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arxiv: 2601.18310 · v3 · submitted 2026-01-26 · ✦ hep-th · gr-qc

Holographic timelike entanglement and subregion complexity with scalar hair

Hadyan Luthfan Prihadi , Muhammad Alifaldi Ramadhan Al-Faritsi , Rafi Rizqy Firdaus , Fitria Khairunnisa , Yanoar Pribadi Sarwono , Freddy Permana Zen This is my paper

Pith reviewed 2026-05-16 10:58 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords holographic timelike entanglement entropyscalar hairAdS black holessubregion complexityboundary deformationblack hole interior
0
0 comments X

The pith

Scalar hair in AdS black holes breaks the time-interval invariance of the imaginary part of holographic timelike entanglement entropy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines holographic timelike entanglement entropy and subregion complexity for a thermal CFT deformed by a relevant scalar operator, whose bulk dual is a hairy black hole. It finds that the deformation causes the imaginary part of the HTEE to depend on the boundary time interval Δt, in contrast to the constant value seen in pure AdS3 and BTZ geometries. Analytic continuation from CFT temporal entanglement entropy does not reproduce the holographic results, even in two dimensions. The timelike subregion complexity remains real-valued, and for the BTZ case its UV-finite term arises entirely from the black-hole interior.

Core claim

A relevant scalar deformation of the boundary CFT produces a hairy AdS black hole whose extremal surfaces for timelike subsystems yield an imaginary HTEE that varies nontrivially with Δt. This breaks the invariance present in undeformed geometries. Analytic continuation of CFT temporal entropy fails to match the holographic values. Within the complexity=volume prescription the timelike subregion complexity stays real, and its UV-finite piece for the BTZ black hole is contributed solely by the interior region.

What carries the argument

The prescription that merges spacelike and timelike surfaces at the interior to form an extremal surface homologous to a boundary timelike interval of duration Δt.

If this is right

  • The imaginary component of HTEE acquires a nontrivial dependence on Δt once scalar hair is introduced.
  • Analytic continuation from CFT temporal entanglement entropy no longer reproduces the holographic HTEE under boundary deformation.
  • Timelike subregion complexity remains real-valued for the deformed geometries.
  • In the BTZ case the entire UV-finite contribution to subregion complexity comes from the interior region.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same interior-only finite term may appear for other black-hole solutions with scalar hair in higher dimensions.
  • Similar Δt dependence could appear in other holographic quantities that probe the interior when relevant deformations are added.
  • Direct comparison between HTEE and complexity results in d=3 could reveal how the interior contribution scales with dimension.

Load-bearing premise

Merging spacelike and timelike surfaces at the interior still produces a valid extremal surface when scalar hair is present.

What would settle it

A numerical evaluation of the extremal surface for a fixed nonzero scalar-hair parameter and small nonzero Δt that yields a Δt-independent imaginary HTEE would falsify the central claim.

Figures

Figures reproduced from arXiv: 2601.18310 by Fitria Khairunnisa, Freddy Permana Zen, Hadyan Luthfan Prihadi, Muhammad Alifaldi Ramadhan Al-Faritsi, Rafi Rizqy Firdaus, Yanoar Pribadi Sarwono.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of the AdS [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Relations between boundary time interval ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Real part of the total area versus the boundary time interval ∆ [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Imaginary part of the total area versus the boundary time interval ∆ [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Illustration of the volume enclosed by [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Illustration of the extremal surfaces in charged black [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. ∆ [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Exterior (purple) vs. interior (orange) contributions [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. ∆ [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
read the original abstract

