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arXiv:2602.16331 · Physical review. D/Physical review. D.

Where Multipartite Entanglement Localizes: The Junction Law for Genuine Multi-Entropy

Norihiro Iizuka, Akihiro Miyata

hep-thcond-mat.stat-mechquant-phSubmitted Feb 18, 2026• Announced Feb 19, 2026• 1 citation
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Abstract

We uncover a "junction law" for genuine multipartite entanglement, suggesting that in gapped local systems multipartite entanglement is controlled and effectively localized near junctions where subsystem boundaries meet. Using the Rényi-2 genuine multi-entropy $\mathrm{GM}^{(\mathtt{q})}_2$ as a diagnostic of genuine $\mathtt{q}$-partite entanglement, we establish this behavior in $(2+1)$-dimensional gapped free-fermion lattices with correlation length $ξ$. For partitions with a single junction, $\mathrm{GM}^{(\mathtt{q})}_2$ exhibits a universal scaling crossover in $L/ξ$, growing for $L\llξ$ and saturating to a $ξ$-dependent constant for $L\ggξ$, up to $\mathcal{O}(e^{-L/ξ})$ corrections. In sharp contrast, for partitions without a junction, $\mathrm{GM}^{(\mathtt{q})}_2$ is exponentially suppressed in $L/ξ$ and drops below numerical resolution once $L\ggξ$. We observe the same pattern for $\mathtt{q}=3$ (tripartite) and $\mathtt{q}=4$ (quadripartite) cases, and further corroborate this localization by translating the junction at fixed system size. We also provide a geometric explanation of the junction law in holography. Altogether, these results show that in this gapped free-fermion setting genuine multipartite entanglement is localized within a correlation-length neighborhood of junctions.

Comments
16 pages, 15 figures

Related papers

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We investigate how the junction law for multipartite entanglement is realized in confining holographic backgrounds, using genuine multi-entropy (GM) as our main diagnostic. We first study an AdS$_3$ hard-wall toy model as an analytic benchmark, where multi-way cuts and junction geometries can be analyzed explicitly. In this setup, we classify the relevant saddles, determine the dominant phases, and show that the genuinely multipartite contribution diagnosed by GM is localized near the junction. We also examine how this structure depends on subsystem sizes, asymmetry, and the confinement scale, including phase transitions between competing saddles. We then move beyond the hard-wall benchmark to smooth confining geometries, focusing on the D4-soliton and D3-soliton backgrounds and formulating the corresponding framework also for the Klebanov--Strassler background. In the smooth-cap examples, we find that the junction picture persists, while the detailed phase structure differs from the hard-wall case: in particular, the hard-wall plateau does not survive, and GM instead decreases monotonically and vanishes at a finite critical scale. We also find that the short-distance behavior is background-dependent, with $\mathrm{GM}^{(3)}\sim L^{-4}$ in the D4-soliton background, $\mathrm{GM}^{(3)}\sim L^{-2}$ in the D3-soliton background, and $\mathrm{GM}^{(3)}\sim L^{-2}\cdot (\log L)^{2}$ in the Klebanov--Strassler background. These results clarify which features of the junction-law picture are robust in confining holography and which features of the phase structure and short-distance scaling are background-dependent.

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Can quantum entanglement continue to grow even after the entanglement entropies of subsystems have saturated? We address this question using genuine multi-entropy, which probes multi-partite entanglement structure beyond bipartite entanglement. In a two-sided BTZ black hole, this quantity grows linearly with time after every subsystem entropy of a fixed tripartition has saturated, extending the interval in which entanglement detects Einstein-Rosen bridge growth. The result refines the ER=EPR relation by identifying multi-partite structure invisible to ordinary entanglement entropies. Some spin-chain models and the spatially partitioned Sachdev-Ye-Kitaev (SYK) model exhibit distinct transients, while the late-time plateau values in holography, the SYK model, and Haar-random states share a universal behaviour.

