Reflected entropy in random tensor networks

TL;DR

The paper analyzes reflected entropy in random tensor networks, revealing a holography-like duality to the entanglement wedge cross section and exposing nonperturbative smoothing of Page-like transitions. By applying a two-parameter replica construction and a Schwinger-Dyson resolvent approach, it derives the full reflected entanglement spectrum for a single tensor, showing a two-component structure with a dominant connected-sector peak and a disconnected-sector pole, controlled by non-crossing permutations and $q$-Catalan combinatorics. It introduces a robust analytic continuation prescription for $(m,n)$-Rényi reflected entropies that avoids problematic order-of-limits issues, and extends the analysis to hyperbolic RTNs where a richer phase structure with an $X$-pocket arises. The authors propose an effective description of the canonical purification as a superposition of tensor-network states with an emergent area operator, which explains the non-flat spectrum and provides a quantum-error-correction perspective on the EW geometry. Overall, the work connects RTN entanglement structures to holographic expectations, demonstrates nontrivial tripartite entanglement, and offers a concrete framework to build gravity-inspired geometries from RTNs via iterated canonical purification.

Abstract

In holographic theories, the reflected entropy has been shown to be dual to the area of the entanglement wedge cross section. We study the same problem in random tensor networks demonstrating an equivalent duality. For a single random tensor we analyze the important non-perturbative effects that smooth out the discontinuity in the reflected entropy across the Page phase transition. By summing over all such effects, we obtain the reflected entanglement spectrum analytically, which agrees well with numerical studies. This motivates a prescription for the analytic continuation required in computing the reflected entropy and its Rényi generalization which resolves an order of limits issue previously identified in the literature. We apply this prescription to hyperbolic tensor networks and find answers consistent with holographic expectations. In particular, the random tensor network has the same non-trivial tripartite entanglement structure expected from holographic states. We furthermore show that the reflected Rényi spectrum is not flat, in sharp contrast to the usual Rényi spectrum of these networks. We argue that the various distinct contributions to the reflected entanglement spectrum can be organized into approximate superselection sectors. We interpret this as resulting from an effective description of the canonically purified state as a superposition of distinct tensor network states. Each network is constructed by doubling and gluing various candidate entanglement wedges of the original network. The superselection sectors are labelled by the different cross-sectional areas of these candidate entanglement wedges.

Figures (25)

• Figure 1: (a) A spatial slice of AdS with $A$ and $B$ chosen to be two intervals. The figure depicts the entanglement wedge of $AB$ (gray), the entanglement wedge of $C$ (green), the RT surface $\gamma_{AB}$ and the entanglement wedge cross section $\Gamma_{A:B}$. (b) A spatial slice of the proposed holographic dual to the canonically purified state $\ket{\sqrt{\rho_{AB}}}$. The RT surface for $A\cup A^*$ is given by a doubled copy of the entanglement wedge cross section.
• Figure 2: "Page curve" of the reflected entropy, for a single tripartite random tensor with bond dimensions $\chi_A, \chi_B, \chi_C$. The blue curve corresponds to the infinite bond dimension limit, $\chi \to \infty$. The other curves correspond to large but finite bond dimension, correctly limiting to the blue curve as the bond dimension increases.
• Figure 3: The reflected spectrum in the single tensor model has two superselection sectors corresponding to the disconnected and connected phase respectively.
• Figure 4: (a) We tile the hyperbolic disk isometrically by a graph. Each blue circle represents a random tensor and the connecting edges indicates contraction of the tensors. This tensor network defines a state on the Hilbert space spanned by the dangling legs on the boundary. (b) The Ising domain wall that arises in the calculation of Rényi entropies of region $A$ in the RTN. Minimizing the length of the domain wall gives rise to an entropy formula that corresponds to the RT formula in AdS/CFT.
• Figure 5: A graphical representation of $g_A$ and $g_B$. The individual circles each represents the $m$ replicas of the original tensor and each of the circles is further replicated $n$ times. Going clockwise in each circle increases the $m$ replica number and going to the next circle on the right increases the $n$ replica number. A cycle of the permutation is represented by a closed directed loop. The element $g_A$ can be thought of as cutting open the $m$-circles of $g_B$ from the middle, shifting the bottom half cyclically in the $n$ direction (as the red arrows) and then gluing them back together.

Theorems & Definitions (39)

• Lemma 1
• proof
• Lemma 2
• proof
• Definition A.1: Non-crossing permutations
• Theorem 3
• Lemma 4
• proof
• proof
• Corollary 5