The Gravity Dual of Renyi Entropy

TL;DR

This work generalizes holographic entanglement entropy to Renyi entropies by introducing a cosmic brane prescription in the bulk whose area computes the Renyi entropy via a one-parameter area law. The authors derive the area-law relation through a holographic replica trick, relate a natural Renyi entropy to a bulk free-energy derivative, and show how to include higher-derivative and quantum corrections. They demonstrate practicality by computing the mutual Renyi information for two disks and deriving results that connect to CFT data such as the central charge, including checks in $d=2$. This framework provides an efficient and conceptually meaningful route to studying Renyi entropies in strongly coupled systems and deepens the link between entanglement structures and gravity.

Abstract

A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge/gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Renyi entropies. Here we show that all Renyi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Applying this we provide the first holographic calculation of mutual Renyi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Renyi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.

Figures (1)

• Figure 1: Two examples of cosmic branes as the gravity dual of Rényi entropy. The entangling region $A$ (red) is either (a) connected or (b) disconnected. In each case a strongly coupled QFT on the plane (blue) has a holographic dual description in terms of a gravitational theory in the bulk spacetime above the plane. The cosmic brane $C^{(n)}$ (green) is anchored at the entangling surface $\partial A$ (brown) and backreacts on the bulk geometry $\hat{B}_n$, although the backreaction is difficult to show in the figure. The Rényi entropy is determined by the area of the cosmic brane. As the Rényi index $n$ approaches $1$ the cosmic brane become a non-backreacting minimal surface, reproducing the Ryu--Takayanagi prescription for entanglement entropy.