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arXiv:2510.19922

Probing the Hierarchy of Genuine Multipartite Entanglement with Generalized Latent Entropy

Byoungjoon Ahn, Jaydeep Kumar Basak, Keun-young Kim A, Gwon Bin Koo, Vinay Malvimat, Junggi Yoon

Published Oct 22, 2025• 0 citations
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Abstract

We introduce generalization of the recently proposed \textit{Latent Entropy} (L-entropy) \cite{Basak:2024uwc} as a refined measure of genuine multipartite entanglement (GME) in pure states of $n$-party quantum systems. Generalized L-entropy provides a natural ordering among $k$-uniform states, maximising 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 model. 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.

DOI
10.48550/arxiv.2510.19922

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We introduce generalization of the recently proposed \textit{Latent Entropy} (L-entropy) \cite{Basak:2024uwc} as a refined measure of genuine multipartite entanglement (GME) in pure states of nnn-party quantum systems. Generalized L-entropy provides a natural ordering among kkk-uniform states, maximising for absolutely maximally entangled states (AME), effectively capturing the hierarchical structure of multipartite entanglement. We analyze the behavior of this measure for nnn-party Haar-random states and demonstrate that, in the large local-dimension limit, the maximal L-entropy saturates its upper bound for odd nnn, while for even nnn 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 SYK4_44​, SYK2_22​, mass-deformed SYK, sparse SYK, and N=2\mathcal{N}=2N=2 supersymmetric SYK model. 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.
We advance ``Latent entropy" (L-entropy) as a novel measure to characterize genuine multipartite entanglement in pure states, applicable to quantum systems with both finite and infinite degrees of freedom. This measure, derived from an upper bound on reflected entropy, attains its maximum for three-party GHZ states and n=4,5n=4,5n=4,5-party 222-uniform states. We establish that it satisfies all essential properties of a genuine multipartite entanglement measure, including being a pure-state entanglement monotone. We further obtain an analogue of the Page curve by analyzing the behavior of L-entropy in multiboundary wormholes, emphasizing their connection to multipartite entanglement in random states. Specifically, for n=5n = 5n=5, we show that random states approximate 222-uniform states, exhibiting maximal multipartite entanglement. Extending these ideas to finite temperatures, we introduce the Multipartite Thermal Pure Quantum (MTPQ) state, a generalization of the thermal pure quantum state to multipartite systems, and demonstrate that the entanglement structure in states of the multicopy SYK model exhibits finite-temperature 222-uniform behavior.
Abstract We introduce the latent entropy (L-entropy) as a novel measure to characterize the genuine multipartite entanglement in quantum systems. Our measure leverages the upper bound of reflected entropy and its maximal values attained by 2-uniform states for n-party (n=4,5n= 4,5n=4,5) and by the GHZ state for 3-party quantum systems. We demonstrate that the measure is a nonnegative, local unitary invariant function that vanishes for separable states. We then analyze its interesting characteristics in spin chain models and the Sachdev–Ye–Kitaev (SYK) model. Subsequently, we explore its implications for holography by deriving a Page-like curve for the L-entropy in the conformal field theory (CFT) dual to a multiboundary wormhole model. Furthermore, we examine the behavior of L-entropy in Haar random states, deriving analytical expressions and validating them against numerical results. In particular, we show that for n=5n =5n=5, random states approximate 2-uniform states with maximal multipartite entanglement. Furthermore, we propose a potential connection between random states and multiboundary wormhole geometries. Extending to finite-temperature systems, we introduce the multipartite thermal pure quantum (MTPQ) state, a multipartite generalization of the thermal pure quantum state, and explore its entanglement properties. By incorporating the state-dependent construction of the MTPQ state, we resolve the factorization issue in the random average of the MTPQ state, ensuring consistency with the correlation functions in the holographic dual multiboundary wormhole. Finally, we apply this construction to the multicopy SYK model and examine its multipartite entanglement structure.
