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Entanglement measures for causally connected subregions and holography

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Pith's one-line read This paper claims a holographic duality for entanglement between timelike-separated subregions: the reflected entropy obtained by analytic continuation of twist correlators equals twice the timelike entanglement wedge cross section at leadi

desk verdict A careful, internally consistent proposal for timelike EWCS and reflected entropy, but the central match likely reflects shared analytic-continuation input rather than independent evidence of a new duality. read the letter →

arxiv 2508.05158 v1 pith:ZYMJWFJK submitted 2025-08-07 hep-th gr-qcquant-ph

XiangKun Gong , Wu-zhong Guo , Jin Xu This is my paper
classification hep-thgr-qcquant-ph
keywords timelikeentanglemententropytransitionoperatorSchwinger-Keldyshformalismreal-timereplicamethodwedgecrosssectionreflectedRyu-TakayanagisurfaceAdS3/CFT2
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

The paper tries to show that entanglement is not only a property of spacelike-separated regions: one can define a transition operator $T_{AB}$ for subregions $A$ and $B$ separated in time, compute its von Neumann entropy using a real-time replica method, and give that timelike entanglement entropy a holographic description in terms of extremal surfaces obtained by analytically continuing Euclidean Ryu-Takayanagi surfaces. The main new result is a timelike version of the entanglement wedge cross section, defined by a stationarity condition on a complex area, and a match at leading order in $G$ between the reflected entropy of timelike intervals (computed by analytic continuation of twist-operator correlators) and twice that timelike EWCS. If the paper is right, the holographic dictionary extends beyond the usual Cauchy-surface setup, and dynamical, causal information about states is encoded in complexified bulk geometry.

What carries the argument

The load-bearing object is the transition operator $T_{AB}$, a generally non-Hermitian generalization of the reduced density matrix defined by $\operatorname{Tr}(T_{AB}O_AO_B(t))=\langle O_AO_B(t)\rangle_\psi$, prepared by a Schwinger-Keldysh path integral with cuts on the two time-separated subregions. Its replicated traces $\operatorname{Tr}(T_{AB}^n)$ are evaluated by the real-time replica method and become Lorentzian correlators of twist operators. On the bulk side, the machinery is the analytic continuation of the Euclidean RT surface, $\tau_0\to i t_0$ with the turning point following $z_\tau \to z_t$; this produces complex extremal surfaces whose areas split into real and imaginary pa

What would settle it

Compute the timelike reflected entropy for two intervals in a finite-dimensional system (e.g. a spin chain) directly from a purification of the non-Hermitian $T_{AB}$ via the real-time replica method, and compare with $2E_W$ evaluated on the analytic-continuation branch $\tau_0\to i t_0$; a mismatch, or sensitivity to the $i\epsilon$ ordering of the twist insertions, would rule out the claimed duality.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that a consistent set of entanglement measures can be attached to timelike-separated subregions through the non-Hermitian transition operator $T_{AB}$, and that these measures have concrete holographic counterparts. For a single time interval, the entanglement entropy computed from $T_{AB}$ via the Schwinger-Keldysh replica method reproduces the known timelike entanglement entropy, and the corresponding RT surface is obtained by continuing the Euclidean extremal surface to $\tau_0\to i t_0$, producing a complex turning point. For two intervals, the entanglement wedge is bounded by such complex RT surfaces, and its cross section is defined by station

Load-bearing premise

All the holographic results rest on assuming that the correct bulk surface for a timelike interval is obtained by analytically continuing the Euclidean surface with the turning point following the chosen branch $\tau_0\to i t_0$; the paper adopts this branch from prior work without deriving it, so if another complex saddle is the one dual to the transition operator, the central duality collapses.

