⚛️ high-energy theory

Relative entropy of entanglement and tripartite minimal surface

This paper proposes and rigorously supports a holographic dual for the relative entropy of entanglement in a boundary tripartition, equating it to the difference between the areas of a minimal bulk tripartition surface and the standard minimal surface homologous to the subsystems, with proofs provided for both general random tensor networks and specific Haar random tensor configurations.

Original authors: Takato Mori, Beni Yoshida

Published 2026-10-01

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Takato Mori, Beni Yoshida

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to understand how the universe is woven together, physicists have long suspected that the fabric of spacetime itself emerges from a deeper layer of quantum connections. This idea, known as holography, suggests that the three-dimensional world we experience is a projection of information stored on a distant, two-dimensional boundary. For decades, the primary tool for exploring this connection has been entanglement, a strange quantum phenomenon where particles remain linked regardless of distance. Scientists have successfully mapped how the amount of entanglement between two regions relates to the area of a surface separating them in the hidden bulk geometry. However, a significant gap remained in our understanding: while we could measure simple connections between two parties, the rules for more complex, three-way relationships were far less clear. Specifically, a measure called the relative entropy of entanglement, which quantifies how distinguishable a quantum state is from a completely unconnected one, had no agreed-upon geometric shape to represent it.

A team of researchers has now proposed a concrete geometric answer to this missing piece, suggesting that the relative entropy of entanglement for three distinct regions is determined by a specific, minimal surface that divides the hidden bulk space into three parts. To test this idea, they turned to a simplified model of the universe built from random mathematical blocks, known as a tensor network. In this model, the complex quantum state is generated by connecting random mathematical tensors, much like a vast, intricate web of nodes and links. The researchers calculated the relative entropy of entanglement for a boundary split into three sections and found that the value is directly proportional to the difference between two areas: the area of the usual minimal surface connecting two of the sections, and the area of a new, more complex surface that separates all three sections. This new surface, which can form a Y-shaped junction in the center of the bulk, acts as the geometric dual to the information-theoretic distance between the quantum state and the set of unconnected states.

The team proved this relationship by constructing a specific type of mathematical state that is guaranteed to be unconnected, or separable, across the three regions. They showed that the "cost" to transform their random quantum state into this unconnected state is minimized when the transformation follows the path of this new three-way dividing surface. By rigorously analyzing networks containing one, two, and three random tensors, they demonstrated that this geometric rule holds true even when the internal structure of the network creates a non-trivial junction, resembling a Mercedes logo, where the three regions meet. Their work establishes that the most efficient way to disconnect the three regions is not simply by cutting the links between pairs, but by finding the optimal three-way split that minimizes the total area of the cut.

This discovery is significant because it links a difficult-to-calculate measure of quantum correlation directly to a tangible geometric object. In the past, calculating the relative entropy of entanglement was considered an extremely hard problem, often requiring complex optimizations that were difficult to solve even for simple systems. By showing that this value corresponds to the area of a minimal tripartition surface, the researchers provide a clear, visual way to understand how multipartite entanglement is organized in holographic theories. They also established that no other arrangement of unconnected states could produce a smaller value, confirming that this specific geometric surface is indeed the correct dual. The proof relied on a method of sequential optimization, where the researchers systematically tested different ways to break the network apart, effectively performing a local search to find the global minimum. This approach allowed them to rule out the possibility that a more complicated, finely tuned unconnected state could mimic the behavior of the random network with a lower cost.

The implications of this finding extend to the structure of spacetime itself. In the geometric picture, the minimal surface that separates three regions can take on different shapes depending on the size and arrangement of the boundary regions. As the boundary regions grow, the geometry of the hidden bulk undergoes distinct transitions. First, the regions become connected in a way that allows for mutual information. Then, at a larger size, the minimal three-way surface changes its topology, shifting from a disconnected configuration to a connected one with a central junction. Finally, at an even larger size, a different geometric feature, related to the ability to extract useful information from the system, undergoes its own transition. These three transitions occur at different points, revealing a rich hierarchy of geometric changes that were previously hidden. This suggests that the emergence of spacetime is not a single event but a layered process, where different types of quantum correlations become dominant at different scales.

