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On subregion action complexity in AdS$_3$ and in the BTZ black hole

Roberto Auzzi, Stefano Baiguera, Andrea Legramandi, Giuseppe Nardelli, Pratim Roy, Nicolo Zenoni

arXiv:1910.00526v2hep-th

TL;DR

The paper asks how holographic subregion complexity relates to entanglement entropy for simple and disconnected regions. It analytically computes CA and related quantities for BTZ segments and two-segment AdS3 regions, finding a simple entropy-linked structure for one segment but additional finite structure for two segments.

  • Problem

    The paper investigates whether the simple relation between subregion complexity and entanglement entropy persists beyond a single segment.

    Source: §1 · p. 2, §3.5 · p. 14, §6 · p. 20

  • Method

    The paper analytically computes subregion CA and CV 2.0 for a segment in BTZ and AdS3, then computes action and mutual complexity for two disjoint AdS3 segments.

    Source: §1 · p. 3, §6 · p. 20

  • Results

    For one segment, complexity is directly related to entanglement entropy; for two segments, it depends more intricately on lengths and separation, and mutual complexity is not proportional to mutual information.

    Source: §6 · p. 20

  • Takeaways & Limitations

    Subregion complexity carries information distinct from entanglement entropy for two disjoint segments.

    Source: §6 · p. 20

    The physical meaning of the null-boundary counterterm scale ˜L and its relation to the field-theory side remain unclear.

    Source: §6 · p. 21

Abstract

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We analytically compute subsystem action complexity for a segment in the BTZ black hole background up to the finite term, and we find that it is equal to the sum of a linearly divergent term proportional to the size of the subregion and of a term proportional to the entanglement entropy. This elegant structure does not survive to more complicated geometries: in the case of a two segments subregion in AdS$_3$, complexity has additional finite contributions. We give analytic results for the mutual action complexity of a two segments subregion.

1 Introduction

The paper studies holographic subregion complexity through CA and CV 2.0, focusing on analytic results for segments and two disjoint segments. A single segment in BTZ has a simple relation to entanglement entropy, whereas two segments introduce additional finite structure.

1 Introduction

  • Computational complexity is introduced holographically to probe quantum-information aspects of gravity beyond entanglement entropy.The motivation includes the longer growth timescale of the Einstein-Rosen bridge compared with entanglement-entropy thermalization.Source: §1 · p. 2
  • The paper applies subregion CA and CV 2.0 to asymptotically AdS3 geometries, including BTZ and disjoint boundary segments.Subregion CA uses the action of the WDW-patch and entanglement-wedge intersection, while CV 2.0 uses its spacetime volume.Source: §1 · p. 3
  • For a BTZ segment, the analytic subregion complexity contains a linear divergence and a term proportional to the RT entanglement entropy.The result also applies to AdS3, and a similar expression is obtained for CV 2.0.Source: §1 · p. 3
  • The simple segment relation arises through cancellations in which logarithmic divergences and part of the finite contribution combine into the entropy term.Both the linear and logarithmic divergent coefficients depend on the counterterm scale ˜L.Source: §1 · p. 3
  • The counterterm scale ˜L remains physically obscure and is left for further investigation.Positivity requires ˜L > L, which makes the entropy-term coefficient negative.Source: §1 · p. 3
  • For two disjoint AdS3 segments, complexity retains linear and logarithmic divergences but develops additional finite contributions when the separation is small.The paper provides analytic results for this case and studies mutual action complexity.Source: §1 · p. 3

2 Subregion complexity for a segment in AdS3

The AdS3 single-segment calculation evaluates the action of the intersection between the WDW patch and entanglement wedge using regulated bulk, null-boundary, counterterm, and joint contributions. The resulting complexity has divergences organized by subregion length and entanglement entropy.

2 Subregion complexity for a segment in AdS3

  • The AdS3 setup uses Poincaré coordinates with curvature R = −6/L2 and relates the AdS scale to the CFT central charge.The RT surface is a spacelike geodesic, and the entanglement wedge is bounded by null surfaces.Source: §2 · pp. 4–5
  • The calculation considers a boundary strip of length l and evaluates the WDW-patch and entanglement-wedge intersection with null surfaces, joints, and counterterms included.The analysis mostly uses regularization B, in which the WDW patch is built from z = ε, and compares regularization A separately.Source: §2 · pp. 4–5

2.1 Bulk term

  • The bulk action is split across the two regions determined by the intersection of the WDW and entanglement-wedge boundaries, using zint(x), zRT(x), and the cutoff ε.The integrations differ according to whether the WDW patch or entanglement wedge provides the time and radial bounds.Source: §2.1 · p. 6