Editorial extensions

If this is right

  • If correct, the timelike entanglement entropy defined from $T_{AB}$ is a bona fide boundary quantity whose holographic dual is the analytic continuation of the Euclidean RT surface, not just a formal Wick rotation.
  • If correct, the reflected entropy of timelike intervals is holographically dual to the timelike entanglement wedge cross section, extending the spacelike reflected-entropy duality.
  • If correct, the timelike EWCS is well defined and positive in AdS$_3$ examples even though the entangling surface itself is complex.
  • If correct, the phase structure of two-interval timelike entanglement entropy corresponds to distinct RT-surface configurations, so the transition operator knows about bulk phase transitions.
  • If correct, the real-time replica method yields a route to other timelike measures such as logarithmic negativity, with the adjacent-interval limit tied to timelike mutual information.

Reading between the lines

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

  • Editorial inference: if the equality $S_R(T_1:T_2)=2E_W(T_1:T_2)$ survives beyond vacuum and thermal states, the stationarity condition on the complex area can serve as a working definition of a timelike entanglement wedge even where the usual notion of minimal surface is undefined.
  • Editorial inference: the paper adopts the analytic-continuation branch $\tau_0\to i t_0$ for the turning point from earlier work; a direct real-time replica evaluation of $\operatorname{Tr}(T_{AB}^n)$ on the same backgrounds would test whether that branch is selected by the path integral rather than imposed.
  • Editorial inference: because $T_{AB}$ encodes time evolution and scattering data, a holographic duality for its entropies suggests some dynamical information may be extractable from complexified bulk geodesics, though the paper does not demonstrate this.
  • Editorial inference: the observed parity pattern (imaginary timelike entanglement entropy in AdS$_4$, real in AdS$_5$) points to a dimensional dichotomy in complexified holography that could be probed in other odd/even bulk dimensions.
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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 / 4 minor

Summary. The paper extends entanglement measures from spacelike-separated to causally connected (timelike-separated) subregions in QFT and holography. It constructs a transition operator T_AB using the Schwinger-Keldysh path integral, computes its Rényi/von Neumann entropy via the real-time replica method, and shows that, in the examples considered, the result reduces to correlation functions of twist operators with timelike separations. On the holographic side, the paper adopts the procedure of analytically continuing Euclidean RT surfaces to select Lorentzian saddles, and verifies the resulting areas against exact analytic expressions and a numerical shooting method in AdS3, AdS4, AdS5, and BTZ geometries. It then proposes a timelike generalization of the entanglement wedge cross section by a stationarity condition on a complex area, and computes it in pure AdS3, global AdS3, and BTZ. Finally, by analytically continuing the spacelike reflected-entropy formula, it obtains a timelike reflected entropy and observes that at leading order in G it equals twice the timelike EWCS (Eq. (111)). The paper explicitly notes that no canonical purification for the non-Hermitian transition operator is known and that the timelike reflected entropy is only a candidate definition.

Significance. If the proposed duality were established, it would meaningfully extend holographic entanglement relations to causally connected regions and connect them to the transition-operator framework and pseudo-entropy. The paper contains several useful concrete results: a Schwinger-Keldysh path-integral representation of T_AB, explicit real-time replica computations, a numerical shooting method that agrees with analytic RT-area formulas, and closed-form expressions for the proposed timelike EWCS in multiple backgrounds. The authors are also transparent about the conceptual open problems, particularly the absence of a purification-based definition of timelike reflected entropy. However, the central equality S_R = 2E_W rests on two quantities that are themselves defined by analytic continuation, so as it stands the match is a consistency check among proposed definitions rather than an independent verification of a holographic duality. The paper is valuable as an exploratory proposal, but its main claim would need either a first-principles definition of the timelike reflected entropy or a substantially more cautious framing.

