The Speed of Quantum Information Spreading in Chaotic Systems

Brian Swingle; Josiah Couch; Phuc Nguyen; Shenglong Xu; Stefan Eccles

REVIEW 3 major objections 5 minor 2 cited by

The Speed of Quantum Information Spreading in Chaotic Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that one information speed, $v_I = v_E/(1-f)$, unifies the entanglement and butterfly speeds in chaotic many-body systems.

desk verdict A novel and mostly credible unification of entanglement and butterfly speeds, but the central formula rests on an unproven linear-growth assumption for strip entanglement. read the letter →

arxiv 1908.06993 v2 pith:PKDYT3IB submitted 2019-08-19 cond-mat.stat-mech gr-qchep-thquant-ph

Josiah Couch , Stefan Eccles , Phuc Nguyen , Brian Swingle , Shenglong Xu This is my paper · ORCID

classification cond-mat.stat-mechgr-qchep-thquant-ph

keywords quantuminformationscramblingentanglementspeedbutterflyHayden-PreskillprotocolgrowthholographicentropyAdS/CFTchaoticmany-bodysystems

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 argues that in chaotic quantum many-body systems, initially localized quantum information spreads ballistically at a single information speed for strip-shaped regions, $v_I$, which interpolates between two previously separate quantities: the entanglement speed $v_E$ (how fast entanglement grows after a quench) and the butterfly speed $v_B$ (how fast perturbations spread). The proposed relation is $v_I = v_E(f,\varepsilon)/(1-f)$, where $f$ is the fraction of thermal entanglement present in the initial state, so $f=0$ gives $v_E$ and $f\to 1$ gives $v_B$. The authors support this relation with a quantum-information argument based on a generalized Hayden-Preskill protocol, with numerical simulations of a chaotic spin chain, and with holographic calculations in AdS/CFT using charged Vaidya black brane geometries. If correct, the formula turns two seemingly different scrambling speeds into limits of one underlying process, with entanglement generation rather than operator growth as the usual bottleneck.

What carries the argument

The load-bearing object is a generalized Hayden-Preskill protocol applied to a spatially local chaotic system: a reference is entangled with a few thermal cells, and a region A can recover that entanglement whenever it holds both enough scrambled output and enough entanglement with the complement. The quantitative input is the assumed entanglement growth law for strips, $S[A,\psi(t)] = \min\{ f s |A| + v_E(f) s |\partial A| t,\, s|A|\}$, linear growth at speed $v_E(f)$ followed by immediate saturation; this gives the saturation time $t_{\text{sat}} = R(1-f)/v_E(f)$ and hence $v_I = v_E/(1-f)$. On the holographic side the same physics is carried by HRT extremal surfaces in Vaidya-AdS-RN backgrounds: the smallest entanglement wedge containing the infalling reference particle expands at rate $v_I$, and the analytic appendix derives $v_E(f)$ from the extremal-surface area growth for strips at finite entanglement fraction.

What would settle it

In a chaotic system with a wide available region, prepare a state with known entanglement fraction $f$ (say $f=0.5$), track the smallest region that can recover a reference qubit over time, and independently measure the strip entanglement speed $v_E(f)$; if the measured information speed differs from $v_E(f)/(1-f)$ by more than finite-size error, the central relation fails.

Watch

Extended reading notes

Core claim

The central discovery is that the speed at which a growing strip-shaped region can first recover initially local quantum information is set by the speed of entanglement generation, not by direct operator spreading. Concretely, after a quench in which a reference system is locally entangled with the chaotic system, the smallest half-width $R$ of a strip that can recover the reference grows as $R(t) = v_I t$ with $v_I = v_E(f,\varepsilon)/(1-f)$, where $f$ is the initial entanglement fraction, $\varepsilon$ the energy density, and $v_E$ the entanglement speed for that fraction. The argument identifies the recoverable region with the region whose entanglement entropy has just saturated, and assumes this saturation happens abruptly for strips. In the limit $f\to 1$ the formula yields $v_I \to v_B$, the butterfly speed, so the entanglement and butterfly speeds are unified as the two endpoints of one information speed; the paper also states a refined version $v_I = \min(v_E/(1-f), v_B)$ when operator growth could be the bottleneck. The relation is checked numerically in a 22-site spin chain at infinite temperature and holographically by locating the smallest HRT surface containing an infalling particle entangled with the reference in single- and two-sided AdS-Reissner-Nordstrom-Vaidya geometries.

Load-bearing premise

The formula relies on the assumption that a strip's entanglement entropy grows linearly at rate $v_E(f)$ and then saturates abruptly, with no rounded crossover; if real strips saturate continuously, the relation between saturation time and $v_E$ changes.

