Entanglement inequalities, black holes and the architecture of typical states

TL;DR

This work analyzes typical pure states in large-$N$ holographic CFTs and shows that two energy- and charge-determined scales, $L_{ m UV}$ and $L_{ m IR}$ with $L_{ m IR}>L_{ m UV}$, organize the state into corona, buffer, and interior regions in the BTZ geometry. Saturation of the Araki–Lieb inequality for boundary intervals up to $L_{ m UV}$ enforces an effective factorization of the buffer and a purification relation between the corona and UV sectors, yielding a universal part of the UV factor determined solely by energy $E$ and momentum $J$ up to exponentially small corrections $O(e^{-S_<})$. The paper derives an explicit typical-state ansatz $ig|\psi\big>=(\sum_i\sqrt{p_i}\,ig|\tilde{\chi}_i^{\mathfrak{I}}\big>\otimes\big|\chi_i^{\mathfrak{L}}\big>\big)\otimes|\phi^{\mathfrak{KC}}\rangle+O(e^{-S_<})$ and shows the UV factor is state-independent while the IR sector carries $S_{BH}$, with corrections and dynamical implications discussed. These results imply a mechanism by which AdS black holes decouple from the asymptotic region via a corona, and point to ETH-like behavior and potential quantum-hair structures near the horizon, with avenues for generalization to higher dimensions and dynamical setups such as BTZ-Vaidya and tensor-network models.

Abstract

Using holographic realizations of the Araki-Lieb (AL) inequality, we show that typical pure states in large-N holographic conformal field theories (CFTs) possess two characteristic length scales determined solely by energy and conserved charges: a microscopic L_UV and an infrared L_IR > L_UV. Degrees of freedom between these scales effectively factorize -- one purifying the ultraviolet (scales < L_UV) and the other the infrared sector (scales > L_IR). Remarkably, the pure-state factor including the ultraviolet sector is determined solely by energy and conserved charges up to exponentially suppressed corrections. Our results imply that all black holes in anti-de Sitter space can be isolated from an asymptotic region, the "corona", formed by the inclusion of entanglement wedges for which the AL inequality is saturated, and that an effective factorization emerges in the buffer region between the corona and the outer horizon.

Figures (2)

• Figure 1: The bulk geodesics ending on the boundary interval $R$ (red) are $\mathcal{C}_1$ (green) and $\mathcal{C}_2$ (dashed red). $\mathcal{C}_1\cup \mathcal{H}$ and $\mathcal{C}_2$ are homologous to $R$ with $\mathcal{H}$ the outer horizon (dashed magenta). $\mathcal{C}_1$ and $\mathcal{C}_2\cup \mathcal{H}$ are homologous to $\overline{R}$ (black), the complement of $R$. Here the BTZ black hole shown with compactified radial coordinate $r\to \cot\rho$ has $\mu_+=\mu_-=0.25$ (AdS radius $\ell =1$).
• Figure 2: Two non-overlapping intervals $R_1$ and $R_2$ at the boundary are shown above with two non-vanishing gaps $\mathbf{L}_{g1}$ and $\mathbf{L}_{g2}$ between them. The mutual information between $R_1$ and $R_2$ is non-vanishing if the bulk entanglement wedge is connected (the dotted region) bounded by the dotted red bulk geodesics and the outer horizon instead of the union of the two disconnected regions bounded by the green bulk geodesics. Here the BTZ black hole shown with compactified radial coordinate $r\to \cot\rho$ has $\mu_+=\mu_-=0.25$ (AdS radius $\ell =1$).