The researchers also explored how their findings relate to other measures of entanglement, such as the entanglement of formation, which describes the resources needed to create a specific quantum state. In certain regimes, their results confirm that the entanglement of formation is indeed given by the area of a specific cross-section of the entanglement wedge, a proposal that had been debated in the literature. However, they also identified a regime where the relative entropy of entanglement is strictly smaller than the entanglement of formation, indicating that these two measures, while related, capture different aspects of the quantum connection. This distinction is crucial for understanding the operational limits of quantum information processing in holographic systems.

Ultimately, this work provides a new lens through which to view the quantum foundations of gravity. By identifying the minimal tripartition surface as the geometric dual of the relative entropy of entanglement, the researchers have bridged a gap between abstract information theory and concrete geometry. The result is a clearer picture of how complex, multi-party quantum correlations are encoded in the shape of spacetime. While the model used is a simplified version of the real universe, the principles uncovered here offer a robust framework for understanding how the intricate web of quantum entanglement gives rise to the smooth, continuous geometry we observe. The discovery of the Y-shaped junction as a fundamental building block of this geometry suggests that the universe's structure is more nuanced than previously thought, with different types of connections forming the scaffolding of reality at different levels.

Technical Summary: Relative Entropy of Entanglement and Tripartite Minimal Surface

Problem Statement
The paper addresses the challenge of identifying a holographic dual for the relative entropy of entanglement (REE), $E_R(\rho_{AB})$ , a fundamental entanglement measure defined as the minimum relative entropy between a state $\rho_{AB}$ and the set of separable states across the bipartition $A:B$ . While holographic duals for entanglement entropy (Ryu-Takayanagi formula) and related quantities like logarithmic negativity are well-established, measures involving optimization over separable states (such as $E_R$ , entanglement of formation $E_F$ , and entanglement of purification $E_P$ ) have remained analytically intractable in holographic settings. The authors propose a specific geometric dual for $E_R$ in the context of a boundary tripartition $A:B:C$ .

Methodology
The authors employ Haar random tensor networks (RTNs) as a controlled toy model for holography. The methodology proceeds in two main phases: establishing an upper bound and establishing a matching lower bound.

Upper Bound Construction:
- The authors construct a specific family of separable states $\sigma_{ABC}^\Gamma$ by "dephasing" all maximally entangled bonds crossing a chosen bulk tripartition surface $\Gamma$ . This dephasing converts entangled EPR pairs into classical mixtures in the Schmidt basis, rendering the state separable across $A:B:C$ .
- They evaluate the relative entropy $D(\rho_{AB} \| \sigma_{AB}^\Gamma)$ using the replica method. This involves calculating the partition function of a spin model on the tensor network graph.
- The dephasing of bonds on $\Gamma$ modifies the interaction terms in the replica model. The authors demonstrate that the resulting optimization problem reduces to a binary spin model they term the Ising–OR model. In this model, undephased edges carry standard Ising domain-wall costs, while dephased edges on $\Gamma$ carry an "OR" interaction cost (nonzero cost if at least one endpoint is in the excited state).
- By analyzing the ground state energy of this Ising–OR model, they prove that the relative entropy is minimized when $\Gamma$ is the minimal bulk tripartition surface ( $\Gamma_{\min}$ ), which separates the bulk regions associated with boundary regions $A$ , $B$ , and $C$ .
Lower Bound Construction:
- To prove the lower bound, the authors utilize a known inequality relating $E_R$ to the maximal product-state overlap $\Lambda(\Psi_{A:B:C}) = \max_{\alpha, \beta, \gamma} |\langle \alpha, \beta, \gamma | \Psi \rangle|^2$ . Specifically, $E_R(\rho_{AB}) \geq -S(\rho_{AB}) - \log_2 \Lambda$ .
- The core technical challenge is to show that the maximal product overlap is achieved (at leading order) by a product state constructed from the minimal tripartition surface $\Gamma_{\min}$ , where bonds crossing $\Gamma_{\min}$ are replaced by fixed product states (e.g., $|0\rangle$ ).
- The authors rigorously prove this conjecture for tensor networks consisting of one, two, and three independent Haar random tensors.
- The proof technique involves sequential optimization over the boundary product states ( $\alpha, \beta, \gamma$ ). By optimizing over one subsystem at a time, the problem reduces to maximal product overlap problems for the remaining sub-networks. This sequential process is shown to effectively perform a "local search" for the minimal tripartition surface.
- The probabilistic arguments rely on $\epsilon$ -net discretization of the continuous product state space and Gaussian concentration inequalities to bound the overlap for fixed tensors, followed by union bounds over the net.

