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arXiv:2509.08209 · Journal of High Energy Physics

Tripartite correlation signal from multipartite entanglement of purification

Ning Bao, Keiichiro Furuya, Joydeep Naskar

Published May 20, 2026• 2 citations
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Abstract

A bstract We propose a signal $$ {\Delta }_p^{(3)} $$ ∆ p 3 for tripartite correlations in finite-dimensional quantum systems and $$ {\Delta }_w^{(3)} $$ ∆ w 3 for holographic systems. We prove that $$ {\Delta }_p^{(3)} $$ ∆ p 3 is non-negative for any tripartite entangled mixed states and vanishes for mixed product states. It is a correlation signal because it is generally nonzero for a separable state containing a classical mixture. Based on the conjecture, the equality between an entanglement wedge cross section E w and entanglement of purification E p , i.e., E w = E P in the semiclassical limit, we apply the tripartite correlation signal to study the structures of tripartite entanglement in AdS 3 /CFT 2 , especially for pure AdS 3 . We comment on a generalization to n -partite correlation signals $$ {\Delta}_p^{(n)}\left({A}_1:\cdots :{A}_n\right) $$ Δ p n A 1 : ⋯ : A n .

DOI
10.1007/jhep05(2026)236

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A bstract We propose a signal Δp(3){\Delta }_p^{(3)}Δp(3)​ ∆ p 3 for tripartite correlations in finite-dimensional quantum systems and Δw(3){\Delta }_w^{(3)}Δw(3)​ ∆ w 3 for holographic systems. We prove that Δp(3){\Delta }_p^{(3)}Δp(3)​ ∆ p 3 is non-negative for any tripartite entangled mixed states and vanishes for mixed product states. It is a correlation signal because it is generally nonzero for a separable state containing a classical mixture. Based on the conjecture, the equality between an entanglement wedge cross section E w and entanglement of purification E p , i.e., E w = E P in the semiclassical limit, we apply the tripartite correlation signal to study the structures of tripartite entanglement in AdS 3 /CFT 2 , especially for pure AdS 3 . We comment on a generalization to n -partite correlation signals Δp(n)(A1:⋯:An){\Delta}_p^{(n)}\left({A}_1:\cdots :{A}_n\right)Δp(n)​(A1​:⋯:An​) Δ p n A 1 : ⋯ : A n .
A bstract We introduce a new information-theoretic measure of multipartite correlations Δ P , by generalizing the entanglement of purification to multipartite states. We provide proofs of its various properties, focusing on several entropic inequalities, in generic quantum systems. In particular, it turns out that the multipartite entanglement of purification gives an upper bound on multipartite mutual information, which is a generalization of quantum mutual information in the spirit of relative entropy. After that, motivated by a tensor network description of the AdS/CFT correspondence, we conjecture a holographic dual of multipartite entanglement of purification Δ W , as a sum of minimal areas of codimension-2 surfaces which divide the entanglement wedge into multi-pieces. We prove that this geometrical quantity satisfies all properties we proved for the multipartite entanglement of purification. These agreements strongly support the Δ P = Δ W conjecture. We also show that the multipartite entanglement of purification is larger than multipartite squashed entanglement, which is a promising measure of multipartite quantum entanglement. We discuss potential saturation of multipartite squashed entanglement onto multipartite mutual information in holographic CFTs and its applications.
A bstract We argue that holographic CFT states require a large amount of tripartite entanglement, in contrast to the conjecture that their entanglement is mostly bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp conflict with two well- supported conjectures about the entanglement wedge cross section surface EW . If EW is related to either the CFT’s reflected entropy or its entanglement of purification, then those quantities can differ from the mutual information at O(1GN). O\left(\frac{1}{G_N}\right). O(GN​1​). O 1 G N . We prove that this implies holographic CFT states must have O(1GN). O\left(\frac{1}{G_N}\right). O(GN​1​). O 1 G N . amounts of tripartite entanglement. This proof involves a new Fannes-type inequality for the reflected entropy, which itself has many interesting applications.