major comments (3)
  1. [§5.2 and §5.2.1, Eqs. (106), (111)] The central equality S_R = 2E_W is load-bearing but both sides are proposed definitions obtained by analytic continuation. Equation (106) defines timelike reflected entropy as the spacelike twist-correlator formula evaluated at τ_i → i t_i + ϵ_i, and §5.2.1 explicitly states that no canonical purification for the non-Hermitian T_A1A2B1B2 is known and that (106) is 'merely a candidate'. The right-hand side is the stationarity proposal (70). Since both sides are obtained from the same Euclidean spacelike duality by the same τ → it continuation and the same large-c conformal-block approximation, the match in (111) is a consistency check, not independent evidence. The claim 'supporting a holographic duality' should be either backed by a first-principles definition of the timelike reflected entropy or explicitly downgraded to a conjecture/consistency check.
  2. [§4.2, Eq. (70), footnote 2, Appendix D.1] The selection rule for the timelike EWCS is under-specified. Stationarity of a complex area can have multiple saddles; footnote 2 says to choose the 'appropriate surface' by 'physical or geometric principles', and Appendix D.1 lists several saddles, including (±Δ1, ±Δ2) and (±Δ1, ∓Δ2), before selecting (0,0) as 'minimal'. Since the area is complex-valued, the notion of 'minimal' is not defined without a prescribed ordering. Moreover, in §4.2.3 the BTZ complement example uses a different criterion, min{L^(1), L^(2)} in Eq. (83). This ambiguity affects the uniqueness of E_W(T1:T2) and therefore the comparison in Eq. (111).
  3. [§3.1, Eqs. (35)-(37)] The Lorentzian RT surface for a timelike boundary interval is obtained by the analytic continuation τ0 → i t0, z_τ → z_t from the Euclidean saddle. This branch choice is adopted from the procedure of Ref. [30], not derived from first principles. It is load-bearing because every subsequent RT area, the timelike entanglement wedge, and the EWCS depend on z_t. The agreement with the CFT computation is not a branch-selection test, since the CFT side (e.g., Eq. (15)) uses the same τ → it continuation of Euclidean twist correlators. Please state what observable would discriminate between different branches, or provide an independent derivation of the branch choice from the Schwinger-Keldysh replica path integral.
minor comments (4)
  1. [§1 and §3] There are several typos and small errors: 'labtory' in the Introduction; 'Talking into account' in §3.4.2; 'firslty' in Appendix A; 'extreme surface' used for 'extremal surface' in several places. In §3, after Eq. (25), G is described as 'the cosmological constant'; it should be Newton's constant.
  2. [Eq. (94)] The timelike logarithmic negativity for adjacent intervals is given up to an additive constant. Since the constant is not fixed, the reported relation (95) to the mutual information is only up to an undetermined shift; this should be stated more prominently.
  3. [§4.2.3, Eq. (83)] The definition of the EWCS for the complement intervals T1 = [-t,0] and T2 = (-∞,-t] ∪ [0,∞) uses a min{L^(1), L^(2)} rule. This is a different extremality/minimality prescription from the stationarity condition (70) used elsewhere, and the relation between the two prescriptions deserves comment.
  4. [Appendix A] The large-c conformal-block approximation is applied after analytic continuation to complex cross-ratios η_t and ar η_t. The validity of keeping only the vacuum block in this Lorentzian regime is not discussed; a comment on the relevant OPE limits and possible branch issues would help.

Circularity Check

2 steps flagged · score 6.0 of 10

Central timelike SR=2EW equality is the Wick rotation of the known spacelike duality; bulk RT branch is a load-bearing self-citation.

