Bulk curves from boundary data in holography

Abstract

We embed spherical Rindler space -a geometry with a spherical hole in its center -in asymptotically AdS spacetime and show that it carries a gravitational entropy proportional to the area of the hole. Spherical AdS-Rindler space is holographically dual to an ultraviolet sector of the boundary field theory given by restriction to a strip of finite duration in time. Because measurements have finite durations, local observers in the field theory can only access information about bounded spatial regions. We propose a notion of differential entropy that captures uncertainty about the state of a system left by the collection of local, finite-time observables. For two-dimensional conformal field theories we use holography and the strong subadditivity of entanglement to propose a formula for differential entropy and show that it precisely reproduces the areas of circular holes in AdS 3 . Extending the notion to field theories on strips with variable durations in time, we show more generally that differential entropy computes the areas of all closed, inhomogenous curves on a spatial slice of AdS 3 . We discuss the extension to higher dimensional field theories, the relation of differential entropy to entanglement between scales, and some implications for the emergence of space from the RG flow of entangled field theories.

Comments on Jacobson’s “entanglement equilibrium and the Einstein equation”

“…Entanglement is now recognized to play an important role in the emergence of space (and space-time) in quantum gravity with calculations from a variety of different approaches [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Of course, one of the most natural frameworks where this connection can be investigated is the AdS/CFT correspondence [15,16].…”

“…That is, H would be given by some complex and generally nonlocal expression involving collections of operators restricted in a finite region, i.e., the sphere of radius R. However, if it is only examined with long wavelength probes, we can effectively approximate H by a sum of local operators. 10 In fact, reducing the flat space vacuum of a CFT to a sphere of radius R yields a remarkably simple expression for the modular Hamiltonian [26] …”

Exploring the tensor networks/AdS correspondence

“…2 We note that conversely, providing only the data about the number of legs p in each tensor and the number q of nodes being adjacent to each other does not uniquely specify the tessellation. For example, the code in [33] could have been equally well described by [5,5], although the map between vertices, edges and tensors and their contraction would have to follow some different rules. This is connected to the subtlety of what isometries are in fact preserved in the specific tessellation.…”

“…Consider for example the four leg tensor considered in [33]. The tensor structure there is describable by the Coxeter group [4,5], if we inspect the arrangement of the self-repeating units. However, within each unit, the tensors are arranged in a way that is essentially flat.…”

“…This inspired many subsequent works that attempt to make further concrete connection between the AdS/CFT and the tensor network. It is for example observed that the MERA tensor network, that originally inspired Swingle's observation [3], is connected to the space of geodesics in AdS space, termed the kinematic space [4][5][6][7][8][9][10][11][12]. 1 More recently, [31] proposes a concrete construction of a tensor network such that the entanglement entropy of the physical degrees of freedom is given precisely by the size of a minimal surface that cuts through the bulk of the tensor network.…”