Bounds on CFT correlations from the thermal partition function

TL;DR

This work builds upper bounds on the mutual information between disjoint spherical regions in a CFT vacuum using the $L_2$ modular nuclearity framework, linking these bounds to the finiteness of the thermal partition function and to bulk holographic phase transitions. By deriving two bounds—one in terms of the partition function $Z$ and a stronger one in terms of $\log Z(\beta)$—the authors connect boundary information measures to thermal and geometric data in AdS/CFT, particularly the Hawking-Page and entanglement-wedge transitions. They relate cross-ratio data of nested causal diamonds to the boundary temperature and angular velocity, derive high-temperature asymptotics for mutual information and partition functions, and show that MI transitions occur above the Hawking-Page temperature. Extending these ideas, they propose a conjectural boundary quantity that computes the minimal distance between entanglement wedges via heavy-state traces and Berry holonomy in the bulk, and outline a broader program to translate nuclearity data into bulk geometric probes, including non-AdS settings. The results offer a novel bridge between operator-algebraic constraints and bulk geometric diagnostics of entanglement structure in holographic theories.

Abstract

We discuss upper bounds on the mutual information for disjoint spherical regions of the CFT vacuum. To prove our bounds, we utilize the modular nuclearity condition, which is in turn related to finiteness of the thermal partition function of the CFT. Our bounds are satisfied by the conjectured geometric duals of these correlation measures in AdS/CFT, where they link the Hawking-Page phase transition with the entanglement wedge phase transition. We use these results to conjecture a new boundary theory quantity that computes the minimal distance between two general entanglement wedges.

Figures (5)

• Figure 1: Nested causal diamonds. Invariant cross-ratios are determined from the tips of the lightcones
• Figure 2: Figure adapted from Jokela_2019. Entanglement wedges corresponding to the two spherical boundary regions (equal-time solutions) associated with diamonds $\mathcal{A}$ and $\mathcal{B}'$. We have the two disconnected spheres solution for $I(\mathcal{A}:\mathcal{B}) =0$ and the deformed annulus solution for $I(\mathcal{A}:\mathcal{B}) >0$. The green arrow represents the minimum distance between the entanglement wedges of $\mathcal{A}$ and $\mathcal{B}$, which also coincides with the inverse temperature $\beta$ for the thermal partition function. The entanglement wedge cross-section, on the other hand, is the minimum area between the blue surfaces.
• Figure 3: Right figure: Hawking-Page vs mutual information transition in terms of inverse temperatures as a function of dimension of the CFT, starting at $d=1+1$. Due to the spherical symmetry of the solutions $(d>2)$ we work with, we have $\beta_L = \beta_R$. Left figure: for $d=2$, $\beta_L$ and $\beta_R$ are different, the green curve is a cartoon for Hagedorn transition, which happens at the highest temperature
• Figure 4: Mutual Information and $\log Z(\beta)$ as a function of temperature for various dimensions $(d=2,3,4,5)$. Both quantities are zero before their respective critical temperatures $T_{HP}$ and $T_{MI}$. We set the factors $L_{AdS}$ and $4G_{N}$ to 1.
• Figure 5: The parallel transport of an orthonormal frame along the minimal geodesic (green) between the entanglement wedges. The normal vectors are colored in red. Dashed vectors indicate time-reversed frame with respect to the entangling cut.