Key Results
The primary result is the proposal and rigorous verification (for small networks) of the following holographic dual for the relative entropy of entanglement in the large bond-dimension limit ( $n \to \infty$ ):

$E_R(\rho_{AB}) = \frac{1}{4G_N} \left[ \text{Area}(\Gamma_{\min}) - \text{Area}(\gamma_{AB}) \right] + O(1)$

Where:

$\gamma_{AB}$ is the standard Ryu-Takayanagi minimal surface homologous to the boundary region $AB$ .
$\Gamma_{\min}$ is the minimal bulk tripartition surface that separates the three boundary regions $A, B,$ and $C$ . This surface can form nontrivial tri-junctions (resembling a "Mercedes" logo) in the bulk, distinct from the connected RT surface.
The term $\frac{1}{4G_N}\text{Area}(\gamma_{AB})$ corresponds to the von Neumann entropy $S(\rho_{AB})$ . Thus, the proposal can be equivalently stated as the minimization of $-\text{Tr}[\rho_{AB} \log \sigma_{AB}]$ over separable states being equal to $\frac{1}{4G_N}\text{Area}(\Gamma_{\min})$ .

Specific Findings in Tensor Networks:

One Tensor: The minimal tripartition surface corresponds to the smallest cut separating the three legs, and the result matches the known $E_R$ for Haar random states.
Two Tensors: The authors identify four candidate tripartition surfaces and show the minimal one dictates the $E_R$ .
Three Tensors (Mercedes Regime): In the regime where the internal tri-junction surface is smaller than any boundary-pair cut, the minimal tripartition surface forms a nontrivial tri-junction. The authors prove that the maximal product overlap is dominated by the state associated with this tri-junction, confirming the formula.

Geometric Transitions in AdS $_3$
The paper analyzes the behavior of these surfaces in vacuum AdS $_3$ with two symmetric boundary intervals $A$ and $B$ . They identify a hierarchy of three distinct geometric phase transitions as the interval size $\theta$ increases:

$\theta_{\text{wedge}}$ : The entanglement wedge of $AB$ becomes connected (mutual information becomes non-zero).
$\theta_{\Gamma}$ : The minimal tripartition surface $\Gamma_{\min}$ transitions from a disconnected configuration to a connected one with a tri-junction. At this point, $E_R$ deviates from the mutual information $I(A:B)$ .
$\theta_{\text{cross}}$ : The entanglement wedge cross section (dual to $E_P$ or $E_F$ ) undergoes a transition.
The paper establishes the ordering $\theta_{\text{wedge}} < \theta_{\Gamma} < \theta_{\text{cross}}$ .

Significance and Claims
The authors claim that this work provides the first rigorous analytic derivation of the holographic dual for the relative entropy of entanglement in a controlled setting.

Geometric Insight: The result links a genuine entanglement measure (defined by optimization over separable states) to a specific geometric object: the minimal tripartition surface. This suggests that multipartite entanglement structures in holographic states are encoded in the topology of bulk tri-junctions.
Methodological Contribution: The paper introduces a recursive proof technique based on sequential optimization and $\epsilon$ -net arguments, which successfully handles the variational nature of $E_R$ in random tensor networks.
Implications for $E_F$ : The authors note that in specific parameter regimes of the two-tensor network, their result for $E_R$ implies that the entanglement of formation $E_F$ matches the area of the entanglement wedge cross section, providing a non-trivial check on the $E_F$ holographic proposal.
Limitations: The authors explicitly state that their lower-bound proof is currently restricted to networks with one, two, and three tensors. Extending this to general random tensor networks with arbitrary topology remains an open technical challenge due to the combinatorial complexity of optimization orders. Furthermore, a continuum gravity interpretation of the "dephased" state and the associated bulk defect remains an open problem.

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