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Causality & holographic entanglement entropy

Matthew Headrick, Veronika E. Hubeny, Albion Lawrence, Mukund Rangamani

arXiv:1408.6300v2hep-thgr-qc

TL;DR

The paper identifies the causality conditions that entanglement entropy must satisfy and tests HRT in a setting where extremal surfaces can enter causally disconnected bulk regions. The authors translate field-theory causality into requirements on extremal surfaces and prove them using bulk causal geometry, null geodesic congruences, and the null energy condition. HRT satisfies both causality requirements under the bulk null energy condition, and its extremal surface lies outside the causal wedge, never in its interior.

  • Problem

    The paper identifies the causality conditions that entanglement entropy must satisfy and tests HRT in a setting where extremal surfaces can enter causally disconnected bulk regions.

    Source: §1 · pp. 3, 5

  • Method

    The authors translate field-theory causality into requirements on extremal surfaces and prove them using bulk causal geometry, null geodesic congruences, and the null energy condition.

    Source: §1 · p. 3, §4 · p. 26

  • Results

    HRT satisfies both causality requirements under the bulk null energy condition, and its extremal surface lies outside the causal wedge, never in its interior.

    Source: §1 · p. 6, §3.4 · pp. 24–25

  • Takeaways & Limitations

    The result provides additional evidence for HRT and identifies the entanglement wedge as the most natural bulk region corresponding to a boundary reduced density matrix.

    Source: §1 · p. 6, §5 · p. 36

    The discussion examines whether geometric deformations of the metric change ρA.

    Source: §5 · p. 37

Abstract

from arXiv · show

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.

1 Introduction

The paper tests whether covariant holographic entanglement entropy respects field-theory causality, a nontrivial issue because HRT surfaces can enter causally disconnected bulk regions. It proves consistency under the bulk null energy condition and introduces the entanglement wedge as the natural bulk counterpart of a boundary reduced density matrix.

1 Introduction

  • CFT causality requires entanglement entropy to depend only on a region’s domain of dependence and to be unaffected by Hamiltonian perturbations supported in D[A] ∪ D[Ac].The first condition makes entropy a wedge observable; the second applies for evolution toward both the future and the past.Source: §1 · pp. 3–4
  • These field-theory requirements translate holographically into a common extremal surface for regions with the same domain of dependence and placement of that surface in the causal shadow.The causal shadow consists of bulk points spacelike-separated from D[A] ∪ D[Ac].Source: §1 · pp. 3–4
  • Time dependence makes the HRT test essential because extremal surfaces can penetrate behind horizons and into regions causally disconnected from the boundary.Static RT has stronger support from entanglement properties and Euclidean-gravity arguments, whereas HRT has passed fewer checks and lacks a first-principles derivation.Source: §1 · pp. 3, 5
  • Simple examples show that causality can hold only marginally: arbitrarily small deformations of extremal surfaces in pure AdS, eternal Schwarzschild-AdS, or Vaidya-AdS can threaten boundary causality.In static geometries, extremal surfaces avoid event horizons; this protection fails in dynamical settings.Source: §1 · pp. 5–6, Figure 1 · p. 5
  • The HRT prescription is consistent with field-theory causality when the bulk metric obeys the null energy condition.The proof establishes both causality requirements for the extremal surface and provides additional evidence for HRT.Source: §1 · pp. 3, 6
  • The entanglement wedge is a codimension-zero bulk region bounded by D[A] and the extremal surface, proposed as the natural bulk region associated with ρA.The paper also relates its proof to Gao-Wald boundary causality and studies causal shadows, including cases where the shadow remains nontrivial for an entire boundary Cauchy slice.Source: §1 · pp. 4, 6

2 Causal domains and entanglement entropy

This section derives causality constraints on entanglement entropy and examines how HRT extremal surfaces satisfy them under suitable bulk conditions. It introduces spacelike homology and the entanglement wedge, and analyzes the causal-shadow localization of extremal surfaces for disconnected boundary regions, including symmetric cases and extensions beyond spherical symmetry.

