Holographic Subregion Complexity in General Vaidya Geometry

Chao Niu; Cheng-Yong Zhang; Yikang Xiao; Yi Ling; Yuxuan Liu

REVIEW 4 major objections 4 minor 1 cited by

Holographic Subregion Complexity in General Vaidya Geometry

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Holographic subregion complexity after a sudden quench grows in three stages: linear growth, slower linear growth, then either linear growth or linear decrease depending on the transition.

desk verdict Genuinely useful analytic three-stage computation of a subregion volume in Vaidya, but the paper never shows the surface is an extremum of the volume functional. read the letter →

arxiv 1908.06432 v2 pith:DBGY2AMY submitted 2019-08-18 hep-th

Yi Ling , Yuxuan Liu , Chao Niu , Yikang Xiao , Cheng-Yong Zhang This is my paper · ORCID

classification hep-th

keywords holographicsubregioncomplexityCVconjectureVaidya-AdSspacetimethermalquenchHRTsurfaceLloydboundgrowth

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 establish that, in a general Vaidya-AdS spacetime dual to a sudden quench, the holographic subregion complexity of a large strip-shaped boundary region evolves through three distinct stages. Under the subregion CV (complexity-equals-volume) conjecture, the early stage shows linear growth whose rate is fixed by the asymptotic metric coefficient; an intermediate stage near critical extremal surfaces shows a slower linear growth; and the late stage is either continued linear growth or linear decrease depending on whether the transition is discontinuous or continuous. The authors also compare the growth rates with the Lloyd bound: with a particular choice of length scale, the early-time rate saturates the bound while the intermediate rate always lies below it. Finally, they argue that subregion CV complexity remains different from global CV complexity even when the strip covers the whole boundary, because the tip of the extremal surface still settles to a finite equilibrium value.

What carries the argument

The central object is the codimension-one extremal volume $\Gamma_A$ whose boundary is the strip $A$ on the AdS boundary together with the HRT surface $\gamma_A$; the subregion CV conjecture sets $C_A = V(\Gamma_A)/(R G_N)$. The argument is carried by the conserved quantity $z^{d-1} L_S = C$, the matching conditions at the null shell $v'_+=v'_-$ and $z'_+ = (1-\tfrac12 g(z_c))z'_+$, and the critical HRT surface at $z_c=z_c^*(1-\epsilon)$ near which the function $H(z)$ develops a double zero at $z_m$. This double zero produces a logarithmic divergence in $\epsilon$ that links time, strip width, and the tip depth $z_t$; expanding time and on-shell volume in $1/z_t$ and $\epsilon$ around that critical configuration yields the three linear rates.

What would settle it

Compute the full $\Delta V(t)$ numerically in $d=2$ Schwarzschild-AdS for a very large strip half-width $l$ near the critical configuration $z_c=z_c^*(1-\epsilon)$ and check whether the slope tends to zero as $l\to\infty$; a nonzero surviving linear slope would invalidate the claim that the leading rate vanishes, while a zero slope would confirm that the linear growth is carried by uncontrolled subleading terms.

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Extended reading notes

Core claim

On its own terms, the central claim is that in the large-size limit the change in on-shell volume $\Delta V(t)$ of the extremal surface $\Gamma_A$ bounded by the strip and its HRT surface follows three analytic regimes. At early times, $\Delta V(t) = A_A R_{\mathrm{AdS}}^d (\omega/2) t + \cdots$, independent of the strip size, with $\omega$ fixed by $g(z)\sim\omega z^d$ near the boundary. At the intermediate stage, when the HRT surface is near its critical configuration $z_c=z_c^*(1-\epsilon)$ with $l\gg t\gg z_h$, the growth is again linear; for Schwarzschild-AdS in $d\ge 3$ it reads $\Delta V(t)=A_A R_{\mathrm{AdS}}^d \sqrt{d(d-2)/(2(d-1))}\, z_h^{-d}\, t$. At late times, if the transition is discontinuous the growth remains linear all the way to equilibrium, while if it is continuous the subtracted volume decreases linearly with coefficient $C_s$ in Eq. (116), which the paper argues is negative. The reason subregion CV differs from global CV even for an infinite strip is that in the global case the maximal-volume surface is always a Cauchy surface, whereas the subregion surface's tip always descends to a finite equilibrium value $z_s$.

