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arxiv: 2604.11158 · v1 · submitted 2026-04-13 · ✦ hep-th · gr-qc

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Entanglement inequalities for timelike intervals within dynamical holography

Gaurav Katoch , Debajyoti Sarkar , Bhim Sen
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:14 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords timelike entanglementholographic entanglement entropyAdS-Vaidyastrong subadditivityAraki-Lieb inequalitymutual informationweak monotonicity
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The pith

Holographic calculations show that strong subadditivity is violated for timelike intervals in dynamical AdS-Vaidya geometries.

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

This paper examines entanglement inequalities for timelike subregions on the boundary in AdS3-Vaidya holography, extending prior analysis to pairs of intervals. It confirms that timelike mutual information remains positive and weak monotonicity holds when intervals do not overlap. For overlapping cases, examples demonstrate that while subadditivity and the Araki-Lieb inequality continue to hold, the strong subadditivity inequality is generally violated. A sympathetic reader would care because this highlights how entanglement properties can differ between spacelike and timelike separations in time-dependent holographic spacetimes, potentially affecting models of quantum information in gravitational systems.

Core claim

In the AdS3-Vaidya setup, the holographic entanglement entropy for timelike boundary intervals satisfies the subadditivity inequality and the Araki-Lieb inequality even when intervals overlap, but explicit computations show violations of the strong subadditivity inequality in these dynamical geometries.

What carries the argument

The holographic entanglement entropy prescription applied to timelike intervals in the AdS3-Vaidya bulk geometry, used to compute quantities like mutual information and test inequalities.

If this is right

  • Timelike mutual information is positive for non-overlapping timelike subregions.
  • Weak monotonicity holds for non-overlapping intervals.
  • The Araki-Lieb inequality remains valid for overlapping timelike intervals.
  • Subadditivity holds for timelike intervals, but strong subadditivity does not in general.

Where Pith is reading between the lines

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

  • This suggests that standard quantum information inequalities may need timelike-specific adjustments in holographic duals of dynamical spacetimes.
  • Similar violations might appear in other time-dependent holographic models, affecting how we model information scrambling during black hole formation.
  • These findings could motivate searches for analogous behaviors in lattice models or condensed matter systems with time-dependent couplings.

Load-bearing premise

The holographic prescription for entanglement entropy applies directly to timelike boundary intervals in the Vaidya geometry without requiring additional modifications.

What would settle it

A calculation in the dual boundary field theory that explicitly verifies whether strong subadditivity holds or is violated for timelike intervals in a similar dynamical setup.

Figures

Figures reproduced from arXiv: 2604.11158 by Bhim Sen, Debajyoti Sarkar, Gaurav Katoch.