We investigate the holographic timelike entanglement entropy (HTEE) and timelike subregion complexity of a thermal CFT$_d$ deformed by a relevant scalar operator $\phi_0$, dual to a hairy black hole in AdS$_{d+1}$. We employ the prescription of merging spacelike and timelike surfaces at the interior, constructing an extremal surface homologous to a boundary timelike subsystem with a time interval $\Delta t$. Consequently, this deformation breaks the invariance of the imaginary component of HTEE observed in pure AdS$_3$ and BTZ geometry, introducing a nontrivial dependence on $\Delta t$. At small $\Delta t$, we derive analytical expressions that are in agreement with numerical results, and observe partial consistency with analytic continuation to temporal or spacelike entanglement entropy at the level of the near-boundary expansion. However, analytic continuation of CFT temporal entanglement entropy fails to reproduce the HTEE calculations under boundary deformation, even in $d=2$. Furthermore, we extend the numerical calculations to higher dimensions ($d=3$). In addition, we study holographic timelike subregion complexity within the complexity=volume conjecture and find that it remains real-valued, providing a complementary geometric probe of the black hole interior. In particular, for the BTZ black hole, we analytically show that the UV-finite term of the subregion complexity receives its entire contribution from the interior region alone.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates holographic timelike entanglement entropy (HTEE) and timelike subregion complexity for a thermal CFT_d deformed by a relevant scalar operator, dual to a hairy black hole in AdS_{d+1}. Employing a prescription that merges spacelike and timelike extremal surfaces at an interior point to construct a surface homologous to a boundary timelike interval of duration Δt, the authors find that the scalar deformation breaks the Δt-invariance of the imaginary part of HTEE observed in pure AdS3 and BTZ geometries, leading to a nontrivial Δt dependence. Analytical expressions at small Δt agree with numerical results, though analytic continuation of CFT temporal entanglement entropy does not reproduce the HTEE results even in d=2. The study extends to d=3 and examines holographic timelike subregion complexity via the complexity=volume conjecture, showing it remains real-valued, with the UV-finite term for the BTZ case receiving its entire contribution from the interior region.

Significance. If the central construction is valid, this work provides a concrete example of how relevant deformations affect timelike entanglement measures and offers a geometric probe of the black hole interior through subregion complexity. The analytic demonstration for BTZ that the UV-finite complexity comes solely from the interior is a notable result, as is the agreement between small-Δt analytics and numerics. These findings could inform studies of entanglement in non-conformal theories and the role of interior geometry in holographic complexity.

major comments (2)
  1. [Construction of extremal surfaces (method section)] The headline results on the nontrivial Δt dependence of Im(HTEE) and the interior-only contribution to UV-finite complexity rely on the merged spacelike-timelike surface remaining extremal in the presence of scalar hair. The manuscript states that it employs the merging prescription but does not provide an explicit verification that the first variation of the area functional (including the scalar field contribution) vanishes at the junction point on the deformed metric. This check is necessary to confirm the surface is indeed extremal after backreaction.
  2. [BTZ subregion complexity calculation] For the BTZ analytic result, the claim that the UV-finite term of the subregion complexity receives its entire contribution from the interior region alone should be supported by showing the explicit cancellation or vanishing of any boundary contributions in the regularized volume integral.
minor comments (2)
  1. [Abstract] The abstract states 'partial consistency with analytic continuation... at the level of the near-boundary expansion' without specifying which expansion terms agree or disagree; adding this detail would improve clarity.
  2. [Numerical results section] In the numerical results for d=3, including quantitative error bars or convergence tests for the agreement between small-Δt analytics and numerics would strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate the requested clarifications and explicit calculations in the revised version.

read point-by-point responses

  1. Referee: [Construction of extremal surfaces (method section)] The headline results on the nontrivial Δt dependence of Im(HTEE) and the interior-only contribution to UV-finite complexity rely on the merged spacelike-timelike surface remaining extremal in the presence of scalar hair. The manuscript states that it employs the merging prescription but does not provide an explicit verification that the first variation of the area functional (including the scalar field contribution) vanishes at the junction point on the deformed metric. This check is necessary to confirm the surface is indeed extremal after backreaction.

    Authors: We agree that an explicit verification strengthens the construction. In the revised manuscript we will add a dedicated paragraph in the methods section computing the first variation of the area functional (including the scalar-field contribution) and showing that it vanishes at the junction point for the backreacted hairy metric. This confirms that the merged surface remains extremal. revision: yes

  2. Referee: [BTZ subregion complexity calculation] For the BTZ analytic result, the claim that the UV-finite term of the subregion complexity receives its entire contribution from the interior region alone should be supported by showing the explicit cancellation or vanishing of any boundary contributions in the regularized volume integral.