Genuine multipartite entanglement (GME) is a valuable resource for quantum technologies, but quantifying it in many-body systems can be demanding since measures defined through optimizations over bipartitions are often assumed to require large joint measurements and many measurement settings. We introduce the concepts of GME evaluation length and GME evaluation number, which capture, respectively, the largest subsystem that must be jointly measured and the number of distinct subsystems that need to be measured to evaluate multipartite entanglement measures defined via bipartition optimizations. Taking the genuine multipartite concurrence (GMC) as a canonical example, we show that, for NNN-qubit Haar-random pure states, the GMC can be evaluated using only NNN single-qubit purity measurements. Thus, for typical many-body states, evaluating the GMC requires considering only the NNN single-site bipartitions, rather than the exponentially many distinct bipartitions. We further corroborate this behavior numerically for physically accessible states generated by local random quantum circuits. Our results show that the practical cost of quantifying GME can be dramatically lower than suggested by its formal definition, establishing measurement complexity as a complementary operational lens for multipartite entanglement.
Multipartite entanglement is a fundamental aspect of quantum mechanics, crucial to advancements in quantum information processing and quantum computation. Within this field, Genuinely Multipartite Entanglement (GME), being entangled in all bipartitions, and Absolutely Maximally Entanglement (AME), maximally entangled in all bipartitions, represent two significant types of entanglement with diverse applications. In this work, we introduce a new measure called the GME-AME multipartite entanglement measure, with a non-zero value representing the GME states and the maximum value is reached only by the AME states. The measure is applied to study the multipartite entanglement of four partite systems using the operator to state mapping, and the four partite permutation qutrit states are classified according to the measure. With various examples, we show that our measure is robust in classifying the four partite entangled states.
We investigate the structure of quantum magic in interacting disordered fermionic systems, quantifying non-stabilizerness via the fermionic stabilizer Rényi entropy (SRE). To resolve the distribution of magic across different scales, we introduce a multipartite non-local magic functional, constructed from an inclusion-exclusion combination of subsystem contributions. This measure serves as a fine-grained diagnostic, isolating genuinely global contributions and revealing nontrivial interactions between local and collective supports of magic. We illustrate the measure on paradigmatic multipartite states and apply these diagnostics to the Sachdev-Ye-Kitaev model and its variants. Crucially, for thermal/typical ensembles, we observe a marked disparity between Thermal Pure Quantum (TPQ) states and the thermal density matrix. This reveals a concealed complexity: the immense computational hardness characterizing the unitary evolution is encoded in the specific microstructure of the black hole microstates, while being washed out in the coarse-grained thermodynamic description. Furthermore, in N=2\mathcal N=2N=2 supersymmetric SYK, we show that while fortuitous BPS states exhibit intermediate stabilizer complexity, the multipartite measure unveils a rich, sector-dependent pattern of global correlations, distinguishing them from generic chaotic states.
Understanding and classifying multipartite entanglement is fundamental to quantum information processing. This work focuses on absolutely maximally entangled (AME) states, a class of highly entangled states characterized by their maximal entanglement across any bipartitions. To analyze and classify AME states, we employ the minimal decomposition entropy, defined as the minimum Rényi entropy SqS_qSq​ associated with the state's decomposition over all local product bases. This quantity identifies the product bases in which the state is maximally localized, thereby yielding optimal representations for analyzing properties of AME states. We develop an efficient algorithm for computing the minimal decomposition entropy for finite q>1q>1q>1 and compare AME and Haar-random states for q=2 q = 2 q=2 and q=∞ q = \infty q=∞ in qubit, qutrit, and ququad systems. For q=2 q = 2 q=2, AME states of four qutrits and ququads show lower minimal entropy than generic states, indicating sparser optimal forms. For q=∞ q = \infty q=∞ -- related to the geometric measure of entanglement -- AME states exhibit higher entanglement. The algorithm also simplifies known AME states into sparser representations, aiding in distinguishing genuinely quantum AME states from those constructible from classical combinatorial designs. Our results advance the classification of AME states and demonstrate the utility of minimal decomposition entropy as both a local unitary invariant and a tool for state simplification.