Figure 1
Figure 1. Figure 1: Evolution of TEE as the function of boundary time. The different cases correspond [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Timeline of various disconnected and connected phases with respect to [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: D(1, 1) − C(1, 1) geodesic configurations valid through −∞ < T1 < −(2τ + t). The critical subsystem separation tc for which the phase transition takes place can simply be figured out by equating the value of TEE for disconnected phase to the connected phase, i.e. the moment TMI equates to zero. For the case at hand, due to the simplicity of the mathematical expressions it is possible to solve for tc. Howev… view at source ↗
Figure 4
Figure 4. Figure 4: C(1, 1) → D(1, 1) phase transition in TMI occurring at tc = 41.4 for T1 = −250. This is precisely the value one obtains by equating the mutual information (11) to zero, which is tc = (√ 2 − 1)τ. 3.2 Second Configuration (D(2, 1) − C(2, 1)) As the boundary subsystems are translated along the boundary time, the second configuration arises, in which the upper subsystem crosses the shell and its upper end poin… view at source ↗
Figure 5
Figure 5. Figure 5: D(2, 1) − C(2, 1) geodesic configurations valid through −(2τ + t) < T1 < − [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: C(2, 1) → D(2, 1) phase transition in TMI occurring at tc = 50.3 for T1 = −200.01. 3.3 Third Configuration (D(3, 1) − C(2, 1)) As is clear from figure 7, the next configuration appears when the lower subregion A is still in the AdS part of the geometry, while the upper subregion B has transitioned into case 3. Therefore, the corresponding disconnected phase should appropriately be labelled by D(3, 1). This… view at source ↗
Figure 7
Figure 7. Figure 7: D(3, 1) − C(2, 1) geodesic configurations valid through − [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Phase transition C(2, 1) → D(3, 1) at critical separation tc = 5.9 for T1 = −149.01. Given the ranges of time intervals, it is clear that when considering the connected phase, the corresponding configuration is still the trailing one C(2,1). Hence, the total TEE for the connected phase is still given by (18). The resulting plot for the mutual information has been provided in figure 8. From the plot, we see… view at source ↗
Figure 9
Figure 9. Figure 9: D(4, 1) − C(2,2) (left and middle panels), D(4, 1) − C(3, 3) (left and right panels) geodesic configurations valid through −(τ + t) < T1 < −(τ + t 2 ) and − [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Left panel: Phase transition from C(2, 2) → D(4, 1) at tc = 6.94 for T1 = −99.01. Right panel: Phase transition from C(3, 3) → D(4, 1) occurring at tc = 2.74 for T1 = −98.01. However, there are two connected configurations that arise in the above-mentioned region of validity −(τ + t) < T1 < −τ , i.e. for the same disconnected phase D(4,1). All of these config￾urations have been illustrated in figure 9. Th… view at source ↗
Figure 11
Figure 11. Figure 11: D(4, 2) − C(3, 4) geodesic configurations valid through −τ < T1 < − τ 2 . 15 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The phase transition from C(3, 4) → D(4, 2) occurs immediately at tc = 6.5 × 10−8 for T1 = −99.01. It should be noticed that this configuration is a very sensitive one in which the phase tran￾sition happens immediately at an extremely small separation. 3.6 Sixth configuration (D(4, 3) − C(3, 4)) In the next disconnected phase, T1 is still below the null shell while T2 > |T1|. This resulting configuration … view at source ↗
Figure 13
Figure 13. Figure 13: D(4, 3) − C(3, 4) geodesic configurations valid through − τ 2 < T1 < 0. The total timelike entanglement entropy of this disconnected phase can be worked out from (13) for (n, m) = (4, 3) to be S˜ 6D = c 2 log  β πϵ sinh  π β τ  + c 6 log τ ϵ  + c 6 Φ4(T3, T4, β) + c 2 iπ . 10 20 30 40 50 t -2 2 4 6 8 10 I  [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Phase transition from C(3, 4) → D(4, 3) occurring at tc = 32.6 for T1 = −49. Whereas, the corresponding connected phase does not switch and retains the former con￾figuration C(3,4) (in other words, S˜ 7C = S˜ 6C; see the right panel of figure 13) for the entire duration of −τ < T1 < 0 . The resulting plot for TMI and the critical time tc is given in figure 14. 3.7 Seventh configuration (D(4, 4) − C(4, 4))… view at source ↗
Figure 15
Figure 15. Figure 15: D(4, 4) − C(4, 4) geodesic configurations valid through T1 > 0. The corresponding connected phase is also characterized by T1 > 0 and pertains to C(4,4) configuration with S˜ 8C = c 3 log  β πϵ sinh  π β (2τ + t)  + c 3 log  β πϵ sinh  π β t  + c 6 Φ4(T1, T4, β) + c 6 Φ4(T2, T3, β) + cπi 2 . (27) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 I  [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 16
Figure 16. Figure 16: The phase transition from C(4, 4) → D(4, 2) occurring at tc = 0.98 for T1 = 0.001. Once again, figure 15 gives a schematic arrangement of all the surfaces in the Penrose diagram and figure 16 depicts the corresponding transition of TMI at the critical point tc. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Time evolution of ˜I C ≥ 0 in the connected phase when subsystem separation is chosen to be t = 10−8 . Vertical partitions are same as the ones in figure 2. In the above section, we have listed out all the possible configurations and arrangements for the pair of timelike intervals, and we have shown that in all of those configurations, the timelike mutual information behaves as expected. Namely, even for … view at source ↗