We identify a non-negative and upper-bounded entanglement signal in holography which is defined as a combination of entanglement wedge cross sections (EWCS) for a tripartite mixed state ABEABEABE: EIΔ(A:B∣E)=EWCS(A:EB)+EWCS(B:EA)−EWCS(E:AB)\mathrm{EI}_Δ(A:B|E) = \mathrm{EWCS}(A:EB) + \mathrm{EWCS}(B:EA) - \mathrm{EWCS}(E:AB)EIΔ​(A:B∣E)=EWCS(A:EB)+EWCS(B:EA)−EWCS(E:AB). This quantity is an analogue of conditional mutual information (CMI) and shares similar mathematical structures in both quantum information theory and holography. We show that CMI is upper bounded by a quantum information quantity, the entanglement of assistance, which quantifies the entanglement that can be generated between two parties AAA and BBB, given assistance from a third party EEE. We prove that EIΔ\mathrm{EI}_ΔEIΔ​ is also upper bounded by the entanglement of assistance in the canonical purification state. We analyze its upper bound by maximizing EIΔ(A:B∣E)\mathrm{EI}_Δ(A:B|E)EIΔ​(A:B∣E) over all configurations of the auxiliary subsystem EEE in AdS3_33​/CFT2_22​. The maximized EIΔ\mathrm{EI}_ΔEIΔ​ displays a rich phase structure governed by the cross ratio XABX_{AB}XAB​: it vanishes below a critical threshold and, beyond a second phase transition point, saturates the bound of entanglement of assistance. We comment on the interpretation of EIΔ\mathrm{EI}_ΔEIΔ​ as characterizing the assisted bipartite quantum entanglement between AAA and BBB with the help of EEE.
Many of the tools and ideas of quantum information theory have been employed to shed light on the holographic principle.The entanglement entropy in the AdS/CFT correspondence tells us that information of the bulk geometries is encoded in the boundary correlations.This thesis is aimed to deepen our understanding of this direction by studying various correlation measures in holography.We first derive a generalization of the holographic entanglement of purification conjecture for multipartite states.It turns out that this proposal is consistently supported in arbitrary n-partite states.We next study the entanglement of purification in quantum many body systems by numerical methods.That reveals a remarkable non-monotonicity of the entanglement of purification, attributed to the different sensitivities to quantum and classical correlations.We finally discuss the holographic duals of two classes of correlation measures: optimized correlation measures and axiomatic entanglement measures.The results suggest that classical correlation also plays an important role in encoding the geometrical information.i
A bstract We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section. We generalize both quantities and prove several information theoretic inequalities involving them. These include upper bounds on conditional mutual information and tripartite information, as well as a lower bound for tripartite information. These inequalities are proven both holographically and for general quantum states. In addition, we use the cyclic entropy inequalities to derive a new holographic inequality for the entanglement wedge cross-section, and provide numerical evidence that the corresponding inequality for the entanglement of purification may be true in general. Finally, we use intuition from bit threads to extend the conjecture to holographic duals of suboptimal purifications.
Black hole as a multipartite entangler: multi-entropy in AdS3{}_33​/CFT2{}_22​
We study multipartite entanglement in typical pure states holographically dual to pure BTZ black holes, using multi-entropy and its ``genuine'' version. In the bulk, these quantities are computed by minimal geodesic networks (so-called Steiner trees). We find that at sufficiently high temperature, the genuine tripartite multi-entropy exhibits a volume-law scaling in sharp contrast to vacuum AdS3_33​, where the genuine contribution is universal and size-independent. Moreover, we find another phase: once one subsystem exceeds half of the total system, the leading genuine tripartite entanglement vanishes and reduces to that for global AdS3{}_33​. This transition is indeed consistent with recent arguments for distillable EPR pairs in tripartite Haar-random states. Motivated by finite-cutoff holography, we further study the radial cutoff dependence of multi-entropy and show that genuine multi-entropy acquires nontrivial size dependence even for the tripartite case in AdS3{}_33​. As a byproduct, we also observe an intriguing ``area-law'' contribution to multi-entropy that is relevant to vacuum AdS3{}_33​.