    Authors: We thank the referee for this suggestion. In the revised version we will present the explicit regularized volume integral for the BTZ case, including the boundary terms, and demonstrate their cancellation, thereby confirming that the UV-finite contribution originates entirely from the interior region. revision: yes

Circularity Check

0 steps flagged

No circularity: standard holographic extremal-surface and volume calculations on deformed metric

full rationale

The derivation constructs extremal surfaces by merging spacelike and timelike segments homologous to a timelike boundary interval, then evaluates the area functional and volume on the hairy black-hole metric. The resulting nontrivial Δt dependence of Im(HTEE) and the interior-only UV-finite complexity term follow directly from integrating the corrected metric and scalar profile; they are not obtained by fitting parameters to the target observables or by renaming prior results. Analytic-continuation comparisons are performed as consistency checks rather than as inputs. No load-bearing self-citation, self-definitional step, or ansatz smuggling is present in the reported chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard AdS/CFT duality, the validity of the merged-surface prescription for timelike subsystems, and the scalar deformation parameter; without the full text these cannot be audited in detail.

free parameters (1)
  • scalar vev φ0
    The strength of the relevant scalar operator deformation that sources the hairy black hole.
axioms (2)
  • domain assumption AdS/CFT correspondence holds for the deformed thermal state
    The duality between the hairy black hole geometry and the boundary CFT is assumed throughout.
  • ad hoc to paper Merged spacelike-timelike extremal surface prescription is valid
    The construction used to define the holographic timelike entanglement entropy is introduced in the paper.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Holographic complexity of conformal fields in global de Sitter spacetime

    hep-th 2026-04 unverdicted novelty 5.0

    Holographic complexity of CFTs in global dS_d is computed via volume and action prescriptions in AdS foliation and brane setups, then compared to results from static and Poincare patches.

Reference graph

Works this paper leans on

80 extracted references · 80 canonical work pages · cited by 1 Pith paper · 3 internal anchors

  1. [1]

    In the numerical calculations, we chooseL= 1

    These will give us two different boundary condi- tions for the scalar field. In the numerical calculations, we chooseL= 1. The relation betweenr 0 and ∆tfor various value of ˜ϕ0 =T −d+∆ϕ0 can be seen in figure 3, which recovers the analytical solution ford= 2 andϕ(r) = 0. The deformation parameter ˜ϕ0 modifies the relation between r0 and ∆tfor hairy black...

    work page

  2. [2]

    − ϕ2 0 √πΓ( 1 4) 16Γ( 3

    work page

  3. [3]

    , and A−(γT ) Vd−2Ld−1 ≈ i r0 2√πΓ( 3 4) Γ( 1

    r0 (38) + ϕ2 0 4r⋆ r2 0 +. . . , and A−(γT ) Vd−2Ld−1 ≈ i r0 2√πΓ( 3 4) Γ( 1

    work page

  4. [4]

    − iϕ2 0 √πΓ( 1 4) 16Γ( 3

    work page

  5. [5]

    (39) Finally, for (d,∆) = (3,1.5), the area functional with the first-order correction can be evaluated as follows: A+(γT ) Vd−2Ld−1 ≈2 ε − 2√πΓ( 3 4) r0Γ( 1

    r0 + iϕ2 0 4r⋆ r2 0 +. . . .(39) Finally, for (d,∆) = (3,1.5), the area functional with the first-order correction can be evaluated as follows: A+(γT ) Vd−2Ld−1 ≈2 ε − 2√πΓ( 3 4) r0Γ( 1

    work page

  6. [6]

    , (40) and A−(γT ) Vd−2Ld−1 ≈ i r0 2√πΓ( 3 4) Γ( 1

    + 3ϕ2 0 8 r2 0 ln r0 r⋆ √ 2 +. . . , (40) and A−(γT ) Vd−2Ld−1 ≈ i r0 2√πΓ( 3 4) Γ( 1

    work page

  7. [7]