Figure 18
Figure 18. Figure 18: The fact that the connected phase has surfaces homologous to ( [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Timeline for small t (not to scale) with respect to T1. In this progression of cases, the magenta line shows the range − 3 2 τ + t < T1 < −τ + t  of timelike SSA violation. 4.1.1 Early time behaviour for small overlap In this subsection we will show that the first two configurations for small t obey SSA inequalities over the range −∞ < T1 < − 3 2 τ + t. The first configuration appearing here is D(1, 1) −… view at source ↗
Figure 20
Figure 20. Figure 20: D(1, 1) − C(1, 1) geodesic configurations valid through −∞ < T1 < −2τ + t. As the collective region A ∪ B is evolved further upwards in time, the disconnected config￾uration switches over to D(2, 1) and the associated configuration representing the RHS of (30) simultaneously switches over to C(2, 1). The Penrose diagram of this configuration is depicted in figure 21. This combination of configurations las… view at source ↗
Figure 21
Figure 21. Figure 21: D(2, 1) − C(2, 1) geodesic configurations valid through −2τ + t < T1 < − 3 2 τ + t. The combined plot for these first two configurations have been depicted clearly in figure 22. The switch-over between the two configurations has been denoted by the vertical dashed line. In this plot (and the ones to follow), the dashed red curve corresponds to S˜D(n,m) and the blue curve corresponds to S˜ C(n′ ,m′) . The … view at source ↗
Figure 22
Figure 22. Figure 22: Evolution of boundary intervals in the range [PITH_FULL_IMAGE:figures/full_fig_p023_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: D(3, 1) − C(2, 1) geodesic configurations valid through − 3 2 τ + t < T1 < −τ [PITH_FULL_IMAGE:figures/full_fig_p024_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: D(3, 2) − C(2, 2) (left and middle panels), D(3, 2) − C(3, 3) (left and right panels) geodesic configurations valid through −τ < T1 < −τ + t 2 and −τ + t 2 < T1 < −τ +t respectively. T1 > − 3 2 τ +t and persists as long as T1 < −τ. The schematic Penrose diagrams appear in figure 23. As these pairs of subregions are evolved further by increasing T1, the configurations switches from D(3, 1) − C(2, 1) to D(3… view at source ↗
Figure 25
Figure 25. Figure 25: Evolution of SSA during the range − 3 2 τ + t < T1 < −τ + t for τ = 100 and overlap t = 35. Since, the blue curve for S˜ C(n′ ,m′) immediately jumps above the red curve representing S˜D(n,m) , SSA is clearly violated in this range. One may further inquire whether TEE in the same range also reflects a violation in the slightly weaker Araki-Lieb and triangle inequality. It turns out that the answer is negat… view at source ↗
Figure 26
Figure 26. Figure 26: D(4, 2) − C(3, 4) (left and right panels), D(4, 3) − C(3, 4) (middle and right panels) geodesic configurations valid through −τ + t < T1 < − τ 2 and − τ 2 < T1 < 0 respectively. Finally, the configuration D(4, 2) − C(3, 4) switches over to D(4, 4) − C(4, 4), giving rise to 25 [PITH_FULL_IMAGE:figures/full_fig_p025_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Evolution of boundary intervals in the range [PITH_FULL_IMAGE:figures/full_fig_p026_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Plots of S˜(A) + S˜(B) (in red), S˜(A) ∪ S˜(B) (in blue) and |S˜(A) − S˜(B)| (in green) showing the validity of subadditivity like statement and the Araki-Lieb inequality (31) for all T1. We have used t = 40 and τ = 100. Having presented the clear violation of SSA for TEE, one might wonder about the status of the weaker Araki-Lieb and triangle inequality given by (31). This is an important question, as th… view at source ↗
Figure 29
Figure 29. Figure 29: Timeline for large t (not to scale) as function of T1. In this progression of events, the magenta line shows the range − 3 2 τ + t < T1 < − τ 2  of timelike SSA violation [PITH_FULL_IMAGE:figures/full_fig_p027_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: D(2,2), D(3,3) geodesic configurations valid through −τ < T1 < − 3 2 τ + t and − τ 2 < T1 < −τ + t respectively. As shown in the schematic plot of figure 29, the region of violation of SSA is now contained within the interval − 3 2 τ + t to − τ 2 . Following the simple procedure outlined in the previous sec￾tion, one can numerically compare the collective TEE of the various configurations, and arrive at t… view at source ↗
Figure 31
Figure 31. Figure 31: Evolution of boundary intervals for all range of [PITH_FULL_IMAGE:figures/full_fig_p028_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Plots of S˜(A) + S˜(B) (in red), S˜(A) ∪ S˜(B) (in blue) and |S˜(A) − S˜(B)| (in green) showing the validity of subadditivity like statement and the Araki-Lieb inequality (31) for all T1. We have used t = 60 and τ = 100. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Evolution of boundary interval with varying overlap. The red and the dashed blue [PITH_FULL_IMAGE:figures/full_fig_p029_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Evolution of boundary intervals with varying overlap showing the violation of SSA [PITH_FULL_IMAGE:figures/full_fig_p030_34.png] view at source ↗
read the original abstract

