensity indicates system size, scaling from light ($N=18$) to dark ($N=26$). Dashed black lines denote power-law fits extracted from the largest converged dataset for each metric: $N=26$ for MI and $N=24$ for GMN.
• Figure 5: Large-$N$ convergence of multiparty exponents. Finite-size estimates of the decay exponents for genuine multiparty negativity ($\alpha_k^{\mathrm{GMN}}$) and multiparty mutual information ($\alpha_k^{\mathrm{MI}}$) plotted versus $1/N$, using the fitted values in Table. \ref{['tab:entanglement_exponents']}. For mutual information, the exponents show negligible drift up to $N=26$, and we therefore plot the expected asymptotic constants $\alpha_k^{\mathrm{MI}}=k+2$. For GMN, the $k=2,3$ exponents retain visible finite-size drift and are extrapolated with $\alpha_k^{\mathrm{GMN}}(N)=C-A e^{b/N}$ (dashed lines) for $N\geq 20$. For $\alpha_4^{\mathrm{GMN}}$, the available system sizes are insufficient to support a reliable extrapolation.