Deriving covariant holographic entanglement

TL;DR

This paper provides a gravitational derivation supporting the covariant holographic entanglement entropy proposal, linking extremal surfaces in the bulk to boundary entanglement in time-dependent states using a replica and Schwinger-Keldysh approach.

Contribution

It introduces a gravitational argument for the covariant holographic entanglement entropy, utilizing replica geometries and Schwinger-Keldysh contours to justify extremal surfaces in dynamic settings.

Findings

Replica geometries lead to a consistent calculation of Renyi entropies.

Extremal surfaces emerge as saddle points in the replica limit.

Supports the covariant proposal for holographic entanglement entropy.

Abstract

We provide a gravitational argument in favour of the covariant holographic entanglement entropy proposal. In general time-dependent states, the proposal asserts that the entanglement entropy of a region in the boundary field theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate Schwinger-Keldysh contour to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the field theory Renyi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this construction.