The gravity dual of Rényi entropy

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 Rényi entropies. Here we show that all Rényi 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 Rényi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Rényi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.

Introduction

One of the most remarkable discoveries in fundamental physics is that black holes carry entropy with an amount equal to a quarter of the horizon area in Planck units1,2,3:

Here GN denotes Newton’s constant. The property that gravitational entropies satisfy an area law, however, is not restricted to black holes. It was elegantly generalized by Ryu and Takayanagi4 in the context of gauge/gravity duality, an exact equivalence between certain strongly coupled quantum field theories (QFTs) and weakly coupled gravitational theories in one higher dimensions5,6,7. In this context, they proposed that the von Neumann entropy, also known as the entanglement entropy, of any spatial region A at a moment of time-reflection symmetry in the boundary QFT is determined by the area of a codimension-2 minimal surface in the dual spacetime:

The minimal surface is constrained to be at a moment of time-reflection symmetry in the bulk and homologous to the entangling region A. In particular, this means that the minimal surface is anchored at the entangling surface ∂A. Here we follow the standard terminology of referring to the dual spacetime in which the gravitational theory lives as the bulk, and identify the spacetime in which the QFT lives with the asymptotic boundary of the bulk spacetime.

This elegant prescription for holographic entanglement entropy was initially proven in the special case of spherical entangling regions in the vacuum state of a conformal field theory (CFT), by employing a U(1) symmetry to map the problem to one of finding the thermal entropy of the CFT on a hyperboloid8. The latter problem was then solved by gauge/gravity duality, which tells us that the thermal state of the CFT is dual to a hyperbolic black hole in the bulk, and the thermal entropy is given by the area of the black hole horizon according to equation (1). Not surprisingly, this horizon is mapped back to the Ryu–Takayanagi minimal surface in the original problem.

In more general cases, there is no U(1) symmetry to facilitate such a derivation. Nonetheless, Lewkowycz and Maldacena9 overcame this difficulty and showed that the Ryu–Takayanagi prescription (2) follows from gauge/gravity duality, by applying the replica trick and generalizing the Euclidean method developed in refs 2, 10 of calculating gravitational entropies to cases without a U(1) symmetry. Similar techniques were used in refs 11, 12, 13 to generalize the Ryu–Takayanagi prescription to cases where the bulk theory involves higher derivative gravity, and used in ref. 14 to find quantum corrections to the prescription.

Discussions of area laws such as equations (1) and (2) have so far been constrained to the von Neumann entropy. The main goal of this paper is to generalize these laws to Rényi entropies15, which are labelled by an index n and defined in terms of the density matrix ρ of the entangling region as

In the n→1 limit, we recover the von Neumann entropy S≡−Tr(ρ ln ρ).

Although Rényi entropies are often introduced as a one-parameter generalization of the von Neumann entropy, they are much easier to experimentally measure and numerically study (see, for example, ref. 16for recent progress in measurements). They also contain richer physical information about the entanglement structure of a quantum state. In particular, the knowledge of Rényi entropies for all n allows one to determine the whole entanglement spectrum (the set of eigenvalues ofρ). Rényi entropies have been extensively studied by numerical methods17, in spin chains18, in tensor networks19, in free field theories20, in two-dimensional CFTs21,22,23,24,25,26,27,28 or higher29,30,31,32,33,34,35,36,37, and in the context of gauge/gravity duality38,39,40,41,42,43,44,45. They have also been generalized to charged46 and supersymmetric cases47,48,49.

In this paper, we show that all Rényi entropies satisfy a similar area law in holographic theories. As we will see, the gravity dual of Rényi entropy is a cosmic brane. This provides a simple geometric prescription for holographic Rényi entropies, generalizing the Ryu–Takayanagi prescription (2) for entanglement entropy. It is important to distinguish our area law from a universal feature of entanglement and Rényi entropies, which is that the most ultraviolet divergent part of these entropies is proportional to the area of the entangling surface in any QFT50,51 (see also ref. 52 and references therein for area-law bounds on entanglement entropy). Our area law reproduces and goes beyond this universal feature in that it is an exact prescription that applies to both finite and divergent parts of Rényi entropies, and it serves as a useful criterion in distinguishing theories with a gravitational dual from those without.

Results

An area-law prescription for Rényi entropy

Our main result is that a derivative of holographic Rényi entropy Sn with respect to the Rényi index n satisfies an area law. It is given by a quarter of the area in Planck units of a bulk codimension-2 cosmic brane homologous to the entangling region:

Here the subscript n on the cosmic brane denotes that its brane tension as a function of n is given by

As shown in Fig. 1, the cosmic brane is analogous to the Ryu–Takayanagi minimal surface, except that it backreacts on the ambient geometry by creating a conical deficit angle53

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 ∂A (brown) and backreacts on the bulk geometry , 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.