2.2 Null boundary counterterms

  • Affine parametrization makes the null-boundary term vanish, while the null counterterm is retained to ensure reparameterization invariance.The counterterm depends on the expansion scalar and the arbitrary scale ˜L.Source: §2.2 · p. 7

2.3 Joint terms

  • Joint contributions arise at the cutoff, RT surface, and intersections of WDW and entanglement-wedge null boundaries.Summing joint and counterterm terms cancels double logarithms and removes dependence on the null-normal normalization α and β.Source: §2.3 · pp. 7–8

2.4 Complexities

  • The final AdS3 complexities contain a term proportional to the segment’s entanglement entropy, alongside divergences and a constant finite contribution.This structure motivates testing the corresponding BTZ calculation.Source: §2.4 · pp. 8–9

3 Subregion complexity for a segment in the BTZ black hole

The paper evaluates subregion action complexity for a boundary segment in the planar BTZ geometry by integrating the WDW–entanglement-wedge intersection and its bulk, null-boundary, counterterm, and joint contributions. The resulting analytic complexity reproduces the expected divergences and admits checks against limiting cases and related volume complexity.

3 Subregion complexity for a segment in the BTZ black hole

  • The segment geometry is determined by the BTZ metric, its RT geodesic, and the entanglement wedge formed by null geodesics from the boundary causal diamond.Because z∗ < zh for every segment length, the RT geodesic remains outside the event horizon.Source: §3 · pp. 9–10

3.1 Bulk contribution

  • Analytic integration of the bulk region yields the CV 2.0 complexity for a segment in the BTZ background.The integration region is split as in the AdS calculation before evaluating the total bulk action.Source: §3.1 · p. 11

3.2 Null normals

  • The null-normal construction uses affine geodesics with conserved quantities, while the WDW-boundary geodesics have J = 0 and an arbitrary normalization α.The resulting tangent vectors and normal one-forms provide the data needed for the null-boundary and joint terms.Source: §3.2 · p. 11

3.3 Null boundaries and counterterms

  • The null-boundary counterterm vanishes on the entanglement-wedge boundaries because Θ = 0, whereas the WDW boundaries require a direct expansion calculation.The affine parametrization also removes the term associated with the null-boundary parametrization.Source: §3.3 · p. 12

3.4 Joint contributions

  • The joint contributions at the cutoff, RT surface, and WDW–entanglement-wedge intersections are evaluated with the cutoff ε regulating all such terms.The analysis also includes a codimension-3 joint at the boundary, which can at most shift the action by an overall constant.Source: §3.4 · pp. 12–13

3.5 Complexities

  • The final BTZ complexity has the same divergences as the AdS result, agrees with the known BTZ complexity in the l ≫ zh limit, and loses its log ε divergence in that limit.The paper also compares the action result with interval volume complexity, which is independent of temperature.Source: §3.5 · pp. 13–14

4 Subregion complexity for two segments in AdS3

For two disjoint AdS3 segments, the authors evaluate action and spacetime-volume subregion complexity across disconnected and connected extremal-surface configurations, finding more complicated finite terms than for one segment.

4 Subregion complexity for two segments in AdS3

  • The extremal surface is disconnected for d > d0 and connected for d < d0, with d0 fixed by equal competing surface lengths.The disconnected configuration uses the individual RT surfaces, while the connected configuration joins the two subregions.Source: §4 · p. 14

4.1 Bulk contribution

  • The connected configuration is evaluated using symmetry and by splitting the relevant bulk region into seven analytically integrable parts.The calculation includes the WDW patch, entanglement-wedge null boundaries, counterterms, and joint contributions.Source: §4 · p. 14, §4.1 · p. 16

4.2 Counterterms

  • The null-boundary counterterms and several joints contribute to the action, including joints at the cutoff, on RT1 and RT2, and along their null-boundary intersection.These terms are assembled with the bulk contribution to obtain the total action complexity.Source: §4.2 · p. 16, §4.3 · p. 17

4.4 Complexities

  • Both CA and CV 2.0 have the same leading divergences as the single-segment case, including subleading divergences proportional to the entanglement entropy.The finite contribution is instead a more complicated function of the segment length l and separation d.Source: §4.4 · p. 18

5 Mutual complexity

Mutual complexity is finite and vanishes for disconnected RT surfaces, but action mutual complexity can change sign with the counterterm scale and separation, unlike CV 2.0.

