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Time-like Entanglement Entropy: a top-down approach

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proposes that inserting a parameter λ=±1 in the bulk metric and extremizing a single area functional computes time-like entanglement entropy for higher-dimensional holographic QFTs without analytic continuation, reproducing…

desk verdict A plausible but under-derived top-down prescription for tEE that matches known conformal results and offers new confining background computations; needs the companion paper and a direct confrontation of the surface-choice issue. read the letter →

arxiv 2505.20388 v2 pith:TQSGNADE submitted 2025-05-26 hep-th

Carlos Nunez , Dibakar Roychowdhury This is my paper · ORCID
classification hep-th
keywords time-likeentanglemententropyholographicAdS/CFTanalyticcontinuationconfinementcentralchargephasetransitionstabilitycriterion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims to compute time-like entanglement entropy (tEE) for higher-dimensional holographic QFTs by a direct top-down prescription that avoids analytic continuation from Euclidean to Lorentzian signature. The trick is to insert a parameter $\lambda = \pm 1$ in front of the time component of the bulk metric; setting $\lambda = +1$ gives the ordinary holographic entanglement entropy, while $\lambda = -1$ gives the time-like one. The authors test the approach on CFTs in several dimensions, reproducing known results for slab and spherical entangling regions, and on two confining models, where they find approximations, a stability criterion $Z(u_0) < 0$, and first-order phase transitions tied to confinement. They also show that the entropy computed this way yields the Liu-Mezei central charge (an entanglement-based measure of degrees of freedom) of the dual CFT, connecting real-time entanglement to the number of degrees of freedom.

What carries the argument

The central device is a sign parameter $\lambda$ inserted into the time component of the bulk metric, together with the eight-manifold $\Sigma_8$ defined by $v = \text{constant}$ and $t = t(u)$. After integrating out the internal coordinates, the area functional reduces to $S_{\rm tEE} = \frac{N}{4G_{10}} \int du \sqrt{G^2(u) + F^2(u) t'^2}$, whose first integral gives the time separation $T$ and the entropy; the approximate time separation $T_{\rm app} = \pi G(u)/F'(u)$ at the turning point $u_0$ and its derivative $Z(u_0) = \frac{d}{du}\left(\pi G/F'\right)\big|_{u_0}$ serve as stability diagnostics. For spherical entangling regions, the same machinery with the embedding $t(u) = \sqrt{R^2 u^2 - \lambda}/u$ yields a regulated entropy whose logarithmic term encodes the Liu-Mezei central charge $c_{\rm LM} = \lambda \hat{N}/(8G_{10})$, defining an entanglement-based measure of the number of degrees of freedom of the dual CFT.

What would settle it

In Witten's confining model, carry out the extremization of the full ten-dimensional area without factorising the integrand, and check whether the first-order phase transition (the swallow tail in the parametric plot of the entropy versus $|T|$) still appears at the same turning point $u_0$ where $Z(u_0)$ changes sign; if the location or existence of the transition changes, the surface selection or the stability criterion fails.

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Extended reading notes

Core claim

The central claim is that time-like entanglement entropy for holographic QFTs of dimension $d \geq 3$ is obtained by extremizing the area functional (4) over an eight-manifold $\Sigma_8$ with $v = \text{constant}$ and $t = t(u)$, after replacing the metric's time component by $\lambda dt^2$ with $\lambda = -1$. The same functional with $\lambda = +1$ reproduces the ordinary Euclidean holographic entanglement entropy, so the signature never needs to be changed during the computation. All Lorentzian information is carried by $\lambda$: the time separation $T$ becomes purely imaginary for $\lambda = -1$, the tEE is real for even $d$ and imaginary for odd $d$ when written in terms of $T$, and the stability of the embedding is governed by the sign of $Z(u_0) = \frac{d}{du}\left(\pi G/F'\right)$ evaluated at the turning point. The paper verifies the prescription by matching the bottom-up results of [10] for strips and spheres in arbitrary dimensions, and by showing that the approximate formulas for the time separation and the entropy agree with exact and numerical evaluations in Witten's confining model and in the Anabalón-Ross model.