Editorial extensions

If this is right

The entanglement speed and butterfly speed are not independent: both are the $f=0$ and $f\to1$ limits of one information speed, so a measurement of $v_E(f)$ across fractions determines $v_B$ whenever the bound $v_E/(1-f)\le v_B$ saturates.
In the strip geometry, entanglement generation is the bottleneck for information scrambling: information cannot be recovered by a region faster than that region can become maximally entangled with its complement.
With a suitable decoding operation, information can be read out from outside a radius $v_I t$ around its initial location, even though the information is not locally detectable.
In the $f=1$ traversable-wormhole setup with a spatially local coupling, a locally detectable signal propagates at speed $v_B$.
In the large-spin random circuit limit the formula predicts $v_I=v_B=1$ for all $f$, since $v_E(f)=1-f$, matching the expectation that information propagates as fast as possible in that case.

Reading between the lines

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

Editorial inference: if the strip formula is confirmed at larger sizes, the same quench protocol could convert entanglement-speed measurements into butterfly-speed estimates without needing OTOC measurements.
Editorial inference: the sharp-saturation assumption is likely the main obstruction to generality, since spherical regions already show $v_I=2v_E$ at $f=0$ and saturation at $v_B$ above a critical fraction, suggesting a shape-dependent analog of the formula.
Editorial inference: wavefront broadening should leave the asymptotic information speed unchanged as long as it is sub-ballistic, so mutual-information contour measurements in larger systems could cleanly separate the asymptotic speed from finite-time rounding.
Editorial inference: because the protocol requires dynamically generated entanglement, it may serve as a diagnostic of weak versus strong scrambling, distinguishing systems that scramble operators quickly but generate entanglement slowly.

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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

The paper does something genuinely new: it defines a single information speed v_I and argues, via a generalized Hayden-Preskill protocol, that v_I = v_E(f)/(1-f), interpolating between the entanglement speed at f=0 and the butterfly speed at f=1. The derivation is clear, the holographic calculation is nontrivial, and the spin-chain numerics support the formula at small f. That is a real contribution, and the paper is honest about its assumptions.

The load-bearing step is Eq. (6): the strip entanglement entropy is assumed to grow linearly at speed v_E(f) until a sharp, first-order saturation. The formula for v_I follows immediately from the saturation time, so everything hinges on that law. The authors state it as a general expectation, not a theorem, and they explicitly note it fails for spherical regions. I think the concern in the stress-test note is real but proportionate: this is a conjecture, not a derivation, and the holographic evidence for it comes from one family of models (AdS-RN-Vaidya). Still, the holographic calculation is a genuine independent check, and it works across all f, including the limit v_E/(1-f) → v_B. That is more than a toy model.

The spin-chain data is the weakest link: at large f the system is too small to extract v_E reliably, and the plotted v_E/(1-f) actually exceeds v_B near f=1, forcing the min-formula. The paper attributes this to finite-size effects, which is plausible, but the data don't cleanly test Eq. (1) in that regime. Also, no code or data is provided, and the velocity extractions lack error bars. These are standard issues for numerical work of this kind, so I'd call them minor, not fatal.

There is a mild circularity in identifying the recoverable region with the just-saturated region, but v_E is independently measured in both the spin chain and the holographic setting, so the relation is not purely definitional. The Hayden-Preskill argument is doing real work.

Who should read this: people working on entanglement dynamics, scrambling, and holographic thermalization. It will likely be cited as the reference for the interpolation formula, and it deserved a serious referee. I would send it to peer review, asking the authors to state explicitly that Eq. (6) is a conjecture for general systems, to discuss when it might fail (beyond the sphere example), and to provide error bars or a stronger finite-size analysis for the spin chain at large f. The central claim is probably right for strips; it just needs a more careful statement of its domain of validity.

Referee Report

3 major / 5 minor

Summary. The paper proposes a general characterization of quantum information spreading in spatially local chaotic systems. It defines an information speed v_I through a quench-type experiment: a reference is entangled locally with the system, and v_I is the asymptotic growth rate of the smallest centered strip region that can recover near-maximal mutual information with the reference. The central result is Eq. (1), v_I = v_E(f, ε)/(1-f), where f is the initial entanglement fraction and v_E(f) is the entanglement speed. The paper argues that this formula interpolates between the standard entanglement speed at f = 0 and the butterfly speed at f = 1. The evidence consists of a generalized Hayden-Preskill argument (Sec. II), Krylov-space spin-chain numerics (Sec. III), holographic HRT-surface calculations in single- and double-sided AdS-RN-Vaidya spacetimes (Sec. IV and Appendix A), and a decoding/traversable-wormhole analysis (Sec. V). The paper is explicit that Eq. (1) is a conjecture for strip-like regions and that the underlying linear-growth-plus-first-order-saturation ansatz is an assumption rather than a theorem.

