Macroscopic theory of multipartite correlations in permutation-invariant open quantum systems

TL;DR

This work addresses the scaling of multipartite correlations in permutation-invariant open quantum systems by developing a phase-space, symmetry-based method that expresses the asymptotic behavior of the stationary mutual information $I_M$ entirely in terms of the mean-field drift. The central result is $\lim_{N\to\infty} \frac{I_M}{N} = S(\overline{\rho_{\xi}}) - \overline{S(\rho_{\xi})}$, linking macroscopic correlations to infinite-time averages over the drift dynamics. It shows that extensive scaling of $I_M$ is tied to the presence of time-dependent attractors (e.g., limit cycles) and is not robust for fixed-point relaxation, with the driven-dissipative Lipkin-Meshkov-Glick model as a concrete demonstration. This framework provides a practical route to quantify multipartite synchronization and other correlations in large quantum networks and offers avenues for extending quantum thermodynamics and non-equilibrium analyses in permutation-invariant settings.

Abstract

Information-theoretic quantities have received significant attention as system-independent measures of correlations in many-body quantum systems, e.g., as universal order parameters of synchronization. In this work, we present a method to determine the macroscopic behavior of the steady-state multipartite mutual information between $N$ interacting units undergoing Markovian evolution that is invariant under unit permutations. Using this approach, we extend a conclusion previously drawn for classical systems that the extensive scaling of mutual information is either not possible for systems relaxing to fixed points of the mean-field dynamics or such scaling is not robust to perturbations of system dynamics. In contrast, robust extensive scaling occurs for system relaxing to time-dependent attractors, e.g., limit cycles. We illustrate the applicability of our method on the driven-dissipative Lipkin-Meshkov-Glick model.

Figures (4)

• Figure 1: The intensive multipartite mutual information $I_M/N$ in the driven-dissipative LMG model for $\mathcal{J}=3\gamma$ and (a) $h=0$, (b) $h=0.5 \gamma$. The solid green line represents the predictions of our theory and dots represent the master equation results for different system sizes $N$. Dotted lines are added for eye guidance.
• Figure 2: The intensive multipartite mutual information $I_M/N$ in the ground state of the ferromagnetic ($\mathcal{J}<0$) LMG model for different system sizes $N$. The inset presents a smaller range of $h$. Calculation details are presented in Appendix \ref{['app:ground']}.
• Figure 3: (a) Scaled total entropy $S(\rho_T)/N$ and (b) local entropy of a single unit $S(\rho_i)$ as a function of $\Gamma/\gamma$. The inset in (b) shows the derivative $dS(\rho_i)/d\Gamma$ for a smaller range of $\Gamma/\gamma$. Parameters and notations as in Fig. 1(b) in the main text. For $N \rightarrow \infty$, we apply our theory as $S(\rho_T)/N = \overline{S(\rho_{\boldsymbol{m}_t})}$, $S(\rho_i)=S(\overline{\rho_{\boldsymbol{m}_t}})$.
• Figure 4: (a) Longitudinal magnetization $\langle m_z \rangle$ and (b) transverse magnetization $m_{xy}=\sqrt{\langle m_{x} \rangle^2+\langle m_{y} \rangle^2}$ as a function of $\Gamma/\gamma$ (note different scales on $y$ axis). Parameters and notations as in Fig. 1(b) in the main text. For $N \rightarrow \infty$, we apply the mean-field theory as $\langle m_\alpha \rangle=\overline{(\boldsymbol{m}_{t})_\alpha}$.