Load-bearing premise

The prescription stands on the unproven choice that the surface with $v = \text{constant}$ and $t = t(u)$ is the true holographic surface for time-like entanglement entropy, and on the assumption that the bulk integrand factorises into a product of a function of the internal coordinates and a function of the radial coordinate.

Editorial extensions

If this is right

  • Time-like entanglement entropy for any CFT with a metric of the form (10) follows from a single extremisation with $\lambda = -1$; no Euclidean-to-Lorentzian analytic continuation is needed.
  • The stability of a tEE embedding is decided by the sign of $Z(u_0) = \frac{d}{du}(\pi G/F')$ at the turning point: negative means stable, positive means unstable and signals a possible phase transition.
  • tEE supplies central charges: for spherical entangling regions $c_{\rm LM} = \lambda \hat{N}/(8G_{10})$, and for slabs $c_{\rm slab}$ is proportional to the free energy of the dual CFT, so the time-like entropy carries the same information about degrees of freedom as the Euclidean one.
  • In confining holographic models (Witten's D4-brane model and the Anabalón-Ross model), the time-like entropy as a function of separation is double-valued and develops a swallow tail, indicating a first-order phase transition that does not occur in CFTs.
  • The tEE definition is invariant under U-duality, so the same result is obtained regardless of whether the computation is done in the ten- or eleven-dimensional supergravity description.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the imaginary values found for tEE in odd $d$ (when expressed in terms of $T$) are likely genuine physical features of transition matrices rather than artifacts of the chosen contour; a field-theoretic derivation in $d = 5$ would test that.
  • A testable point the paper leaves open is the factorization assumption: for a warped background where the internal metric couples to the radial coordinate, the reduction to the generic form (4) may fail, and computing tEE directly would show how restrictive 'it usually occurs' is.
  • If the phase-transition prediction holds, the location where $Z(u_0)$ changes sign could serve as a holographic definition of the confinement scale, providing a diagnostic the paper does not develop.
  • The $\lambda$-prescription is formulated for metrics with a time isometry; extending it to time-dependent or cosmological backgrounds would clarify whether the same extremal surface continues to compute the correct time-like entropy.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a holographic prescription for time-like entanglement entropy (tEE) by inserting a parameter λ=±1 into the metric and extremizing the area functional of an eight-manifold Σ8 with v=constant and t=t(u). This is intended to compute tEE directly in Lorentzian signature, bypassing the analytic continuation used in previous approaches. The authors derive analytic expressions for tEE and time separation in slab and spherical regions for conformal backgrounds, reproduce prior results of [10], propose Liu–Mezei central charges, and apply the formalism to two confining holographic models, Witten's D4-brane model and the Anabalon–Ross model, where they find stability criteria and first-order-like phase transitions.

Significance. If established, the prescription would offer a simpler, top-down route to tEE in higher-dimensional QFTs and provide new tools (a stability criterion, approximate formulas, and central-charge extraction) that could be widely used. The paper reproduces known conformal results and its analytic approximations are likely useful. However, the central claim of a 'robust' prescription is currently undermined by load-bearing assumptions that are either deferred to an unpublished companion paper or explicitly left unaddressed.

