A Holographic Approach to Spacetime Entanglement

TL;DR

This paper reviews and extends the holographic relation between spacetime horizon entropy and boundary entanglement entropy, providing explicit methods to construct boundary regions and bulk curves in three-dimensional settings.

Contribution

It offers a detailed review and new explicit constructions for relating bulk spacetime regions to boundary entanglement structures in holography.

Findings

Proven holographic relation for Einstein gravity with planar symmetry

Explicit methods for constructing boundary intervals from bulk curves

Extension of constructions to higher dimensions

Abstract

Recently it has been proposed that the Bekenstein-Hawking formula for the entropy of spacetime horizons has a larger significance as the leading contribution to the entanglement entropy of general spacetime regions, in the underlying quantum theory [2]. This `spacetime entanglement conjecture' has a holographic realization that equates the entropy formula evaluated on an arbitrary space-like co-dimension two surface with the differential entropy of a particular family of co-dimension two regions on the boundary. The differential entropy can be thought of as a directional derivative of entanglement entropy along a family of surfaces. This holographic relation was first studied in [3] and extended in [4], and it has been proven to hold in Einstein gravity for bulk surfaces with planar symmetry (as well as for certain higher curvature theories) in [4]. In this essay, we review this proof and provide explicit examples of how to build the appropriate family of boundary intervals for a given bulk curve. Conversely, given a family of boundary intervals, we provide a method for constructing the corresponding bulk curve in terms of intersections of entanglement wedge boundaries. We work mainly in three dimensions, and comment on how the constructions extend to higher dimensions.