5 Mutual complexity

  • For d > d0, mutual complexity vanishes because the RT surface is disconnected and the union complexity equals the sum of the individual complexities.The quantity is finite in all three holographic conjectures and can be discontinuous at d = d0.Source: §5 · p. 18
  • CV 2.0 mutual complexity is negative definite, while CA mutual complexity can be either positive or negative.The negative CV 2.0 sign follows from the relative sizes of the bulk regions in the two RT configurations.Source: §5 · pp. 18–20

5.1 Strong super/subaddivity for overlapping segments

  • For overlapping intervals, CA is strongly subadditive when η < 1/2 and strongly superadditive when η > 1/2, while CV 2.0 is strongly superadditive.This generalizes mutual-complexity comparisons beyond disjoint segments.Source: §5.1 · p. 20

6 Conclusions

The paper relates one-segment subregion complexity in AdS3 and BTZ to entanglement entropy, while two-segment complexity depends separately on lengths and separation and yields mutual complexity not proportional to mutual information.

6 Conclusions

  • For one segment, subregion complexity in AdS3 and BTZ is directly related to entanglement entropy.Source: §6 · p. 20
  • For two segments, AdS3 complexity depends more intricately on segment lengths and relative separation, and mutual complexity is not proportional to mutual information.Source: §6 · p. 20
  • The physical meaning of the null-boundary counterterm scale ˜L, including its relation to the field-theory side, remains unclear.Source: §6 · p. 21
  • The CA mutual-complexity sign depends strongly on η = ˜L/L: it is always negative for η ≥ 2.465, but changes from negative at small d to positive at larger d when 1 < η < 2.465.Source: §6 · p. 21
  • The paper leaves higher-dimensional regions of different shapes, such as spheres or strips, for future study.Source: §6 · p. 21

A Another regularization for the action of one segment in BTZ

A second BTZ regularization sends the WDW null boundaries from the true boundary and introduces a timelike cutoff, producing a geometry without codimension-3 joints.

A Another regularization for the action of one segment in BTZ

A.1 Bulk contribution

The bulk action is decomposed into separate contributions, with an additional term required because the initial split does not capture the entire bulk region.

A.1 Bulk contribution

  • The bulk contributions are split into separate terms for evaluation.Source: §A.1 · p. 22
  • An additional bulk term I3 is required because the initial WDW–entanglement-wedge split omits part of the spacetime region.I3 accounts for the region between the transverse-coordinate values xint and xmax.Source: §A.1 · p. 23
  • The resulting bulk contribution is obtained by direct evaluation of the separated terms.Source: §A.1 · p. 23

A.2 Gibbons-Hawking-York contribution

The Gibbons-Hawking-York contribution comes only from the timelike cutoff surface and is separated into WDW-patch and entanglement-wedge parts.

A.2 Gibbons-Hawking-York contribution

  • The GHY term is evaluated using the induced boundary metric and the trace of the extrinsic curvature.Source: §A.2 · p. 23
  • Only the timelike regularizing surface at z = ε contributes a GHY term.Source: §A.2 · p. 23
  • The total GHY contribution combines one part from the WDW patch with another from the entanglement wedge.Source: §A.2 · p. 23

A.3 Null boundaries counterterms

The calculation treats null-boundary counterterms, joint terms, and the final complexity expression, finding that regularizations agree in divergences but differ through finite counterterm-scale dependence.

A.3 Null boundaries counterterms

  • The null-boundary counterterm calculation parallels the earlier treatment, with the entanglement-wedge counterterm vanishing and the WDW counterterm providing the remaining contribution.Source: §A.3 · pp. 23–24

A.4 Joint terms

  • Joint terms are evaluated for null–timelike or null–spacelike intersections using the induced surface measure and outward-directed normals.The auxiliary tangent vector is chosen orthogonal to the joint and outward-directed from the relevant region.Source: §A.4 · p. 24
  • The joint calculation includes cutoff, entanglement-wedge, RT null-null, and WDW–entanglement-wedge intersections, with the latter integrated up to xint(ε).Source: §A.4 · p. 24

A.5 Complexity

  • Adding all contributions yields the final complexity expression for the regularization.Source: §A.5 · p. 25
  • The alternative regularization changes the coefficient of the 1/ε divergence and adds a finite term proportional to log of the counterterm scale.Source: §A.5 · p. 25
  • The two regularizations have identical numerical coefficients for all divergences but differ by a finite piece dependent on the counterterm length scale ˜L.Source: §A.5 · p. 25
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