major comments (3)
  1. [Summary and Conclusions] The paper explicitly states: 'Our approach is not addressing the important issue regarding what is the correct surface (time or space-like) that minimizes the calculation [11,18].' This admission directly contradicts the abstract's claim of a 'robust top-down prescription' for computing tEE. The choice of the eight-manifold with v=constant and t=t(u) is the foundation of eq. (4); without a justification that this surface is the correct holographic dual of the time-like transition matrix, the computed StEE in eqs. (20), (43), (48) and the phase-transition results in the confining section are not established as entanglement entropies. The authors need to either provide a derivation or decisive citation for the surface-selection rule, or explicitly reframe the paper's claims as a conjecture.
  2. [Introduction and General Idea, eqs. (8)–(9)] The stability criterion Z(u0)<0 (eq. (8)) and the approximate tEE formula (eq. (9)) are load-bearing for the confining-model analysis, since they underlie the stability statements and the phase-transition claims in eqs. (49)–(51) and (55), as well as Figures 1–4. However, the text states that the proof of the criterion is 'presented in [25]' and that the derivation of eq. (9) is also 'presented in [25]', where [25] is an unpublished companion paper 'To appear'. As submitted, the reader cannot verify these central steps. The manuscript needs to include the derivations or reference a published, arXiv-numbered source.
  3. [Introduction and General Idea, eq. (4)] The reduction from the ten-dimensional integral (3) to the one-dimensional action (4) assumes that the integrand e^{-4Φ} f^{d-1} det[g_{9-d}] factorizes into a product of a function of the internal coordinates and a function of u. The text states that this 'usually occurs' but does not prove it, and the factorization is verified only in the explicit N=2 quiver example. Since eq. (4) is the starting point for every calculation in the paper, this assumption should be stated as a condition of the formalism and verified for each background considered, especially for the Witten and Anabalon–Ross models where the factorization is not demonstrated.
minor comments (6)
  1. [Eq. (15) and Sec. 'An explicit example'] The notation for L is used ambiguously: in eq. (15) L^(d-2) is the volume of spatial directions, while in eq. (40) N = 64π^2 L^2 P Σ R_k^2 appears to use L as an AdS scale. Please clarify the distinction.
  2. [Eqs. (18)–(22)] The paper states that T and T_app are purely imaginary for λ=-1, and that the tEE is real for even d and purely imaginary for odd d when expressed in terms of T. The branch conventions and the meaning of |T| in eq. (20) should be spelled out to avoid confusion.
  3. [Figures 2 and 3] The caption of Figure 2 says it displays 'the parametric plot of the entropy in terms of the separation', but the figure appears to show only S vs u0. The parametric plot is presumably Figure 3. Please correct the caption.
  4. [Reference [25]] Reference [25] is cited as 'To appear' with no arXiv number. Please update the citation if a preprint is available, or include the essential derivations in the present manuscript.
  5. [Sec. 'A Liu-Mezei central charge'] The definitions of c_LM in eqs. (30)–(31) are given without derivation, and the instruction to 'take the absolute value of the result' for λ=-1 is ad hoc. A brief justification or a reference would improve clarity.
  6. [Sec. 'Witten's model', eq. (47)] The typesetting of eq. (47) is ambiguous: the prefactor could be read as either (2R^{3/2}/√λ) * 1/(u0 sqrt(u0^3-uΛ^3)) or (2R^{3/2}/√λ) * u0/sqrt(u0^3-uΛ^3). Please clarify the expression, since the large-u0 scaling of T depends on this distinction.

Circularity Check

2 steps flagged · score 4.0 of 10

Conformal tEE results are benchmarked independently, but the stability criterion and approximate tEE formulas used for the confining phase-transition claims are deferred to an unpublished same-author companion [25], making those steps self-citation-load-bearing.

  1. self citation load bearing [Introduction, eq. (8) and following paragraph (stability criterion Z(u0))]
    "This criterium for stability can be proven using the formal similarity between the action for a generic Wilson loop and that of the time-like EE in eq.(4). The proof is presented in [25]."

    The stability criterion Z(u0)<0 is used to declare embeddings stable or unstable in both confining models and to infer the location of phase transitions, e.g. 'The function Z(u0) < 0 indicates that the embedding is stable for u0^3 > uLambda^3(1+3sqrt5), but unstable for values of u0 closer to uLambda.' The proof of the criterion for the time-like EE action is deferred entirely to ref. [25], an unpublished companion paper by the same two authors. Within this paper the criterion is therefore an imported, unverified self-citation rather than a derived statement, and the confining phase-transition claims depend on it.