2.1 Causality of entanglement entropy in QFT

  • QFT entanglement spectra depend only on D[A] and are insensitive to local Hamiltonian deformations within D[A] or D[Ac], establishing the required causality constraints.The spacetime decomposes into the domains of dependence of A and Ac plus the causal future and past of the entangling surface.Source: §2.1 · pp. 8–9

2.2 Bulk geometry and holographic entanglement entropy

  • The covariant HRT prescription requires spacelike homology, with EA anchored on ∂A and homologous to A through a bulk Cauchy-slice region RA.This condition resolves the ambiguity in the covariant generalization of static homology for holographic QFTs governed by Einstein gravity with matter satisfying the null energy condition.Source: §2.2 · pp. 9–11
  • The entanglement wedge WE[A] is the bulk domain of dependence of RA, equivalently the region spacelike-separated from EA and connected to D[A].It is a codimension-zero bulk region naturally associated with the boundary reduced density matrix and depends only on D[A].Source: §2.2 · p. 11

2.3 Causality constraints on extremal surfaces

  • The main theorem places EA in the causal shadow of ∂A, preventing perturbations in D[A] or D[Ac] from affecting its area and preserving HRT causality.It also ensures that regions A′ with D[A′] = D[A] are spacelike-homologous to the same EA, so HRT assigns them equal entanglement entropy.Source: §2.3 · p. 14

2.4 Entanglement for disconnected boundary regions

  • For disconnected boundaries, the theorems prevent extremal surfaces from entering the complementary causal wedge even when time-dependent deformations let them cross event or apparent horizons.In the eternal Schwarzschild-AdS example, the full boundary region’s extremal surface is the black-hole bifurcation surface and forms the causal shadow.Source: §2.4 · p. 15
  • For a one-sided perturbation of eternal Schwarzschild-AdS, causality prevents the original extremal surface from moving, though a new extremal surface could potentially form.The surface remains at the bifurcation surface where the relevant regions meet.Source: §2.4 · p. 16
  • Extremal surfaces for A = ΣR must lie in the causal shadow, spacelike-separated from both boundary domains.This follows directly in the time- and space-reflection-symmetric geometry, where EA sits at the center of the causal shadow.Source: §2.4 · p. 17
  • The null energy condition excludes a candidate extremal surface from the causal future of the opposite boundary by preventing its past-directed null congruence from having unbounded area growth.Vanishing initial expansion, nonpositive area evolution, and finite sphere area produce the contradiction.Source: §2.4 · pp. 17–18
  • In the symmetric Vaidya-Schwarzschild-AdS example, EA lies where the future and past apparent horizons meet, the apparent bifurcation surface.Its location in the causal shadow follows from apparent horizons remaining within the corresponding event horizons.Source: §2.4 · p. 18
  • The paper states that later theorems establish the causal-shadow result in full generality beyond spherical symmetry.The subsequent proof strategy studies null congruences from bulk codimension-two surfaces before presenting the general proof.Source: §2.4 · p. 18

3 Null geodesic congruences in AdS3

The AdS3 null congruences from boundary-anchored curves change behavior at a=1: they reach the boundary for a<1, intersect for a>1, and develop caustics associated with divergent expansion. These structures clarify how causal domains depend on the initial surface.

3 Null geodesic congruences in AdS3

  • Boundary-anchored curves are parameterized by a; a=1 gives the semicircular extremal surface, while a<1 and a>1 lie inside and outside the causal wedge, respectively.The corresponding null geodesics are labeled by x0 and affine parameter λ.Source: §3 · p. 19

3.1 Explicit solutions for geodesic congruences

  • For a<1, generators reach the boundary within D+[A] without intersecting; for a>1, they intersect in a crossover seam before reaching the boundary.At a=1, all generators meet at the future tip of D+[A].Source: §3.1 · p. 20, §3.2 · p. 21

3.2 Intersections within congruences

  • The crossover seam terminates on the boundary at the future tip of D+[A], where z=0, x=0, and t=1.The seam is generated by intersections of geodesics with opposite starting positions.Source: §3.2 · p. 20

3.3 Expansion of congruences and caustics

  • Initial expansion is positive for a<1 and negative for a>1, so the corresponding congruences expand toward the boundary or converge toward intersections.At x0=0, Θ0=a(1−a^2), while Θ0 vanishes at x0=±1.Source: §3.3 · pp. 22–23
  • For a>1, the expansion diverges to −∞ at finite λc, signaling caustics where infinitesimally nearby geodesics intersect.The Raychaudhuri equation gives Θ(λ)=Θ0/(1+Θ0λ) in the one-dimensional, shear-free setting.Source: §3.3 · pp. 23–24
  • The caustic seams are null curves that begin at the central intersection and end on the boundary at finite distance, with zc=0, xc=±(1−a^2), and tc=a^2.Their endpoints are parametrized by the starting position x0.Source: §3.3 · p. 24
  • Divergent expansion marks conjugate points, whereas crossover points are nonlocal intersections where the local expansion can remain finite.On the continued congruence surface, constant-λ wavefronts form closed loops beyond the crossover seam and shrink to zero area at the caustic flank.Source: §3.3 · p. 24

3.4 Summary

  • If the initial surface lies inside the causal wedge, the congruence ends inside D[A]; on its boundary it reaches ∂D[A], while outside it develops a crossover seam and later caustics.The seam reaches the boundary at the future tip of ∂D[A], and extremal surfaces lie outside or at most on the causal wedge boundary.Source: §3.4 · pp. 24–25

4 Theorem and proof

The proof establishes that, under the stated holographic assumptions and the bulk null energy condition, extremal surfaces satisfy the causal requirements inherited from boundary entanglement entropy. It does so by controlling null geodesics, matching bulk and boundary causal decompositions, and showing that the extremal surface is a wedge observable.