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We uncover a "junction law" for genuine multipartite entanglement, suggesting that in gapped local systems multipartite entanglement is controlled and effectively localized near junctions where subsystem boundaries meet. Using the Rényi-2 genuine multi-entropy GM2(q)\mathrm{GM}^{(\mathtt{q})}_2GM2(q)​ as a diagnostic of genuine q\mathtt{q}q-partite entanglement, we establish this behavior in (2+1)(2+1)(2+1)-dimensional gapped free-fermion lattices with correlation length ξξξ. For partitions with a single junction, GM2(q)\mathrm{GM}^{(\mathtt{q})}_2GM2(q)​ exhibits a universal scaling crossover in L/ξL/ξL/ξ, growing for L≪ξL\llξL≪ξ and saturating to a ξξξ-dependent constant for L≫ξL\ggξL≫ξ, up to O(e−L/ξ)\mathcal{O}(e^{-L/ξ})O(e−L/ξ) corrections. In sharp contrast, for partitions without a junction, GM2(q)\mathrm{GM}^{(\mathtt{q})}_2GM2(q)​ is exponentially suppressed in L/ξL/ξL/ξ and drops below numerical resolution once L≫ξL\ggξL≫ξ. We observe the same pattern for q=3\mathtt{q}=3q=3 (tripartite) and q=4\mathtt{q}=4q=4 (quadripartite) cases, and further corroborate this localization by translating the junction at fixed system size. We also provide a geometric explanation of the junction law in holography. Altogether, these results show that in this gapped free-fermion setting genuine multipartite entanglement is localized within a correlation-length neighborhood of junctions.
We investigate how the junction law for multipartite entanglement is realized in confining holographic backgrounds, using genuine multi-entropy (GM) as our main diagnostic. We first study an AdS3_33​ hard-wall toy model as an analytic benchmark, where multi-way cuts and junction geometries can be analyzed explicitly. In this setup, we classify the relevant saddles, determine the dominant phases, and show that the genuinely multipartite contribution diagnosed by GM is localized near the junction. We also examine how this structure depends on subsystem sizes, asymmetry, and the confinement scale, including phase transitions between competing saddles. We then move beyond the hard-wall benchmark to smooth confining geometries, focusing on the D4-soliton and D3-soliton backgrounds and formulating the corresponding framework also for the Klebanov--Strassler background. In the smooth-cap examples, we find that the junction picture persists, while the detailed phase structure differs from the hard-wall case: in particular, the hard-wall plateau does not survive, and GM instead decreases monotonically and vanishes at a finite critical scale. We also find that the short-distance behavior is background-dependent, with GM(3)∼L−4\mathrm{GM}^{(3)}\sim L^{-4}GM(3)∼L−4 in the D4-soliton background, GM(3)∼L−2\mathrm{GM}^{(3)}\sim L^{-2}GM(3)∼L−2 in the D3-soliton background, and GM(3)∼L−2⋅(log⁡L)2\mathrm{GM}^{(3)}\sim L^{-2}\cdot (\log L)^{2}GM(3)∼L−2⋅(logL)2 in the Klebanov--Strassler background. These results clarify which features of the junction-law picture are robust in confining holography and which features of the phase structure and short-distance scaling are background-dependent.
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​.
We study genuine multi-entropy as a diagnostic of multipartite entanglement in the toric code, which provides a controlled setting for probing multipartite structures in topologically ordered states. Our main question is whether genuine multi-entropy captures information that is not reducible to conventional lower-party entropic data, such as topological entanglement entropy. We first analyze toric-code ground states that admit a stabilizer-state description, where the relevant quantities can be evaluated exactly. We conjecture that, for stabilizer states and q≥4\mathtt{q}\ge4q≥4, the q\mathtt{q}q-partite genuine multi-entropy at replica index n<qn<\mathtt{q}n<q collapses to a linear combination of multi-entropies involving at most q−2\mathtt{q}-2q−2 parties. We establish this pattern explicitly for q=4\mathtt{q}=4q=4 in the toric code stabilizer sector: for n=2,3n=2,3n=2,3, the genuine multi-entropy is proportional to the tripartite information I3I_3I3​ and, for the Kitaev-Preskill partition, contains no independent genuine four-partite information beyond that captured by the topological entanglement entropy. At n=4n=4n=4, however, this reduction breaks down: the genuine multi-entropy is no longer proportional to I3I_3I3​, but remains a topological invariant of the toric-code stabilizer ground states under Kitaev-Preskill-preserving deformations that do not change whether the numbers of vertex- and plaquette-type junctions coincide. For generic non-stabilizer superpositions within the ground-state manifold and for coherent superpositions of local excitations, the low-nnn reduction also fails. These results show that genuine multi-entropy probes multipartite entanglement structure beyond the tripartite information, and hence beyond the topological entanglement entropy in the Kitaev-Preskill partition, whereas for stabilizer states at low replica index it reduces to lower-partite entropic data.