    (41) 9 In this case, the leading∼ i r0 behavior gets corrected by ∼r 2 0 lnr 0 when the scalar field is turned on at ∆t≪1

    + 3iϕ2 0 8 r2 0 ln r0 r⋆ √ 2 +. . . . (41) 9 In this case, the leading∼ i r0 behavior gets corrected by ∼r 2 0 lnr 0 when the scalar field is turned on at ∆t≪1. We find an agreement between numerical results and an- alytical calculations for bothd= 2 andd= 3 in small ∆tlimit. When the scalar field is turned off, all expres- sions reduce to those obtained ...

    work page

  8. [8]

    − ϕ2 0 4 r0 √πΓ( 1 4) 4Γ( 3

    work page

  9. [9]

    ,(49) for ∆ = 2, and A(γτE) V1L2 =2 ε − 2√πΓ( 3 4) r0Γ( 1

    +. . . ,(49) for ∆ = 2, and A(γτE) V1L2 =2 ε − 2√πΓ( 3 4) r0Γ( 1

    work page

  10. [10]

    ,(50) for ∆ = 1.5

    − 3ϕ2 0 4 πr2 0 8 +. . . ,(50) for ∆ = 1.5. Here, we find similarr 0 scaling behavior between temporal entanglement entropy and HTEE in (d,∆) = (3,2) for the term that does not depends onr ⋆, while the∼r 2 0 term in the (d,∆) = (3,1.5) case can be obtained from taking the reverse analytic continuation r0 → −ir0 on Eq. (38). We therefore remarkably find th...

    work page

  11. [11]

    Frenkel, S

    A. Frenkel, S. A. Hartnoll, J. Kruthoff, and Z. D. Shi, Holographic flows from cft to the kasner universe, Journal of High Energy Physics2020, 10.1007/jhep08(2020)003 (2020)

    work page doi:10.1007/jhep08(2020)003 2020

  12. [12]

    R.-G. Cai, L. Li, and R.-Q. Yang, No inner-horizon the- orem for black holes with charged scalar hairs, Journal of High Energy Physics2021, 10.1007/jhep03(2021)263 (2021)

    work page doi:10.1007/jhep03(2021)263 2021

  13. [13]

    Are´ an, H.-S

    D. Are´ an, H.-S. Jeong, J. F. Pedraza, and L.-C. Qu, Kasner interiors from analytic hairy black holes, Journal of High Energy Physics2024, 10.1007/jhep11(2024)138 (2024)

    work page doi:10.1007/jhep11(2024)138 2024

  14. [14]

    C´ aceres,´A

    E. C´ aceres,´Angel J. Murcia, A. K. Patra, and J. F. Pe- draza, Kasner eons with matter: holographic excursions to the black hole singularity (2024), arXiv:2408.14535 [hep-th]

    work page arXiv 2024

  15. [15]

    Cai, M.-N

    R.-G. Cai, M.-N. Duan, L. Li, and F.-G. Yang, Towards classifying the interior dynamics of charged black holes with scalar hair (2024), arXiv:2312.11131 [gr-qc]

    work page arXiv 2024

  16. [16]

    L.-L. Gao, Y. Liu, and H.-D. Lyu, Internal structure of hairy rotating black holes in three dimensions (2024), arXiv:2310.15781 [hep-th]

    work page arXiv 2024

  17. [17]

    Carballo, A

    J. Carballo, A. K. Patra, and J. F. Pedraza, Diving in- side holographic metals, Journal of High Energy Physics 2025, 10.1007/jhep05(2025)072 (2025)

    work page doi:10.1007/jhep05(2025)072 2025

  18. [18]

    Zhang, X

    X.-K. Zhang, X. Zhao, Z.-Y. Nie, Y.-P. Hu, and Y.-S. An, Diving into a holographic multi-band superconduc- tor, Physics Letters B868, 139684 (2025)

    work page 2025

  19. [19]

    Cai, M.-N

    R.-G. Cai, M.-N. Duan, L. Li, and F.-G. Yang, Clarifying kasner dynamics inside anisotropic black hole with vector hair (2025), arXiv:2408.06122 [gr-qc]

    work page arXiv 2025

  20. [20]