  1. renaming known result [Sec. 5.2, Eqs. (106)-(111); Sec. 4.2.1, Eq. (77)]
    "For the timelike case, the reflected entropy for the timelike case is S_R(T1 : T2) := S_R(A : B)|τi→iti+ϵi, (106) ... by comparing (108) with (77) and (110) with (85), and using the Brown-Henneaux relation c = 3/(2G), we find the following correspondence at leading order in G: S_R(T1 : T2) = 2EW(T1 : T2). (111)"

    The LHS of (111) is defined in (106) to be the spacelike reflected entropy S_R(A:B) of Eq. (103) evaluated at τ_i→i t_i+ε_i. The RHS (77)/(85) is the area of the timelike EWCS, whose defining RT surfaces were themselves obtained by the same τ→i t continuation (Sec. 3.1). In the Euclidean regime the identity S_R=2E_W is the known spacelike result cited to [18]; analytically continuing both sides of that identity gives (111) automatically. Thus the claimed timelike duality is the Wick rotation of the spacelike duality, not an independently derived prediction. The paper itself calls (106) only a 'candidate definition' (Sec. 5.2.1).

  2. self citation load bearing [Sec. 3.1, Eqs. (35)-(37); Sec. 1; Ref. [30]]
    "There remains some ambiguity regarding how to determine the complex RT surface in the complexified geometry. In this section, we present a procedure for constructing the RT surface via analytical continuation from its Euclidean counterpart [30]. ... Under this Wick rotation, the conserved quantity and the turning point transform as p_t := −i p_τ|τ0→it0, z_t := z_τ|τ0→it0. (36)"

    The bulk side of the timelike dictionary rests on the branch choice z_t := z_τ|_{τ0→i t0} in Eq. (36), which the paper adopts from [30] ('we present a procedure ... from its Euclidean counterpart [30]') without derivation. [30] is the authors' own prior work (Guo & Xu). The CFT timelike quantities are defined by the same τ→i t continuation (Eqs. (15),(19),(106)), so the holographic match verifies that both sides use the same continuation prescription, not that the prescription is the unique or correct Lorentzian saddle. The self-citation is load-bearing for every RT area and EWCS computed later.

full rationale

The paper contains no fitted parameters and substantial independent computation (Schwinger-Keldysh construction, replica twist correlators, numerical shooting, and checks against [28]/[29]). However, the headline result (111) is an analytic-continuation identity: both S_R(T1:T2) and E_W(T1:T2) are defined by applying the same τ→i t rule to the two sides of the known spacelike duality S_R=2E_W, so the equality is inherited by construction rather than independently established. Additionally, the timelike RT surface—the foundation of the bulk side—is adopted as a procedure from the authors' own prior work [30] rather than derived, making the holographic dictionary for timelike regions self-referential at its root. These are not fits, but they make the 'timelike duality' a consistency check among proposals. The paper is commendably explicit about these limitations (Sec. 5.2.1), but the central claim still reduces largely to a renaming of the spacelike result.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central derivation rests on the transition-operator construction from [34], the analytic-continuation prescription for complex RT surfaces from [30], the large-c conformal block approximation, and new definitions of timelike EWCS and reflected entropy. No numbers are fitted to data; the only undetermined constant appears in the peripheral negativity formula. The new definitions are flagged by the authors as proposals.

free parameters (1)
  • additive constant in timelike logarithmic negativity (Eq. 94)
    The adjacent-interval negativity is reported up to a constant 'associated with the coupling constant'; it is not computed. This is peripheral to the main EWCS/reflected-entropy claim.
assumptions (5)
  • domain assumption The transition operator T_AB can be represented by the Schwinger-Keldysh path integral with cuts on the forward contour and satisfies Tr_A T_AB = rho_B and Tr_B T_AB = rho_A.
    Section 2.3, Eqs. (9)-(11). This is the foundation for defining timelike entanglement measures; it is constructed to reproduce correlators but is not derived from an independent Hilbert-space definition.
  • domain assumption The physical RT surface for a timelike interval is obtained by Wick rotating the Euclidean extremal surface with tau_0 -> i t_0 and following the resulting complex branch.
    Section 3.1, Eqs. (35)-(37). Adopted from [30]; no proof is given that this branch is the unique or correct Lorentzian saddle.
  • domain assumption Large-c CFT conformal block dominance and the vacuum block formulas (117)-(118) give the dominant twist correlator contributions.
    Appendix A. Standard in holographic CFT, but an approximation that could fail if non-vacuum blocks or subleading corrections dominate.
  • ad hoc to paper The timelike entanglement wedge cross section is defined by stationarity of the complex area, Eq. (70), with the selected saddle chosen by physical or minimality considerations.
    Section 4.2, Eq. (70); Appendix D.1. The notion of minimality is ambiguous for complex areas, and among several saddles the point (0,0) is selected.
  • ad hoc to paper Timelike reflected entropy is defined by analytically continuing the Euclidean twist correlator formula, Eqs. (103)-(106).
    Section 5.2, Eq. (106). The authors state this is a candidate definition; no purification construction for the non-Hermitian T_AB is provided.
invented entities (1)
  • Timelike entanglement wedge and its cross section
    purpose: Bulk quantity conjectured to be dual to timelike reflected entropy.
    Proposed in Section 4.2; the wedge extends into complexified geometry and the cross section is a stationary surface satisfying Eq. (70). No independent falsifiable handle outside the paper yet, and the authors explicitly caution that it lacks the spacelike reconstruction interpretation.