Load-bearing premise

The intermediate- and late-time calculation assumes the hierarchy $z_c^*/z_t$, $z_m/z_t$, and $z_c^*|\log\epsilon|$ are all much smaller than one and that the leading logarithmic terms dominate the expansions of $t$ and $l$; in the $d=2$ SAdS case the leading rate vanishes as $l\to\infty$, so the claimed linear behavior would have to come from subleading terms that the paper does not systematically control.

Editorial extensions

If this is right

For any thin-shell holographic quench whose metric satisfies the stated properties, a large strip's subregion complexity grows linearly at early and intermediate times, with rates fixed only by the asymptotic falloff $\omega$ and the final horizon scale $z_h$.
In Schwarzschild-AdS with $d\ge3$, the early-time rate can saturate the Lloyd bound for a suitable length scale, while the intermediate rate is strictly below the bound for every finite dimension.
Subregion CV complexity is not a large-region limit of global CV complexity: even as the strip covers the whole boundary, the difference persists because the extremal surface's tip behaves differently.
Late-time behavior is controlled by the nature of the transition: a discontinuous jump of the tip gives continued linear growth, while a continuous approach to equilibrium gives a linear decrease near $t_s$.
The equilibrium time of the HRT surface from the referenced analysis applies directly to subregion complexity evolution, since the extremal volume and the surface share the same equilibrium time.

Reading between the lines

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

The paper does not derive the subregion CA (complexity-equals-action) analogue, but the same three-stage structure should appear there without the arbitrary length-scale ambiguity, so a direct CA calculation would be a natural test of the mechanism.
The late-time linear decrease, if taken literally, conflicts with any monotonic second law of complexity for mixed states; this suggests the apparent decrease may encode reference-state dependence or a limitation of the subregion CV measure rather than a true decrease of pure-state complexity.
In the $d=2$ Schwarzschild-AdS case the leading intermediate growth rate vanishes as $l\to\infty$, so the claimed linear behavior must be carried by subleading corrections; exact numerics for large $l$ could show whether a residual linear slope survives or the growth is genuinely sublinear.
Because the early-time rate depends only on $\omega$ while the intermediate rate depends on $z_h$ and the dimension, comparing thermal and electromagnetic quenches in the same spacetime should give different intermediate rates, providing a sharp observable signature of the quench type.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

You should know two things about this paper: it gives the first analytic three-stage description of subregion complexity growth in general Vaidya backgrounds, and it never checks that the surface whose volume it computes is an extremum of the volume functional. The first is genuine progress; the second is a load-bearing gap.

What's new and good: the paper extends the Liu-Suh strategy from entanglement entropy to subregion complexity, computes the early-time rate cleanly (Eq. 55), and derives the intermediate and late-time behavior for large strips, including the discontinuity classification and the explicit difference from global CV. The large-size limit was exactly where the earlier numerics lost control, so this fills a real hole. The derivations are detailed and mostly self-consistent for d ≥ 3.

Where the soft spots are. The main one is extremality. In Sec. 2.5 the volume is evaluated on the HRT profile v_HRT(z) with V_x = 0. But the stationarity condition for the volume functional contains V_x and differs from the HRT area equations (12)-(13); the paper never writes or solves that Euler-Lagrange equation. The ansatz satisfies the boundary conditions, but that's not enough. If the surface is off-shell, Eqs. (55), (83), and (116) describe the volume of a particular surface, not holographic subregion complexity. This is the stress-test point and, on reading the paper, it holds up.

Secondary: the intermediate/late expansions rely on the asymptotic hierarchy in Eq. (61). That's plausible for d ≥ 3, but in d=2 the leading linear rate vanishes as l→∞ (Eq. 91), so the claimed linear growth near the critical configuration has to be carried by subleading terms that aren't controlled. Also minor: Lloyd-bound saturation at early time follows from choosing R_AdS/R = (d-1)/(4π^2) from their earlier paper, so it's an input rather than a prediction.