We show that the Rényi tripartite information I3(α)I_3^{(α)}I3(α)​ of free fermions exhibits a qualitatively ααα-dependent scaling at small Fermi momentum, in sharp contrast to bipartite entropy where only the prefactor changes. In the rank-1 regime (z=kFw≪1z = k_F w \ll 1z=kF​w≪1), I3(α)I_3^{(α)}I3(α)​ receives contributions from two competing channels -- a fractional-moment channel ∼zα\sim z^α∼zα (active for non-integer ααα) and a polynomial channel ∼zm\sim z^m∼zm from the first nonvanishing inclusion-exclusion moment σmσ_mσm​ -- yielding the scaling exponent βm(α)=min⁡(α,m)β_m(α) = \min(α, m)βm​(α)=min(α,m) for mmm-partite information of mmm adjacent strips. Integer Rényi indices α=2,3,…α= 2, 3, \ldotsα=2,3,… are anomalous: the fractional channel closes and the exponent jumps to mmm or higher. A direct consequence is a replica obstruction: Im(n)/Im(1)∼zm−1→0I_m^{(n)}/I_m^{(1)} \sim z^{m-1} \to 0Im(n)​/Im(1)​∼zm−1→0 for all integer n≥2n \geq 2n≥2, so the leading von Neumann signal cannot be reconstructed from integer Rényi data at the level of leading scaling -- a situation with no bipartite analog. Conversely, negativity-based measures (α=1/2α= 1/2α=1/2) give a 20×20\times20× enhanced signal compared to von Neumann. We derive the underlying product formula for the coefficient c(wA,wB,wD)c(w_A, w_B, w_D)c(wA​,wB​,wD​), prove an mmm-partite generating function for the inclusion-exclusion moments, and verify all results numerically to high precision.
Entanglement entropy scaling links geometry to quantum correlations and diagnoses quantum criticality. Fractal lattices extend this link to noninteger dimensions, where boundary and bulk dimensions are independent. It remains open which dimension governs entropy scaling and how strongly gap closing can enhance it. Nonlocal occupied-state projectors and the lack of translation symmetry obstruct standard methods. We overcome these obstacles and classify free-fermion entropy scaling with bounds set by the hopping across the subsystem boundary: an upper bound from its rank and the spectral gap, and a lower bound from how much it lowers the ground-state energy. With only negative-energy states filled and this energy lowering proportional to the boundary, the entanglement entropy obeys the boundary law SX=Θ(LXdb)S_X=Θ\left(L_X^{d_{\mathrm b}}\right)SX​=Θ(LXdb​​) under a uniform gap and satisfies LXdb≲SX≲LXdbln⁡LXL_X^{d_{\mathrm b}}\lesssim S_X\lesssim L_X^{d_{\mathrm b}}\ln L_XLXdb​​≲SX​≲LXdb​​lnLX​ when the gap closes no faster than algebraically. Here LXL_XLX​ is the subsystem linear size and dbd_{\mathrm b}db​ the boundary dimension. Half occupation of the chiral zero modes gives a mixed-state entropy SXmixS_X^{\mathrm{mix}}SXmix​ with max⁡{SX,zXln⁡2}⩽SXmix⩽SX+zXln⁡2\max\left\{S_X,z_X\ln2\right\}\leqslant S_X^{\mathrm{mix}}\leqslant S_X+z_X\ln2max{SX​,zX​ln2}⩽SXmix​⩽SX​+zX​ln2, where zXz_XzX​ is the local zero-mode weight; an extensive zXz_XzX​ thus makes SXmixS_X^{\mathrm{mix}}SXmix​ follow the bulk dimension.
Can quantum entanglement continue to grow even after the entanglement entropies of subsystems have saturated? We address this question using genuine multi-entropy, which probes multi-partite entanglement structure beyond bipartite entanglement. In a two-sided BTZ black hole, this quantity grows linearly with time after every subsystem entropy of a fixed tripartition has saturated, extending the interval in which entanglement detects Einstein-Rosen bridge growth. The result refines the ER=EPR relation by identifying multi-partite structure invisible to ordinary entanglement entropies. Some spin-chain models and the spatially partitioned Sachdev-Ye-Kitaev (SYK) model exhibit distinct transients, while the late-time plateau values in holography, the SYK model, and Haar-random states share a universal behaviour.