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Radial cutoffs and holographic entanglement

Brianna Grado-White, Donald M. Marolf, Sean J. Weinberg

Published Jan 1, 2021• 15 citations
OpenAlex

Abstract

Abstract: \nTensor networks,$$ T\\overline{T} $$TT¯, and broader notions of a holographic principle all motivate the idea that some notion of gravitational holography should persist in the presence of a radial cutoff. But in the absence of time-reflection symmetry, the areas of Hubeny-Rangamani-Takayanagi surfaces anchored to the radial cutoff generally violate strong sub-additivity, even when the associated boundary regions are spacelike separated as defined by both bulk and boundary notions of causality. We thus propose an alternate definition of cutoff-holographic entropy using a restricted maximin prescription anchored to a codimension 2 cutoff surface. For bulk solutions that respect the null energy condition, we show that the resulting areas satisfy SSA, entanglement wedge nesting, and monogamy of mutual information in parallel with cutoff free results in AdS. These results hold even when the cutoff surface fails to be convex.

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Constraints from entanglement wedge nesting for holography at a finite cutoff

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A bstract We explore constraints that arise from associating an entanglement wedge (EW) to subregions of a cutoff boundary at a finite distance in AdS/CFT, using a subcritical end-of-the-world (ETW) brane acting as a cutoff. In particular, we consider the case of two intervals in the holographic dual to a BCFT, with one interval A located at the asymptotic boundary and the second interval B located on the ETW brane. We discuss in detail subtleties that arise near the RT end-points when defining the EW for this configuration, particularly in the connected phase. Entanglement wedge nesting (EWN) requires that $$ {\mathcal{W}}_E(A) $$ W E A ∪ $$ {\mathcal{W}}_E(B) $$ W E B ⊆ $$ {\mathcal{W}}_E\left(A\cup B\right) $$ W E A ∪ B . We demonstrate that already in the simplest example of an AdS 3 bulk geometry, EWN can be violated even if A and B are spacelike separated through the bulk and instead we must require the stronger condition that $$ {\mathcal{W}}_E(A) $$ W E A be spacelike separated from $$ {\mathcal{W}}_E(B) $$ W E B , which highlights the non-local nature of the cutoff theory. Our prescription to associate EWs to subregions on the ETW brane is different from the restricted maximin procedure [1] but will agree within the subset of parameter space where EWN is respected. Additionally, we study EWN in a two sided BTZ black hole geometry with an ETW brane in one of the exteriors. In the BTZ black hole example we find that our condition for EWN disallows configurations where the RT surface goes from the brane to the black hole singularity.

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Holographic entanglement entropy is cutoff-covariant

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Abstract Using the Hubeny-Rangamani-Takayanagi (HRT) conjectured formula for entanglement entropy in the context of the AdS/CFT correspondence with time-dependent backgrounds, we investigate the relation between the bulk null energy condition (NEC) of the stress-energy tensor with the strong sub-additivity (SSA) property of entanglement entropy in the boundary theory. In a background that interpolates between an AdS to an AdS-Reissner-Nordstrom-type geometry, we find that generically there always exists a critical surface beyond which the violation of NEC would naively occur. However, the extremal area surfaces that determine the entanglement entropy for the boundary theory, can penetrate into this forbidden region only for certain choices for the mass and the charge functions in the background. This penetration is then perceived as the violation of SSA in the boundary theory. We also find that this happens only when the critical surface lies above the apparent horizon, but not otherwise. We conjecture that SSA, which is thus non- trivially related to NEC, also characterizes the entire time-evolution process along which the dual field theory may thermalize.