    H. L. Prihadi, R. R. Firdaus, F. Khairunnisa, D. Dwipu- tra, and F. P. Zen, Stationary solution to charged hairy black hole in ads4: Kasner interior, rotating shock waves, and fast scrambling, The European Physical Journal C 85, 10.1140/epjc/s10052-025-14979-0 (2025)

    work page doi:10.1140/epjc/s10052-025-14979-0 2025

  21. [21]

    Maldacena, The large-n limit of superconformal field theories and supergravity, International Journal of The- oretical Physics38, 1113–1133 (1999)

    J. Maldacena, The large-n limit of superconformal field theories and supergravity, International Journal of The- oretical Physics38, 1113–1133 (1999)

    work page 1999

  22. [22]

    Bulk and Transhorizon Measurements in AdS/CFT

    I. Heemskerk, D. Marolf, J. Polchinski, and J. Sully, Bulk and transhorizon measurements in AdS/CFT, Journal of High Energy Physics2012, 165 (2012), arXiv:1201.3664 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  23. [23]

    Hartman and J

    T. Hartman and J. Maldacena, Time evolution of en- tanglement entropy from black hole interiors, Journal of High Energy Physics2013, 10.1007/jhep05(2013)014 (2013)

    work page doi:10.1007/jhep05(2013)014 2013

  24. [24]

    Guo and J

    W.-z. Guo and J. Xu, Duality of ryu-takayanagi sur- faces inside and outside the horizon, Phys. Rev. D112, L101901 (2025)

    work page 2025

  25. [25]

    Ryu and T

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence, Phys. Rev. Lett.96, 181602 (2006)

    work page 2006

  26. [26]

    Ryu and T

    S. Ryu and T. Takayanagi, Aspects of holographic entan- glement entropy, Journal of High Energy Physics2006, 045–045 (2006)

    work page 2006

  27. [27]

    V. E. Hubeny, M. Rangamani, and T. Takayanagi, A covariant holographic entanglement entropy proposal, Journal of High Energy Physics2007, 062–062 (2007)

    work page 2007

  28. [28]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Reviews of Mod- ern Physics81, 865–942 (2009)

    work page 2009

  29. [29]

    Islam, R

    R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement en- tropy in a quantum many-body system, Nature528, 77 (2015)

    work page 2015

  30. [30]

    K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, Timelike entanglement entropy, Journal of High Energy Physics2023, 10.1007/jhep05(2023)052 (2023)

    work page doi:10.1007/jhep05(2023)052 2023

  31. [31]

    K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, Pseudoentropy in dS/CFT and timelike entan- glement entropy, Phys. Rev. Lett.130, 031601 (2023)

    work page 2023

  32. [32]

    Afrasiar, J

    M. Afrasiar, J. K. Basak, and D. Giataganas, Time- like entanglement entropy and phase transitions in non- conformal theories, Journal of High Energy Physics 2024, 10.1007/JHEP07(2024)243 (2024)

    work page doi:10.1007/jhep07(2024)243 2024

  33. [33]

    Afrasiar, J

    M. Afrasiar, J. K. Basak, and D. Giataganas, Holo- graphic timelike entanglement entropy in non-relativistic theories, Journal of High Energy Physics2025, 10.1007/JHEP05(2025)205 (2025)

    work page doi:10.1007/jhep05(2025)205 2025

  34. [34]

    M. P. Heller, F. Ori, and A. Serantes, Geometric interpre- tation of timelike entanglement entropy, Physical Review Letters134, 10.1103/physrevlett.134.131601 (2025)

    work page doi:10.1103/physrevlett.134.131601 2025

  35. [35]

    M. P. Heller, F. Ori, and A. Serantes, Temporal entangle- ment from holographic entanglement entropy, Phys. Rev. X15, 041022 (2025)

    work page 2025

  36. [36]

    Alishahiha, Timelike holographic complexity (2025), arXiv:2510.25700 [hep-th]