Overall: the paper is careful and honest within its chosen framework, but the framework's link to the CV conjecture isn't demonstrated. If the authors explicitly reframe the result as the volume of the region bounded by A and γ_A, the computations stand as useful analytics. If they want to claim it as subregion complexity, they need to face the extremal volume problem.

Recommendation: send to peer review. The gap is real but addressable, and the topic is worth referee time. A serious referee should ask for either a numerical solution of the volume EL equation or a clear redefinition of the quantity.

Referee Report

4 major / 4 minor

Summary. The paper studies holographic subregion complexity (HSC) under the subregion CV conjecture in a general Vaidya-AdS spacetime describing a thin-shell quench. For a strip subregion of large half-width l, it claims that the evolution of HSC has three distinct stages: an early-time linear growth with rate (1/2)A_A R_AdS^d ω t, an intermediate-time linear growth with a dimension-dependent rate (e.g., Eq. (83) for d≥3), and a late-time stage that is either continued linear growth (discontinuous transition) or linear decrease (continuous transition). The paper also compares the growth rates with the Lloyd bound and argues that subregion CV differs from global CV even in the large-size limit.

Significance. If the central claim were established, the paper would provide a useful analytic characterization of mixed-state complexity growth during holographic thermalization, with explicit dimension-dependent rates that could be checked by numerics and field-theoretic models. The general Vaidya setup, the clear three-stage picture, and the step-by-step asymptotic expansions are valuable. However, the central quantity is computed without solving the volume-extremization problem for Γ_A, and the asymptotic control in the intermediate stage is incomplete. These issues currently block the main claim, so the paper needs substantial revision before its results can be accepted as holographic subregion complexity.

major comments (4)

[Sec. 2.5, Eqs. (35)-(38)] The volume functional is evaluated on the HRT profile v(z)=v_HRT(z) without solving the variational problem for the subregion CV surface. Equation (1) requires Γ_A to extremize the volume subject to ∂Γ_A = A ∪ γ_A. For a general embedding v=V(z,x), the induced metric contains V_x-dependent terms, and the stationarity condition for (38) differs from the HRT area equations (12)-(13). The ansatz V_x=0, V=v_HRT(z) satisfies the boundary conditions but is not shown to be a solution of the volume Euler-Lagrange equation. Unless this gap is closed, the rates in Eqs. (55), (83), and (116) are volumes of an arbitrary off-shell surface, not necessarily the HSC defined by the subregion CV conjecture.
[Sec. 3.2, Eqs. (61)-(69) and (90)-(91)] The intermediate-stage derivation relies on the hierarchy in Eq. (61) and on keeping only the leading log ϵ terms in t and l. The paper does not bound the subleading corrections. In d=2 this is not a technicality: Eq. (90)-(91) show that the leading linear growth rate vanishes as l→∞, so the claimed linear growth near the critical configuration must be carried by terms of higher order, which are not computed. The text acknowledges this point for d=2, but the abstract and Sec. 3.2 state the three-stage linear-growth picture without qualification for the general Vaidya setup (d≥2). The d=2 case must either be excluded from the claim or treated to the order that supports the linear rate.
[Sec. 3.3.2, Eqs. (117)-(119)] The late-time linear decrease for the continuous transition follows from the sign of C_s. The sign is decided by the inequality I'(z_s)/F'(z_s) > (1/z_s^{d-1}) ∫_0^1 dz z^{-d}/h(z_s z) in Eq. (119), but I'(z_s) and F'(z_s) are defined as limits of expressions that each contain a 1/θ divergence, and the finite parts are not computed. The inequality between the finite parts is not demonstrated; it appears to be inferred from the integrands rather than derived. Since the negative sign of C_s is the basis of the late-time decrease, this step needs a controlled derivation or a numerical check.
[Sec. 3.1, Eq. (59)] The early-time saturation of the Lloyd bound is enforced by the choice R_AdS/R = (d-1)/(4π²) in Eq. (59). Since R is the free length scale in the CV formula (1), this choice sets the rate to 2M/π by construction. The statement that the Lloyd bound is 'always saturated' at early time is therefore a choice of convention rather than a dynamical result, and the comparison with the Lloyd bound does not provide independent evidence for the proposed growth law.

