Probing the Hierarchy of Genuine Multipartite Entanglement with Generalized Latent Entropy

TL;DR

The paper develops a generalization of Latent Entropy, $\tilde{L}_{gen}$, to quantify genuine multipartite entanglement across arbitrary $n$-party pure states by aggregating purified, balanced bipartite latents from all $k$-party reductions up to $k_{\max}=\max[2,\lfloor n/2\rfloor]$ and normalizing them. It proves that $\tilde{L}_{gen}$ satisfies the standard GME axioms, yields a hierarchical ordering that tracks $k$-uniformity, and reaches 1 for AME states; odd and even $n$ exhibit distinct random-state behavior in the large-$d$ limit. The authors apply the framework to SYK model variants (SYM4, SYK2, mass-deformed, sparse, and N=2 SUSY SYK), showing that $SYK_4$ approaches AME-like saturation with increasing system size while $SYK_2$ remains submaximal, and that deformations shape the late-time growth and eigenstate distribution of $\tilde{L}_{gen}$. The results suggest that $\tilde{L}_{gen}$ is a sensitive probe of multipartite entanglement structure with potential implications for holography and the characterization of highly entangled many-body states. The work opens avenues for optimized state discovery, conformal/holographic analyses, and deeper links between entanglement hierarchies and dynamics in strongly interacting quantum systems.

Abstract

We introduce generalization of the recently proposed Latent Entropy (L-entropy) [1] as a refined measure of genuine multipartite entanglement (GME) in pure states of $n$-party quantum systems. Generalized L-entropy satisfies the axioms required for a valid GME measure and provides a natural ordering among $k$-uniform states maximizing for absolutely maximally entangled states (AME), effectively capturing the hierarchical structure of multipartite entanglement. We analyze the behavior of this measure for $n$-party Haar-random states and demonstrate that, in the large local-dimension limit, the maximal L-entropy saturates its upper bound for odd $n$, while for even $n$ it approaches the bound asymptotically. Furthermore, we apply this framework to examine multipartite entanglement properties of quantum states in several variants of the Sachdev--Ye--Kitaev (SYK) model, including SYK$_4$, SYK$_2$, mass-deformed SYK, sparse SYK, and $\mathcal{N}=2$ supersymmetric SYK. The results demonstrate that the generalized L-entropy serves as a sensitive probe of multipartite entanglement, revealing how deformations influence quantum entanglement structure in such strongly interacting systems.

Figures (20)

• Figure 1: (a) and (b) illustrate the scaling behavior of generalized latent entropy for Haar-random states. The left panel compares systems with different party numbers, while the right panel highlights the even–odd contrast at high dimensions.
• Figure 2: (a) and (b) illustrate the generalized L-entropy for random qudit states with different numbers of parties.
• Figure 3: Time evolution of the generalized L-entropy for the $SYK_2$ model starting from $\ket{000\ldots0}$, averaged over 25 samples. The plots show the variation in late-time saturation with system size.
• Figure 4: Time evolution of the lower L-entropies (characterizing $k$-uniform states) for the $SYK_2$ model with initial state $\ket{000\ldots0}$, averaged over 25 samples.
• Figure 5: Generalized L-entropy distribution across energy eigenstates of the $SYK_2$ model for $N = 18, 20,$ and $22$ (left), and the saturation value of the generalized L-entropy versus $N$ (right).