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Pith. "Pith review of Entanglement measures for causally connected subregions and holography." pith.science (2026). https://pith.science/paper/ZYMJWFJK

@misc{pith2026250805158,
  author       = {Pith},
  title        = {Pith review of: Entanglement measures for causally connected subregions and holography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZYMJWFJK}},
  note         = {Machine review of arXiv:2508.05158}
}
abstract

In this paper, we investigate entanglement for causally connected subregions $A$ and $B$ in quantum field theory and holography. Recent developments have established that a transition operator $T_{AB}$ can be well-defined for such subregions, which is generally non-Hermitian. By employing the Schwinger-Keldysh formalism and the real-time replica method, we show how to construct $T_{AB}$ and compute associated entanglement measures. In certain configurations, this leads to a notion of timelike entanglement entropy, for which we provide explicit quantum field theory computations and propose a holographic dual via analytic continuation from the Euclidean setup. Both analytical and numerical results are compared and found consistent. If entanglement between causally connected subregions is to be meaningful, it should also be able to define other entanglement measures. Motivated by the spacelike case, we propose a timelike extension of the entanglement wedge cross section, though we do not expect it to carry the same physical interpretation. In AdS$_3$/CFT$_2$, we compute explicit examples and find that the timelike entanglement wedge cross section is generally positive. Furthermore, we show that the reflected entropy for timelike intervals -- obtained via analytic continuation of twist correlators -- coincides with twice the timelike entanglement wedge cross section at leading order in $G$, supporting a holographic duality in the timelike case. We also discuss the extension of other entanglement measures, such as logarithmic negativity, to timelike separated regions using replica methods. We highlight conceptual challenges in defining reflected entropy via canonical purification for non-Hermitian operators.

Figures

Figures reproduced from arXiv: 2508.05158 by the authors.