minor comments (4)

[Sec. 2.3, Eq. (15)] There is a typo: 'anothor' should be 'another'.
[Sec. 3.2, Eq. (91)] Equation (91) appears garbled in the typeset version: the square-root symbol is rendered as an integral sign. Please correct the typesetting.
[General] The closely related work by Auzzi et al., 'On volume subregion complexity in Vaidya spacetime' (reference [14]), should be discussed explicitly so that the reader can see the relation and the differences between the two computations.
[Sec. 1 and Sec. 3.1] The notation for the two length scales R and R_AdS is confusing: Eq. (1) uses R, while Eq. (2) and most of the paper use R_AdS. Please clarify the distinction and the role of R consistently.

Circularity Check

1 steps flagged · score 2.0 of 10

The three-stage HSC evolution is derived from the Vaidya metric and HRT equations; only the reported Lloyd-bound saturation is fixed by choosing the arbitrary CV length scale from the authors' earlier work.

self definitional [Sec. 3.1, Eqs. (57)-(60); also Sec. 3.2, Eqs. (84)-(87)]
"In addition, if one sets the length scale as mentioned in [35]: RAdS/R = d−1 4π2 , (59) where we have set ℏ = 1. Then the result reduces to dC/dt = 2M π , (60) which is satisﬁed with the Lloyd bound at early time."

Every complexity rate computed here carries the undetermined prefactor RAdS/R (Eq. (57)). Equation (59) is not a derived relation; it is a choice of the arbitrary length scale R, taken from [35] (whose authors include two coauthors of this paper). Substituting this choice into Eq. (57) makes dC/dt = 2M/π by algebra, so the statement that the Lloyd bound is saturated at early time is logically identical to the chosen normalization. The same choice is reused at the intermediate stage (Eqs. (84)-(87)) to report sub-Lloyd rates. This is circular only for the Lloyd-bound comparison, not for the three-stage volume evolution: Eqs. (55), (83), and (116) are obtained from the metric and extremal-surface equations independently of R.

full rationale

The central chain of the paper is self-contained: with the Vaidya metric (2), the HRT equations (12)-(13), the matching conditions (21)-(22), and the volume functional (38), the early-time rate (55), the intermediate rate (83), and the late-time coefficient (116) are obtained by explicit expansion rather than by fitting or by importing the target result. The critical-surface analysis in Sec. 2.4 follows the independent Liu-Suh strategy [31], not a self-citation chain. No uniqueness theorem or ansatz is smuggled in solely through the authors' prior work. The only construction-by-definition step is the Lloyd-bound saturation: because the CV volume is normalized by an arbitrary scale R, Eq. (59) fixes R so that the prefactor in Eq. (57) equals 2/π, making the resulting 'saturation' an identity rather than a prediction. This is peripheral to the three-stage claim and is acknowledged in the text's sensitivity statement. The separate concern that Sec. 2.5 evaluates the volume on the HRT profile without solving the volume Euler-Lagrange equations is an omitted-extremization proof, not a circularity, so it does not raise the circularity score. Overall the derivation is analytically substantial and only the Lloyd-bound comparison reduces to its input by construction; score 2.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central derivation relies on the CV conjecture, the thin-shell Vaidya metric, and a set of asymptotic scaling assumptions. The only free parameter affecting the comparison with the Lloyd bound is the arbitrary length scale R. No new entities are introduced.

free parameters (1)

Length scale R in the CV formula = R = R_AdS (d-1)/(4π^2) chosen to saturate Lloyd bound
The complexity formula (1) contains an arbitrary length scale R. The paper notes (Sec. 3.1) that whether the Lloyd bound is saturated is sensitive to the choice of R, and the choice R_AdS/R = (d-1)/(4π^2) from [35] is used to obtain dC/dt = 2M/π.

