Entanglement Concentrates at Boundaries in Complex Systems

Researchers have identified a fundamental principle governing the distribution of multipartite entanglement within complex systems. Norihiro Iizuka of the Department of Physics, National Tsing Hua University, and Akihiro Miyata from the Yukawa Institute for Theoretical Physics, Kyoto University, demonstrate that genuine multipartite entanglement localises near junctions where subsystem boundaries converge. This work, conducted in collaboration between National Tsing Hua University, Taiwan, and Kyoto University, Japan, utilises Rényi-2 genuine multi-entropy to reveal a “junction law” governing entanglement behaviour in two-dimensional gapped free-fermion lattices. Establishing this law is significant because it provides insight into how entanglement, a key resource for quantum technologies, is constrained and distributed in realistic, physically relevant systems, potentially informing the development of more robust and efficient quantum devices.

Until recently, pinpointing where complex quantum links form within materials proved elusive. Now, physicists have discovered a fundamental principle governing how these connections concentrate at the boundaries between different regions. This ‘junction law’ explains how entanglement, a key resource for quantum technologies, is naturally confined and managed within these systems.

Scientists have long understood that entanglement is constrained by the locality of physical systems. For gapped quantum states, bipartite entanglement entropy obeys a well-established area law, indicating entanglement concentrates near the surfaces dividing regions. Recent work establishes a “junction law” for genuine multi-entropy, revealing that multipartite entanglement in gapped local systems is controlled and localized near junctions where subsystem boundaries meet.

Utilising the Rényi-2 genuine multi-entropy as a diagnostic tool, researchers investigated this behaviour in two-plus-one-dimensional gapped free-fermion lattices possessing a correlation length, ξ. Their findings demonstrate a universal scaling crossover for partitions containing a single junction. Specifically, the genuine multi-entropy grows with system size, L, when L is smaller than ξ, and then saturates to a constant value dependent on ξ as L becomes much larger than ξ, with only minor corrections.

Strikingly, for partitions lacking a junction, the genuine multi-entropy is exponentially suppressed as L increases, falling below the limits of numerical resolution once L markedly exceeds ξ. These patterns were observed for both tripartite and quadripartite entanglement scenarios. Further confirmation of this localization came from experiments translating the junction’s position while maintaining a fixed system size.

A geometric explanation of the junction law was also developed within the framework of holography. Altogether, these results demonstrate that genuine multipartite entanglement in this specific gapped free-fermion setting is confined to a region within a correlation-length neighbourhood of junctions. Understanding why junctions are special requires considering the limitations imposed by locality and finite correlation lengths.

In local states, connected correlations diminish exponentially with distance, meaning well-separated regions effectively factorize, except for small corrections proportional to the exponential of the distance divided by the correlation length. For a q-partition, any irreducible q-partite correlation must connect all q subsystems. If subsystem boundaries do not intersect, the minimum separation between them is proportional to L, leading to a suppression of these correlations as the exponential of negative L divided by ξ, and in the end causing the genuine multi-entropy to approach zero for sufficiently large L.

At junctions, all q subsystems approach within a distance of order ξ, allowing for a non-negligible contribution to the entanglement. Researchers employed the Rényi-2 genuine multi-entropy, a measure that isolates irreducible q-partite correlations by subtracting contributions from lower-partite entanglement. By studying this quantity in (2+1)-dimensional free-fermion lattice models, they found that for tripartitions and quadripartitions with a single junction, the genuine multi-entropy exhibits a universal crossover behaviour governed by the ratio L/ξ. For L smaller than ξ, the genuine multi-entropy increases with L, while for L much larger than ξ, it plateaus at a constant value determined by ξ and the local angles at the junction.

Lattice partitioning and junction formation for entanglement analysis

A correlation matrix method underpinned this work, allowing analysis of many-body ground states within the half-filled sector of a free-fermion lattice. Specifically, researchers considered Slater determinants obtained by filling negative-energy levels, controlling low-lying excitations via hole creation operators. All resultant states remained Gaussian, enabling the application of this efficient analytical technique.

System partitioning formed a central aspect of the methodology, with lattices divided into subsystems scaling with overall system size L. For tripartite entanglement studies (q = 3), the lattice was partitioned into three subsystems, A, B, and C, where subsystem A comprised two disconnected components separated by the others. In contrast, quadripartite (q = 4) configurations divided the lattice into four quadrants, A, B, C, and D, each occupying one section of the lattice.