    M. Alishahiha, Timelike holographic complexity (2025), arXiv:2510.25700 [hep-th]

    work page arXiv 2025

  37. [37]

    S. S. Jena and S. Mahapatra, A note on the holographic time-like entanglement entropy in lif- shitz theory, Journal of High Energy Physics2025, 18 10.1007/JHEP01(2025)055 (2025)

    work page doi:10.1007/jhep01(2025)055 2025

  38. [38]

    Afrasiar, J

    M. Afrasiar, J. K. Basak, and K.-Y. Kim, Aspects of holographic timelike entanglement entropy in black hole backgrounds (2025), arXiv:2512.21327 [hep-th]

    work page arXiv 2025

  39. [39]

    Jiang, P

    X. Jiang, P. Wang, H. Wu, and H. Yang, Timelike entan- glement entropy andt Tdeformation, Phys. Rev. D108, 046004 (2023)

    work page 2023

  40. [40]

    Basu and V

    D. Basu and V. Raj, Reflected entropy and timelike entanglement in T t-deformed cft 2s, Phys. Rev. D110, 046009 (2024)

    work page 2024

  41. [41]

    Fujiki, M

    K. Fujiki, M. Kohara, K. Shinmyo, Y. ki Suzuki, and T. Takayanagi, Entropic interpretation of einstein equa- tion in ds/cft (2025), arXiv:2511.07915 [hep-th]

    work page arXiv 2025

  42. [42]

    Li, M.-H

    G.-Y. Li, M.-H. Xiao, S. He, and J.-R. Sun, Entanglement first law for timelike entanglement entropy and linearized einstein’s equation (2025), arXiv:2511.17098 [hep-th]

    work page arXiv 2025

  43. [43]

    Kawamoto, R

    T. Kawamoto, R. Maeda, N. Nakamura, and T. Takayanagi, Traversable ads wormhole via non- local double trace or janus deformation, Journal of High Energy Physics2025, 10.1007/jhep04(2025)086 (2025)

    work page doi:10.1007/jhep04(2025)086 2025

  44. [44]

    Harper, T

    J. Harper, T. Kawamoto, R. Maeda, N. Nakamura, and T. Takayanagi, Non-hermitian density matrices from time-like entanglement and wormholes (2025), arXiv:2512.13800 [hep-th]

    work page arXiv 2025

  45. [45]

    Li, Z.-Q

    Z. Li, Z.-Q. Xiao, and R.-Q. Yang, On holographic time-like entanglement entropy, Journal of High Energy Physics2023, 10.1007/jhep04(2023)004 (2023)

    work page doi:10.1007/jhep04(2023)004 2023

  46. [46]

    Jiang, P

    X. Jiang, P. Wang, H. Wu, and H. Yang, Timelike en- tanglement entropy in ds 3/cft2, Journal of High Energy Physics2023, 10.1007/JHEP08(2023)216 (2023)

    work page doi:10.1007/jhep08(2023)216 2023

  47. [47]

    Chu and H

    C.-S. Chu and H. Parihar, Time-like entanglement en- tropy in ads/bcft, Journal of High Energy Physics2023, 10.1007/JHEP06(2023)173 (2023)

    work page doi:10.1007/jhep06(2023)173 2023

  48. [48]

    Grieninger, K

    S. Grieninger, K. Ikeda, and D. E. Kharzeev, Tempo- ral entanglement entropy as a probe of renormaliza- tion group flow, Journal of High Energy Physics2024, 10.1007/JHEP05(2024)030 (2024)

    work page doi:10.1007/jhep05(2024)030 2024

  49. [49]

    Narayan and H

    K. Narayan and H. K. Saini, Notes on time entanglement and pseudo-entropy, European Physical Journal C84, 499 (2024)

    work page 2024

  50. [50]

    He and H.-Q

    P.-Z. He and H.-Q. Zhang, Holographic timelike entan- glement entropy from rindler method, Chinese Physics C 48, 115113 (2024)

    work page 2024

  51. [51]