  2. self citation load bearing [Introduction, eq. (9) and following sentence]
    "For the approximate time-like entanglement entropy of a strip we present an expression that relies on eq.(7). The derivation is presented in [25]"

    Eq. (9) is the approximate tEE formula used throughout the paper to produce Sapp for the conformal, Witten, and Anabalon-Ross cases and to generate the parametric swallow-tail plots that identify phase transitions. The derivation is attributed exclusively to [25], the same authors' unpublished companion, so the claimed 'accurate analytic approximations' are not derivable from the present paper's equations alone. The exact integrals in eqs. (47)-(48) and (56) are also evaluated and plotted, so this is partial rather than total circularity, but the approximate analysis that supports the phase-transition interpretation is not self-contained.

full rationale

The paper does not exhibit definitional circularity of the strongest kind: the conformal tEE expressions are checked against the independent bottom-up results of Doi et al. [10] (the text states 'The expression of eq.(20) ... We compare this expression with those in equations (4.36)-(4.37) in the paper [10], obtaining agreement'), and the sphere/hyperboloid results likewise reproduce [10]. The central-charge sector is a consistency check proportional to known free-energy coefficients, not a fitted prediction. However, two load-bearing ingredients are relegated to [25], an unpublished companion by the same two authors: the proof that Z(u0)<0 is the stability criterion for the tEE action, and the derivation of the approximate tEE formula in eq. (9). Both are used directly in the Witten and Anabalon-Ross analyses to claim stability loss and first-order phase transitions. The exact integrals and approximate quantities are also plotted, so the paper is not fully circular; but for those confining-model claims the reader cannot verify the derivation from the present text. The paper's own disclaimer about surface selection ('Our approach is not addressing the important issue regarding what is the correct surface (time or space-like) that minimizes the calculation [11,18]') is a scope and correctness limitation rather than a circular reduction. Overall score 4: some self-citation is load-bearing, but the central conformal derivation has independent content.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its load-bearing baggage is the unproven surface prescription, the factorization assumption, and the borrowed stability criterion; the free parameter is the undetermined integration constant in the approximate entropies.

free parameters (1)
  • Integration constant in S_app = unspecified
    In eqs. (21) and (51), the approximate time-like entanglement entropy is given up to an integration constant; the paper states 'This expression should be supplemented by an integration constant, that we omitted above.' The constant is not determined by the derivation and shifts the plots.
assumptions (4)
  • domain assumption The area functional on the embedded eight-manifold Σ8 with v = constant and t = t(u) yields tEE.
    Central postulate of the paper, stated in eqs. (2)-(4); no derivation or external benchmark beyond analogy with Ryu-Takayanagi and Wilson loops.
  • ad hoc to paper The integrand e^{-4Φ} f^{d-1} det g_{9-d} factorizes as a function of internal coordinates times a function of u.
    Stated as 'It usually occurs' before eq. (4); used without proof or stated conditions.
  • domain assumption The stability criterion Z(u0) < 0 for the embedding is inherited from Wilson loops [27].
    Quoted from ref. [27]; the formal similarity proof for tEE is deferred to [25].
  • domain assumption Setting λ = -1 gives the Lorentzian tEE without analytic continuation.
    The core method; assumes the extremal action with the signature parameter is the correct tEE while the surface-selection issue is left open.

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Cite this review

Pith. "Pith review of Time-like Entanglement Entropy: a top-down approach." pith.science (2026). https://pith.science/paper/TQSGNADE

@misc{pith2026250520388,
  author       = {Pith},
  title        = {Pith review of: Time-like Entanglement Entropy: a top-down approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQSGNADE}},
  note         = {Machine review of arXiv:2505.20388}
}
read the original abstract

We investigate the concept of time-like entanglement entropy (tEE) within the framework of holography. We introduce a robust top-down prescription for computing tEE in higher-dimensional QFTs, both conformal and confining, eliminating the ambiguities typically associated with analytic continuation from Euclidean to Lorentzian signatures. We present accurate analytic approximations for tEE and time-like separations in slab geometries. We establish a clear stability criterion for bulk embeddings and demonstrate that tEE serves as a powerful tool for computing CFT central charges, extending and strengthening previous results. Finally, we apply our framework to holographic confining backgrounds, revealing distinctive behaviours like phase transitions.

Figures

Figures reproduced from arXiv: 2505.20388 by the authors.

Figure 1
Figure 1. FIG. 1: On the left panel, the exact time separation [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The exact time like entanglement entropy in terms of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The exact time-like entanglement entropy in terms [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The exact time-like entanglement entropy for the An [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

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