Holographic entanglement entropy is cutoff-covariant

Abstract

In the context of the AdS/CFT correspondence, it is often convenient to regulate infinite quantities in asymptotically anti-de Sitter spacetimes by introducing a sharp cutoff at some finite, large value of a particular radial coordinate. This procedure is a priori coordinate dependent, and may not be well-motivated in full, covariant general relativity; however, the fact that physically meaningful quantities such as the entanglement entropy can be obtained by such a regulation procedure suggests some underlying covariance. In this paper, we provide a careful treatment of the radial cutoff procedure for computing holographic entanglement entropy in asymptotically anti-de Sitter spacetimes. We prove two results that are frequently assumed in the literature, but that have not been carefully addressed: (i) that the choice of a "globally minimal surface" among several extremal candidates is independent of the choice of regulator, and (ii) that finite CFT quantities such as the mutual information which involve "divergence-cancelling" sums of entanglement entropies are well-defined under the usual prescription for computing covariant holographic entanglement entropy. Our results imply that the "globally minimal surface" prescription for computing the holographic entanglement entropy is well-posed from the perspective of general relativity, and thus support the widely-held belief that this is the correct prescription for identifying the entanglement wedge of a boundary subregion in AdS/CFT. We also comment on the geometric source of state-dependent divergences in the holographic entanglement entropy, and identify precisely the regime of validity of the "vacuum subtraction" protocol for regulating infinite entanglement entropies in arbitrary states by comparing them to the entanglement entropies of identical regions in the vacuum. Our proofs make use of novel techniques for the covariant analysis of extremal surfaces, which are explained in detail and may find use more broadly in the study of holographic entanglement entropy.

Large-d phase transitions in holographic mutual information

“…[7] for a recent review. Although the entanglement entropy of a spatial region with nonempty boundary suffers from a UV divergence in any local quantum field theory (its bulk dual manifested by the associated HRRT surfaces having infinite proper area since they reach the spacetime boundary), one can nevertheless construct meaningful finite quantities by combining entanglement entropies of several subsystems so that the UV divergences cancel in a cutoff-independent way [8]. The most basic such quantity is the mutual information I(A : B), defined by (1.…”

“…To support the large amount of entanglement in A and B, the complement A ∪ B must have at least as many degrees of freedom as A ∪ B -or, allowing for the states on A and B to be submaximally entangled, perhaps some order-one fraction thereof. 8 Another line of argument following from monogamy of entanglement, however, suggests a very different result: if the local degrees of freedom in the holographic theory are entangled in isotropic fashion, then the higher the dimensionality, the smaller the amount of entanglement which can be spared for any given direction (since it has to be equally distributed amongst all the directions). In the present setup, there is only a single direction (the angle) spanning the separation between A and B; the entanglement in this direction should be greatly suppressed relative to the d − 2 directions along the entangling surfaces ∂A and ∂B.…”

“…From equation (2.3), it is straightforward to check that surfaces of constant ρ are extremal surfaces, as previously claimed. One simply defines a normal vector to any such surface as n a = ∇ a ρ, normalizes it ton a = n a / n b n b , then checks that the unit normal vector satisfies ∇ an a = 0 (2.4) on the surface, which is a necessary and sufficient condition for a surface to be locally extremal [8]. The fact that surfaces of constant ρ are not only extremal, but form a foliation of H d by extremal surfaces, suggests that there exists an isometry which maps between different minimal caps.…”

Holographic scattering requires a connected entanglement wedge

“…While both areas on the left-hand side of this expression are infinite, their difference can be treated consistently in a regulator-independent way using the methods of[42] 19. The factor of two multiplying the area from the cusps comes from the fact that each point in a cusp is formed by the collision of two null generators.…”

Quantum maximin surfaces

“…However, see[58] for an argument that the minimal area extremal surface is cut-off independent in this limit.13 By this we mean the smallest length scale at which bulk excitations exist in the bulk effective field theory.…”