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Entanglement entropy and algebraic holography

Bernard S. Kay

In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.

arXiv:1605.07872•May 25, 2016•0 citations
Abstract: \nTensor networks,ToverlineTT\\overline{T}ToverlineTTT¯, and broader notions of a holographic principle all motivate the idea that some notion of gravitational holography should persist in the presence of a radial cutoff. But in the absence of time-reflection symmetry, the areas of Hubeny-Rangamani-Takayanagi surfaces anchored to the radial cutoff generally violate strong sub-additivity, even when the associated boundary regions are spacelike separated as defined by both bulk and boundary notions of causality. We thus propose an alternate definition of cutoff-holographic entropy using a restricted maximin prescription anchored to a codimension 2 cutoff surface. For bulk solutions that respect the null energy condition, we show that the resulting areas satisfy SSA, entanglement wedge nesting, and monogamy of mutual information in parallel with cutoff free results in AdS. These results hold even when the cutoff surface fails to be convex.
A bstract We explore constraints that arise from associating an entanglement wedge (EW) to subregions of a cutoff boundary at a finite distance in AdS/CFT, using a subcritical end-of-the-world (ETW) brane acting as a cutoff. In particular, we consider the case of two intervals in the holographic dual to a BCFT, with one interval A located at the asymptotic boundary and the second interval B located on the ETW brane. We discuss in detail subtleties that arise near the RT end-points when defining the EW for this configuration, particularly in the connected phase. Entanglement wedge nesting (EWN) requires that WE(A) {\mathcal{W}}_E(A) WE​(A) W E A ∪ WE(B) {\mathcal{W}}_E(B) WE​(B) W E B ⊆ WE(A∪B) {\mathcal{W}}_E\left(A\cup B\right) WE​(A∪B) W E A ∪ B . We demonstrate that already in the simplest example of an AdS 3 bulk geometry, EWN can be violated even if A and B are spacelike separated through the bulk and instead we must require the stronger condition that WE(A) {\mathcal{W}}_E(A) WE​(A) W E A be spacelike separated from WE(B) {\mathcal{W}}_E(B) WE​(B) W E B , which highlights the non-local nature of the cutoff theory. Our prescription to associate EWs to subregions on the ETW brane is different from the restricted maximin procedure [1] but will agree within the subset of parameter space where EWN is respected. Additionally, we study EWN in a two sided BTZ black hole geometry with an ETW brane in one of the exteriors. In the BTZ black hole example we find that our condition for EWN disallows configurations where the RT surface goes from the brane to the black hole singularity.
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.
Abstract Using the Hubeny-Rangamani-Takayanagi (HRT) conjectured formula for entanglement entropy in the context of the AdS/CFT correspondence with time-dependent backgrounds, we investigate the relation between the bulk null energy condition (NEC) of the stress-energy tensor with the strong sub-additivity (SSA) property of entanglement entropy in the boundary theory. In a background that interpolates between an AdS to an AdS-Reissner-Nordstrom-type geometry, we find that generically there always exists a critical surface beyond which the violation of NEC would naively occur. However, the extremal area surfaces that determine the entanglement entropy for the boundary theory, can penetrate into this forbidden region only for certain choices for the mass and the charge functions in the background. This penetration is then perceived as the violation of SSA in the boundary theory. We also find that this happens only when the critical surface lies above the apparent horizon, but not otherwise. We conjecture that SSA, which is thus non- trivially related to NEC, also characterizes the entire time-evolution process along which the dual field theory may thermalize.
We check formally that the Hubeny-Rangamani-Takayanagi prescription for holographic entanglement entropy---when applied to a static black brane spacetime and to a wide class of subregions that do not lie on a constant time slice---gives rise to volume-law entropy in the limit of large subregion. By volume-law entropy, we mean that the entanglement entropy scales with the volume of the projection of the boundary subregion onto a static time slice with respect to the boundary thermal state (with the same coefficient as in the volume law on the static time slice). Our result applies to subregions that have reflection symmetry as well as strips, and we also present field-theoretic arguments in support of our holographic findings.
We identify conditions for the entanglement entropy as a function of spatial region to be compatible with causality in an arbitrary relativistic quantum field theory. We then prove that the covariant holographic entanglement entropy prescription (which relates entanglement entropy of a given spatial region on the boundary to the area of a certain extremal surface in the bulk) obeys these conditions, as long as the bulk obeys the null energy condition. While necessary for the validity of the prescription, this consistency requirement is quite nontrivial from the bulk standpoint, and therefore provides important additional evidence for the prescription. In the process, we introduce a codimension-zero bulk region, named the entanglement wedge, naturally associated with the given boundary spatial region. We propose that the entanglement wedge is the most natural bulk region corresponding to the boundary reduced density matrix.
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.