Figure 1
Figure 1. An illustration of subsystems A and B: (left) A and B are causally disconnected; (right) A and B are causally connected. For causally connected subsystems A and B (Figure on the right of [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. An illustration of the four possible configurations of regions [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. A typical Keldysh contour. As a result, Trρ(t) can be expressed as Trρ(t) = Z [Dϕ]e iSt;t0 (ϕ)−iSt0;t(ϕ) , (6) where St;t0 is the action defined as R t t0 dtL[ϕ]. If we have t0 < t1 < t, the Trρ(t1) could be written as Trρ(t1) = Z [Dϕ]e iSt1;t0 (ϕ)+iSt;t1 (ϕ)−iSt0;t(ϕ) . (7) We could derive it from (4), the time evolution operator U(t, t1) and its Hermitian conjugate U † (t, t1) mutually cancel under the trace. It i… view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: The Schwinger-Keldysh formalism illustrating the path-integral [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: The Schwinger-Keldysh formalism illustrating the path-integral [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: The Schwinger-Keldysh formalism illustrating the path-integral [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: The partial trace of TAB. (a) TrATAB: obtained by gluing the cut along region A. (b) TrBTAB: obtained by gluing the cut along region B. In this case, the resulting Schwinger–Keldysh formalism is equivalent to the geometry without the contribution over region A, as can …
Figure 8
Figure 8. Figure 8: Replica trick with a timelike interval for computing Tr( [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: An illustration of the four subregions considered in Section [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Replica trick with two timelike intervals for computing [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: (b), the surface initially follows the upper branch of a hyperbola in the real Lorentzian geometry. Then it passes through a point at infinity, analytically continues in the imaginary time direction, and reaches a turning point zt on the imaginary branch. After crossi…
Figure 12
Figure 12. Figure 12: Area A of the RT surface as a function of interval length t0 in the timelike AdS3 Poincar´e case. The blue curve represents the analytical result (43), while the red circles are obtained numerically. Panel (a) shows the real component, and panel (b) displays the imagi…
Figure 13
Figure 13. Figure 13: , the numerical results (red circles) perfectly agree with the analytical expression (blue curves). In particular, the surface area exhibits a periodic structure with period 4π due to the dependence sin(t0 2 ). The real part shows a sequence of logarithmic divergences…
Figure 14
Figure 14. Figure 14: Area A of the RT surface as a function of interval length t0 in the timelike AdS4 Poincar´e case. The blue curve represents the analytical result (52), while the red circles are obtained numerically. Panel (a) shows the real component, and panel (b) displays the imagi…
Figure 15
Figure 15. Figure 15: Area A of the RT surface as a function of interval length t0 for the timelike AdS3 black hole case. The blue curve shows the analytical expres￾sion (57), while red circles denote the numerical results. Panel (a) displays the real component; panel (b) shows the imagina…
Figure 16
Figure 16. Figure 16: Area A(0, 0;t0, 0) for the timelike RT surface in the AdS4 black hole background. The red crosses denote numerical results, while the blue curves represent the perturbative analytical result. Panel (a) shows the real part; panel (b) displays the imaginary part. We set…
Figure 17
Figure 17. Figure 17: Regularized area Ar = A(0, 0;t0, 0) − 1 δ 2 as a function of interval length t0 in the AdS5 black hole case. Panel (a) compares the full numerical results (red circles) with the perturbative expressions truncated at order t 2 0 (blue curve) and t 6 0 (black curve). Pa…
Figure 18
Figure 18. Figure 18: Illustration of the entanglement wedge and its cross section for [PITH_FULL_IMAGE:figures/full_fig_p033_18.png]
Figure 19
Figure 19. Figure 19: Illustration of two different phases of the bulk RT surfaces, with [PITH_FULL_IMAGE:figures/full_fig_p034_19.png]
Figure 20
Figure 20. Figure 20: Figure to show the connected phase for two time intervals [PITH_FULL_IMAGE:figures/full_fig_p038_20.png]
Figure 21
Figure 21. Figure 21: The EWCS for the intervals T1 = [−t, 0] and T2 = (−∞, −t] ∪ [0, +∞). Two types of geodesics are shown: Σ1 T1T2 (black dotted line) and Σ 2 T1T2 (green dashed line). constant. See [PITH_FULL_IMAGE:figures/full_fig_p040_21.png]
Figure 22
Figure 22. Figure 22: Real-time replica method for computing the logarithmic nega [PITH_FULL_IMAGE:figures/full_fig_p043_22.png]
Figure 23
Figure 23. Figure 23: Figure illustrating the concept of a timelike tube. The coordi [PITH_FULL_IMAGE:figures/full_fig_p050_23.png]
Figure 24
Figure 24. Figure 24: Illustration of the replica manifold connections. Top: The sheet [PITH_FULL_IMAGE:figures/full_fig_p055_24.png]

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