    B. Liu, H. Chen, and B. Lian, Entanglement entropy of free fermions in timelike slices, Phys. Rev. B110, 144306 (2024)

    work page 2024

  52. [52]

    Nunez and D

    C. Nunez and D. Roychowdhury, Timelike entanglement entropy: A top-down approach, Phys. Rev. D112, 026030 (2025)

    work page 2025

  53. [53]

    Xu and W

    J. Xu and W. Guo, Imaginary part of timelike entan- glement entropy, Journal of High Energy Physics2025, 10.1007/JHEP02(2025)094 (2025)

    work page doi:10.1007/jhep02(2025)094 2025

  54. [54]

    W.-z. Guo, S. He, and Y.-X. Zhang, Relation between time- and spacelike entanglement entropy, Phys. Rev. D 112, 086020 (2025)

    work page 2025

  55. [55]

    Chu and H

    C.-S. Chu and H. Parihar, Timelike entanglement en- tropy with gravitational anomalies, Journal of High En- ergy Physics2025, 10.1007/JHEP08(2025)038 (2025)

    work page doi:10.1007/jhep08(2025)038 2025

  56. [56]

    Giataganas, Timelike entanglement entropy and renormalization group flow irreversibility (2025), arXiv:2512.16499 [hep-th]

    D. Giataganas, Timelike entanglement entropy and renormalization group flow irreversibility (2025), arXiv:2512.16499 [hep-th]

    work page arXiv 2025

  57. [57]

    C´ aceres, A

    E. C´ aceres, A. Kundu, A. K. Patra, and S. Shashi, Trans- ir flows to black hole singularities, Physical Review D (2022)

    work page 2022

  58. [58]

    C´ aceres, A

    E. C´ aceres, A. K. Patra, and J. F. Pedraza, Shock waves, black hole interiors and holographic rg flows, Journal of High Energy Physics2024, 10.1007/jhep07(2024)052 (2024)

    work page doi:10.1007/jhep07(2024)052 2024

  59. [59]

    S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Holo- graphic superconductors, Journal of High Energy Physics 2008, 015–015 (2008)

    work page 2008

  60. [60]

    Hartnoll, G

    S. Hartnoll, G. Horowitz, J. Kruthoff, and J. Santos, Div- ing into a holographic superconductor, SciPost Physics 10, 10.21468/scipostphys.10.1.009 (2021)

    work page doi:10.21468/scipostphys.10.1.009 2021

  61. [61]

    R.-G. Cai, S. He, L. Li, and Y.-L. Zhang, Holo- graphic entanglement entropy in p-wave superconductor phase transition, Journal of High Energy Physics2012, 10.1007/jhep07(2012)027 (2012)

    work page doi:10.1007/jhep07(2012)027 2012

  62. [62]

    R.-G. Cai, S. He, L. Li, and Y.-L. Zhang, Holographic entanglement entropy in insulator/superconductor transition, Journal of High Energy Physics2012, 10.1007/jhep07(2012)088 (2012)

    work page doi:10.1007/jhep07(2012)088 2012

  63. [63]

    Albash and C

    T. Albash and C. V. Johnson, Holographic studies of en- tanglement entropy in superconductors, Journal of High Energy Physics2012, 10.1007/jhep05(2012)079 (2012)

    work page doi:10.1007/jhep05(2012)079 2012

  64. [64]

    A. M. Garc´ ıa-Garc´ ıa and A. Romero-Berm´ udez, Con- ductivity and entanglement entropy of high dimensional holographic superconductors, Journal of High Energy Physics2015, 10.1007/jhep09(2015)033 (2015)

    work page doi:10.1007/jhep09(2015)033 2015

  65. [65]

    Entanglement entropy in a holographic p-wave superconductor model

    L.-F. Li, R.-G. Cai, L. Li, and C. Shen, Entanglement entropy in a holographic p-wave superconductor model (2015), arXiv:1310.6239 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  66. [66]

    H. L. Prihadi, D. Dwiputra, F. Khairunnisa, and F. P. Zen, Scrambling in charged hairy black holes and the kasner interior, The European Physical Journal C85, 10.1140/epjc/s10052-025-14625-9 (2025)

    work page doi:10.1140/epjc/s10052-025-14625-9 2025

  67. [67]