4.1 Holographic setup

  • Under the holographic setup, the bulk obeys the null energy condition, the conformal completion is globally hyperbolic, and the boundary is null-geodesically totally geodesic.These assumptions are supplemented by the stated spacetime embedding and boundary regularity conditions.Source: §4.1 · p. 26

4.2 Congruences of null geodesics

  • The causal-geometric lemmas also imply that distinct boundary components cannot be connected by traversable bulk wormholes.This follows from ˜J+(B1) ∩ B2 = ∅ and is identified as closely related to topological censorship.Source: §4.2 · p. 28
  • Null-geodesic congruence arguments show that an extremal surface satisfies ˜J+[E] ∩ B = J+[E ∩ B], preventing its bulk causal future from reaching new boundary points.Extremality gives zero initial expansion, while the null energy condition controls expansion and shear; orthogonal intersection with the boundary removes the second geodesic type.Source: §4.2 · pp. 28–30

Theorem 12

  • A boundary region A induces a four-way causal decomposition into D[A], D[Ac], and J±[∂A], which the spacelike-homology construction reproduces in the bulk.The bulk decomposition uses the homology region RA and the extremal surface EA, with the boundary reductions matching Ac and ∂A.Source: §4.3 · p. 30, Theorem 12 · pp. 30–31

Theorem 13

  • The theorem proves that the HRT extremal surface is causally disconnected from D[A] and D[Ac], as required by field-theory causality.The result follows from the bulk-boundary causal decomposition and the spacelike-homology construction.Source: Theorem 13 · pp. 31, 33, Theorem 12 · p. 31
  • Theorem 15 shows that regions A and A′ with the same domain of dependence are spacelike-homologous to the same extremal surface, so HRT assigns them equal entanglement entropy.This establishes the HRT counterpart of the field-theory requirement that entropy be invariant under changes of spatial slice preserving the causal domain.Source: Theorem 13 · pp. 32–33

5 Discussion

The paper establishes that HRT is consistent with field-theory causality, then uses this result to analyze entanglement wedges, their relation to reduced density matrices, and the scope of the conclusions.

Causality constraints

  • Theorem 13 shows that the HRT prescription is consistent with field-theory causality.The result is nontrivial because arbitrarily small extremal-surface deformations can otherwise create causal contact with boundary perturbations that cannot affect the entropy.Source: §5 · p. 33

Domain of validity

  • The proof applies to extremal surfaces in two-derivative gravity satisfying the null energy condition, within the strong-coupling planar regime.The argument relies on the Raychaudhuri equation and therefore assumes a macroscopic asymptotically AdS spacetime in a perturbative string regime.Source: §5 · p. 33

Causality constraints

  • Causality alone confines the extremal surface to the causal shadow but generally does not determine its unique codimension-two location.Only special geometries, such as spherical regions in pure AdS or the eternal Schwarzschild-AdS black hole, uniquely localize it.Source: §5 · p. 34

Entanglement wedges

  • The entanglement wedge is the domain of dependence of the homology surface connecting A to its extremal surface, constructed from null boundaries generated by that surface.It is a causally defined bulk region once the oriented extremal surface is specified.Source: §5 · p. 35
  • Equal-area extremal surfaces can produce multiple entanglement wedges for the same boundary region, especially at transitions between competing surfaces or in complicated connected configurations.Candidate wedges can even be proper subsets of one another.Source: §5 · p. 35
  • For mixed states, the bulk decomposition associated with A and its complement generally differs because EA ≠ EAc, requiring the complement of the homology surface within a bulk Cauchy slice.The simpler complementary decomposition is restored when a mixed state is purified by additional boundaries.Source: §5 · p. 36

Dual of ρA

  • The paper proposes the entanglement wedge as the most appropriate bulk dual of the reduced density matrix ρA.This proposal concerns the maximal region whose bulk geometry is naturally associated with fixing ρA, even when entanglement holes occur.Source: §5 · pp. 36–37
  • Unlike the causal wedge, the entanglement wedge can extend behind horizons and deep into the bulk, including regions near singularities or far beyond a black-hole horizon.Deformed Schwarzschild-AdS and shock-deformed eternal black holes illustrate this separation between the two wedges.Source: §5 · pp. 38–39, Figure 11 · p. 39
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