This paper extends our previous work (arXiv:2504.14313) of a single timelike subregion to two, in the framework of AdS$_3$-Vaidya holography. We confirm the positivity of timelike mutual information and the statement of weak monotonicity when the subregions are non-overlapping. We also study entanglement inequalities such as Araki-Lieb inequality and strong subadditivity when the intervals start to overlap. In line with the recent findings in the literature, we provide explicit working examples showing that the timelike version of the strong subadditivity is generally violated in these setups, even though the statements of subadditivity and Araki-Lieb inequality hold true.

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 extends the authors' prior work (arXiv:2504.14313) on timelike entanglement entropy from a single subregion to pairs of subregions in AdS3-Vaidya holography. It confirms positivity of timelike mutual information and weak monotonicity for non-overlapping intervals. For overlapping intervals it examines the Araki-Lieb inequality and strong subadditivity, supplying explicit numerical examples in which the timelike version of strong subadditivity is violated while ordinary subadditivity and the Araki-Lieb inequality continue to hold.

Significance. If the reported computations are robust, the work supplies concrete, falsifiable examples of timelike SSA violation inside a dynamical holographic geometry. This distinguishes timelike from spacelike entanglement inequalities and supplies a useful benchmark for future studies of quantum-information constraints in time-dependent bulk spacetimes. The explicit working examples constitute a clear strength.

major comments (2)
  1. [§2] §2 (holographic prescription for timelike intervals): The central claim rests on applying an extremal-surface formula to timelike boundary intervals in the time-dependent Vaidya geometry. The manuscript does not derive or justify this extension from the variational problem or from causality constraints in a collapsing-shell background; the standard HRT prescription applies to spacelike regions. Without this justification the reported violations could be artifacts of the chosen regularization or surface choice rather than physical.
  2. [§4.2] §4.2 (explicit examples of SSA violation): The numerical results for overlapping timelike intervals are presented without tabulated values of the individual entropies, the precise interval endpoints, the regularization scheme, or error estimates. This makes it impossible to assess whether the observed violation is generic or tied to particular post-hoc choices of parameters.
minor comments (2)
  1. Figure 3 and 4: axis labels and legends should explicitly indicate which curves correspond to timelike versus spacelike intervals and which quantities are plotted (S(A), S(A∪B), etc.).
  2. [Introduction] The relation between the present calculations and the authors' earlier arXiv:2504.14313 paper should be stated more precisely in the introduction so that the new overlapping-interval results are clearly distinguished from prior single-interval results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for recognizing the potential significance of our explicit examples of timelike SSA violation in a dynamical holographic setting. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and additional data.

read point-by-point responses

  1. Referee: §2 (holographic prescription for timelike intervals): The central claim rests on applying an extremal-surface formula to timelike boundary intervals in the time-dependent Vaidya geometry. The manuscript does not derive or justify this extension from the variational problem or from causality constraints in a collapsing-shell background; the standard HRT prescription applies to spacelike regions. Without this justification the reported violations could be artifacts of the chosen regularization or surface choice rather than physical.

    Authors: We agree that a more explicit justification of the timelike prescription would strengthen the manuscript. The prescription used here is a direct extension of the one introduced and motivated in our prior work (arXiv:2504.14313), where the extremal-surface formula was adapted to timelike intervals via analytic continuation of the HRT construction together with consistency checks against causality in the Vaidya geometry. In the revised version we will add a short dedicated paragraph in §2 that recalls this motivation, cites the relevant literature on timelike entanglement, and lists the consistency conditions (analytic continuation, bulk causality, and matching to known limits) that the prescription satisfies. We do not claim a first-principles variational derivation in this paper, as that lies beyond the present scope, but the added discussion should make the assumptions transparent. revision: yes

  2. Referee: §4.2 (explicit examples of SSA violation): The numerical results for overlapping timelike intervals are presented without tabulated values of the individual entropies, the precise interval endpoints, the regularization scheme, or error estimates. This makes it impossible to assess whether the observed violation is generic or tied to particular post-hoc choices of parameters.

    Authors: We concur that the current presentation lacks sufficient numerical detail for independent verification. In the revised manuscript we will insert a new table in §4.2 that reports, for each explicit example: the precise boundary interval endpoints (t1, x1; t2, x2), the individual timelike entanglement entropies S(A), S(B), S(AB), the regularization cutoff, and the estimated numerical uncertainty arising from the surface-finding algorithm. This will allow readers to reproduce the reported violation of timelike strong subadditivity and to judge its robustness. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior single-interval work; central claims on inequality violations are based on independent explicit computations in Vaidya geometry.

full rationale

The paper cites its own previous work (arXiv:2504.14313) to establish the single timelike subregion setup and holographic framework in AdS3-Vaidya, then performs new explicit calculations for overlapping intervals to confirm mutual information positivity, weak monotonicity, Araki-Lieb, subadditivity, and SSA violations. These results consist of direct evaluations of entanglement quantities rather than any reduction of outputs to inputs by construction, fitted parameters renamed as predictions, or self-definitional loops. The self-citation supports the starting framework but is not load-bearing for the new inequality examples, which remain independent computations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the AdS/CFT correspondence applied to timelike regions and the specific choice of the Vaidya metric to model dynamical holography. No new free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • domain assumption The AdS/CFT correspondence holds for the Vaidya geometry in AdS3.
    Standard assumption in holographic calculations of entanglement.
  • domain assumption Entanglement entropy can be computed via the area of extremal surfaces in the bulk for timelike boundary intervals.
    Extension of Ryu-Takayanagi formula to timelike cases as per prior work.

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discussion (0)

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