    Alishahiha, Holographic complexity, Physical Review D92, 10.1103/physrevd.92.126009 (2015)

    M. Alishahiha, Holographic complexity, Physical Review D92, 10.1103/physrevd.92.126009 (2015)

    work page doi:10.1103/physrevd.92.126009 2015

  68. [68]

    Ben-Ami and D

    O. Ben-Ami and D. Carmi, On volumes of subregions in holography and complexity, Journal of High Energy Physics2016, 10.1007/jhep11(2016)129 (2016)

    work page doi:10.1007/jhep11(2016)129 2016

  69. [69]

    Carmi, R

    D. Carmi, R. C. Myers, and P. Rath, Comments on holographic complexity, Journal of High Energy Physics 2017, 10.1007/jhep03(2017)118 (2017)

    work page doi:10.1007/jhep03(2017)118 2017

  70. [70]

    Lebreuilly, Physical Review A96, 10.1103/Phys- RevA.96.033828 (2017)

    D. Stanford and L. Susskind, Complexity and shock wave geometries, Physical Review D90, 10.1103/phys- revd.90.126007 (2014)

    work page doi:10.1103/phys- 2014

  71. [71]

    Susskind, Entanglement is not enough, Fortschritte der Physik 10.1002/prop.201500095 (2016), first pub- lished: 18 January 2016

    L. Susskind, Entanglement is not enough, Fortschritte der Physik 10.1002/prop.201500095 (2016), first pub- lished: 18 January 2016

    work page doi:10.1002/prop.201500095 2016

  72. [72]

    de Haro, K

    S. de Haro, K. Skenderis, and S. N. Solodukhin, Holo- graphic reconstruction of spacetime¶and renormalization in the ads/cft correspondence, Communications in Math- ematical Physics217, 595–622 (2001)

    work page 2001

  73. [73]

    Skenderis, Lecture notes on holographic renormal- ization, Classical and Quantum Gravity19, 5849–5876 (2002)

    K. Skenderis, Lecture notes on holographic renormal- ization, Classical and Quantum Gravity19, 5849–5876 (2002)

    work page 2002

  74. [74]

    S. A. Hartnoll, G. T. Horowitz, J. Kruthoff, and J. E. Santos, Gravitational duals to the grand canonical en- semble abhor cauchy horizons, Journal of High Energy Physics2020, 10.1007/jhep10(2020)102 (2020)

    work page doi:10.1007/jhep10(2020)102 2020

  75. [75]

    Czech, J

    B. Czech, J. L. Karczmarek, F. Nogueira, and M. Van Raamsdonk, The gravity dual of a density ma- trix, Classical and Quantum Gravity29, 155009 (2012)

    work page 2012

  76. [76]

    A. C. Wall, Maximin surfaces, and the strong subaddi- tivity of the covariant holographic entanglement entropy, Classical and Quantum Gravity31, 225007 (2014). 19

    work page 2014

  77. [77]

    Geometric Constraints from Subregion Duality Beyond the Classical Regime

    C. Akers, J. Koeller, S. Leichenauer, and A. Levine, Ge- ometric constraints from subregion duality beyond the classical regime (2016), arXiv:1610.08968 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  78. [78]

    Chapman, H

    S. Chapman, H. Marrochio, and R. C. Myers, Complex- ity of formation in holography, Journal of High Energy Physics2017, 062 (2017)

    work page 2017

  79. [79]

    Auzzi, S

    R. Auzzi, S. Baiguera, A. Legramandi, G. Nardelli, P. Roy, and N. Zenoni, On subregion action complex- ity in ads3 and in the btz black hole, Journal of High Energy Physics2020, 10.1007/jhep01(2020)066 (2020)

    work page doi:10.1007/jhep01(2020)066 2020

  80. [80]

    Xu and W.-Z

    J. Xu and W.-Z. Guo, Imaginary part of timelike entan- glement entropy, Journal of High Energy Physics2025, 094 (2025)

    work page 2025