Crucially, all subsystem boundaries intersected at a single junction, and these junction geometries were the primary focus. To provide a comparative baseline, partitions lacking a common intersection point, referred to as no-junction geometries, were also investigated. Correlation length ξ was determined from the exponential decay of equal-time single-particle correlators, extracting values from a shell-averaged correlator calculated around the junction.

This involved summing over sites within a small neighborhood J surrounding the junction and averaging over a shell of thickness ∆r, accounting for power-law corrections with parameter α. To minimise the influence of lattice anisotropy and precise junction location, the averaging procedure was carefully designed. The observable of interest, the Rényi-2 genuine multi-entropy GM(q)2, was then calculated for q = 3 and q = 4.

Evaluating higher-order multi-entropies presented computational challenges for larger systems. Therefore, a recursion relation was developed, expressing a q-partite Rényi-2 multi-entropy in terms of a (q −1)-partite one through canonical purification. This streamlined computation, generalizing existing q = 3 techniques and detailed in Appendices C and E. Gaussian states, both half-filled ground states and low-lying excitations, were explicitly constructed as described in Appendix F, providing the foundation for all numerical calculations.

Genuine multi-entropy scaling identifies localized entanglement at junction points

Genuine multi-entropy measurements reveal a distinct spatial organization of multipartite entanglement in gapped free-fermion systems. Specifically, for partitions containing a single junction, the Rényi-2 genuine multi-entropy, GM(q) 2, exhibits a universal scaling crossover with system size L and correlation length ξ. At smaller sizes, where L is less than or equal to ξ, GM(q) 2 increases proportionally with L.

Yet, once L becomes larger than ξ, it saturates to a constant value dependent on ξ, with corrections of order exponential decay, e−L/ξ. This behaviour contrasts sharply with partitions lacking a junction, where GM(q) 2 is exponentially suppressed as L increases and falls below the resolution of numerical calculations when L exceeds ξ. The research team observed this pattern consistently for both tripartite (q = 3) and quadripartite (q = 4) configurations.

Further confirmation of this localization came from translating the junction’s position at a fixed system size, demonstrating the entanglement’s confinement to the vicinity of the junction. At a junction, the boundaries of all q subsystems approach within a distance of order ξ, allowing for a non-negligible contribution to the multi-entropy. The magnitude of GM(q) 2 is directly linked to the correlation length, indicating that entanglement is effectively localized within a correlation length neighborhood.

Inside this correlation length neighborhood, the geometry of the junction dictates the constant value to which GM(q) 2 converges, providing a direct link between spatial arrangement and entanglement properties. These findings establish a “junction law” governing genuine multipartite entanglement in these gapped free-fermion settings.

Entanglement concentrates at material boundaries rather than dispersing throughout

Scientists have long sought to understand how entanglement scales up from pairs and triplets to larger, more complex systems. Recent work reveals a surprising constraint on multipartite entanglement within a specific class of materials, suggesting it isn’t freely distributed but instead concentrates around boundaries between different regions. This finding addresses a longstanding difficulty in reconciling theoretical predictions of widespread entanglement with the limited entanglement observed in realistic, imperfect materials.

For years, the expectation was that entanglement would diminish rapidly with increasing system size, yet this research demonstrates a degree of protection at these junctions. Understanding where entanglement resides is as important as knowing how much there is. Instead of being spread evenly, this research indicates entanglement is confined to a narrow zone surrounding junctions, points where different subsystems meet.

Once these junctions are identified, the amount of entanglement appears to stabilise, regardless of how many particles are involved, a result that contrasts with the expected exponential decay seen when junctions are absent. The precise nature of these boundaries and their influence on entanglement remain open to further investigation. At present, these observations are limited to a specific type of material: gapped free-fermion lattices.

Still, the implications extend beyond condensed matter physics, with potential connections to holographic duality, a theoretical framework linking gravity and quantum mechanics. The appearance of infrequent outliers in the data suggests the presence of subtle, yet unexplained, effects at the microscopic level. Future work should focus on identifying their origin and determining whether they represent a fundamental limitation or a pathway to even greater entanglement.

This work proposes a new way to think about entanglement localisation, shifting the focus from overall quantity to spatial distribution. Unlike previous approaches that treated entanglement as a bulk property, this research highlights the importance of geometry and boundaries. Inside the area of quantum computing, this could inform the design of more stable and scalable quantum devices, where protecting entanglement from environmental noise is a major challenge. Since the observed behaviour is tied to a correlation length, future studies might explore how manipulating this length affects entanglement distribution and durability.