# The dual of non-extremal area: differential entropy in higher dimensions

## TL;DR

This paper explores the concept of differential entropy for non-extremal surfaces in higher-dimensional AdS spaces, proposing a new integral formula based on shape derivatives of entanglement entropy, with implications for c-functions in field theories.

## Contribution

It introduces a novel higher-dimensional differential entropy formula using shape derivatives, extending the 2D formalism and addressing UV divergence issues.

## Key findings

- Proves a no-go theorem for candidate higher-dimensional differential entropy expressions.
- Proposes a new integral formula based on shape derivatives of entanglement entropy.
- Tests the formula in AdS4 and suggests a candidate c-function for arbitrary dimensions.

## Abstract

The Ryu-Takayanagi formula relates entanglement entropy in a field theory to the area of extremal surfaces anchored to the boundary of a dual AdS space. It is interesting to ask if there is also an information theoretic interpretation of the areas of non-extremal surfaces that are not necessarily boundary-anchored. In general, the physics outside such surfaces is associated to observers restricted to a time-strip in the dual boundary field theory. When the latter is two-dimensional, it is known that the differential entropy associated to the strip computes the length of the dual bulk curve, and has an interpretation in terms of the information cost in Bell pairs of restoring correlations inaccessible to observers in the strip. A general realization of this formalism in higher dimensions is unknown. We first prove a no-go theorem eliminating candidate expressions for higher dimensional differential entropy based on entropic c-theorems. Then we propose a new formula in terms of an integral of shape derivatives of the entanglement entropy of ball shaped regions. Our proposal stems from the physical requirement that differential entropy must be locally finite and conformally invariant. Demanding cancellation of the well-known UV divergences of entanglement entropy in field theory guides us to our conjecture, which we test for surfaces in $AdS_4$. Our results suggest a candidate c-function for field theories in arbitrary dimensions.