assumptions (6)

domain assumption Subregion CV conjecture: complexity of a boundary region A is V(Γ_A)/(R G_N) (Eq. 1).
The entire calculation interprets the on-shell volume as holographic subregion complexity; this is a conjecture, not a proven theorem.
domain assumption Thin-shell Vaidya metric with f(v,z)=1-θ(v)g(z) (Eq. 3) accurately models a sudden quench.
The paper assumes vanishing shell thickness and a step function encoding the quench.
domain assumption Properties of g(z): g(z_h)=1, monotonic increase, g(z)→ω z^d as z→0 (Sec. 2.1).
These properties define the class of black hole backgrounds considered.
domain assumption HRT surface prescription and the matching conditions (Eqs. 21-22) across the null shell.
The extremal surface is constructed piecewise in AdS and black hole regions with continuity; the matching derivatives are integrated from the EOM.
ad hoc to paper Asymptotic expansions in the large-size limit, specifically the hierarchy Eq. (61) and dominance of log ϵ terms.
These scaling assumptions are introduced to extract leading-order behavior; the d=2 case shows they can fail to control the leading rate.
standard math Calculus of variations and conserved quantities from translation invariance of the action (Eqs. 12-15).
The EOM and conserved quantities C and E are derived by standard variational methods.

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Pith. "Pith review of Holographic Subregion Complexity in General Vaidya Geometry." pith.science (2026). https://pith.science/paper/DBGY2AMY

@misc{pith2026190806432,
  author       = {Pith},
  title        = {Pith review of: Holographic Subregion Complexity in General Vaidya Geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DBGY2AMY}},
  note         = {Machine review of arXiv:1908.06432}
}

read the original abstract

We investigate general features of the evolution of holographic subregion complexity (HSC) on Vaidya-AdS metric with a general form. The spacetime is dual to a sudden quench process in quantum system and HSC is a measure of the ``difference'' between two mixed states. Based on the subregion CV (Complexity equals Volume) conjecture and in the large size limit, we extract out three distinct stages during the evolution of HSC: the stage of linear growth at the early time, the stage of linear growth with a slightly small rate during the intermediate time and the stage of linear decrease at the late time. The growth rates of the first two stages are compared with the Lloyd bound. We find that with some choices of certain parameter, the Lloyd bound is always saturated at the early time, while at the intermediate stage, the growth rate is always less than the Lloyd bound. Moreover, the fact that the behavior of CV conjecture and its version of the subregion in Vaidya spacetime implies that they are different even in the large size limit.

Figures

Figures reproduced from arXiv: 1908.06432 by the authors.

**Figure 1.** For zc = z ∗ c > z0 c and zt → ∞, H(zm) = 0, and H(z) decreases from z = z ∗ c to z = zm as shown in (a), which means z(x) increases monotonically from zc and asymptotes to zm. The corresponding configuration of the critical surface is shown in (b). deriving the on-shell volume. In the next subsection, we tend to describe the configuration of critical HRT surfaces. Taking (3 + 1)-dimensional SAdS case as an example,… view at source ↗

**Figure 2.** For z 0 c < zc = z ∗ c (1 − ) where 0 < 1 and zt → ∞, H(zr) = 0, where zr is slightly less than zm. H(z) decreases from z = zc to z = zr as shown in (a), which means z(x) increases from zc to a maximum zr and then decreases to zero. The corresponding configuration of the critical surface is shown in (b). z 0 + > 0. Thus there exists a value z 0 c such that g(z 0 c ) = 2. Here we find that z 0 c = 21/3m−1/3 . Then… view at source ↗

**Figure 3.** Different red lines represent the subsystems with different half-width [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

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

On volume subregion complexity in Vaidya spacetime
hep-th 2019-08 conditional novelty 6.0 of 10

In the AdS3 Vaidya geometry, the extremal volume defining holographic subregion complexity is genuinely x-dependent during the quench, so the standard x-independent ansatz fails at intermediate times; early and late t...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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