Significance. If the central formula is correct, it provides a conceptually valuable unification of two previously separate velocity scales, the entanglement speed and the butterfly speed, and makes a quantitative, experimentally accessible prediction for scrambling fronts in chaotic systems. The paper has several genuine strengths: the quantum information argument is transparent; the holographic derivation in Appendix A gives a closed-form expression for v_E(f) that is checked numerically; the spin-chain protocol independently measures v_E and v_B; and Sec. IV demonstrates a parametric separation between the speeds using near-extremal charged black holes, which makes the comparison sharp. The claim is falsifiable through the spin-chain mutual-information protocol and through holographic entanglement-wedge calculations. The main weakness is that the central relation is conditional on an unproved entanglement-growth ansatz, so the scope of the claimed universality remains uncertain.

major comments (3)

[Sec. II, Eq. (6)] The central formula Eq. (1) is derived from the assumed entanglement-growth law S[A, ψ(t)] = min{fs|A| + v_E s|∂A| t, s|A|}, which posits linear growth at constant speed v_E(f) followed by sharp first-order saturation. This is the load-bearing step: Eq. (10) is simply the saturation-time consequence of this ansatz, and Eq. (11) then gives v_I = v_E/(1-f). The paper states the ansatz as a 'general expectation' rather than a theorem, and explicitly notes in Sec. IV.D that it is false for spherical subregions. The only clean verification across all f is the holographic AdS-RN-Vaidya family. Since there is no independent criterion for when strip-like chaotic systems exhibit this linear-first-order behavior, the central claim needs either an additional argument establishing the regime of validity of Eq. (6) or a substantially more cautious statement of the theorem's scope.
[Sec. II.B, Eqs. (9)-(11)] The Hayden-Preskill step identifies the just-saturated region as the 'system' and the complement as the dynamically generated 'memory', concluding that any unsaturated region can recover the information and any saturated region cannot. This identification assumes that the dynamically generated entanglement functions like the maximally entangled memory of the standard Hayden-Preskill protocol. The paper acknowledges that the entanglement is not Bell-pair-like and that decoding may be more difficult, but it asserts that the recovery criterion is unaffected. That assertion is not immediate: Hayden-Preskill recovery requires a near-maximally entangled memory and scrambling dynamics of a specific type, and the just-saturated region need not a priori be the minimal recoverable region. A more rigorous decoupling argument, or a numerical test of the recoverability criterion, would be needed to make this step load-bearing.
[Sec. III, Fig. 2(b)] The spin-chain test is one of the three pillars of evidence, but at large f the extracted v_E(f)/(1-f) exceeds v_B, forcing the ad hoc transition to the min formula Eq. (2). This discrepancy is attributed to finite-size effects, but no quantitative finite-size analysis (e.g., scaling of v_E and v_I with L) or error bars are provided. Since the holographic result in Fig. 5 instead satisfies v_E(f)/(1-f) < v_B for all f, the empirical confirmation of Eq. (1) is incomplete precisely in the regime where the formula's behavior is most distinctive. The manuscript should either supply a finite-size scaling estimate supporting the attribution or soften the claim that the spin-chain data verify Eq. (1) at large f.

minor comments (5)

[Sec. III, Fig. 2] The figure would benefit from error bars and a precise description of how v_I and v_E are extracted from the mutual-information wavefront and from the linear-growth regime; the inset showing v_E(f) is mentioned but its extraction procedure is not described in the text.
[Eq. (29)] The quantity z_m appearing in Eq. (29) is defined only in Appendix A.8; the main text should define it or explicitly refer the reader to that definition.
[Abstract and Sec. I] The abstract states that as the entanglement density varies from zero to one the information speed 'varies from the entanglement speed to the butterfly speed.' In the strip holographic calculation v_I is strictly below v_B for all f < 1 and only approaches v_B as f → 1; the wording should be adjusted to 'approaches the butterfly speed' to match Eqs. (3)-(4).
[Sec. V.B] The traversable-wormhole discussion is presented for BTZ where v_B is the speed of light and for f = 1; the text should state more explicitly that the light-cone result shown in Fig. 8 is obtained in this special limit and does not by itself demonstrate propagation at the general information speed v_I = v_E/(1-f) for f < 1.
[Sec. III, Eq. (14)] The text uses 'compliment' in 'the compliment of A'; this should be 'complement'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (1) follows from an explicitly stated entanglement-growth assumption, and v_E is measured independently in both the spin-chain and holographic tests.

full rationale

The central formula v_I = v_E(f)/(1-f) is derived in Sec. II from the stated entanglement-growth law (Eq. 6), which posits linear growth at speed v_E(f) followed by sharp first-order saturation, giving t_sat = R(1-f)/v_E (Eq. 10); combining this with the definition of v_I as R_sat/t (Eq. 5) yields Eq. (1). This is a transparent conditional derivation from an assumption, not a circular reduction: v_E(f) is not defined in terms of v_I, and it is independently measured or computed in both testbeds. In the spin-chain calculation, v_I is extracted from mutual-information wavefronts while v_E comes from entanglement-growth rates and v_B from OTOCs. In the holographic calculation, v_I is read off the slope of the equal-area 'green' curve in the HRT phase diagram, whereas v_E(f) is obtained from the analytic expression (29) and also from an independent fixed-width entanglement-growth computation; the two agree. The paper is explicit that Eq. (6) is an assumption and that it fails for spherical subregions (Sec. IV D), which restricts the claimed regime of validity but does not amount to circularity. The traversable-wormhole and bounding arguments rely on standard external results, not on a self-citation chain that carries the main claim. No step reduces to its own input by construction or by fitted-parameter renaming.

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

The central formula is supported by physical assumptions rather than derived from first principles with no inputs. The most important is the linear-growth-plus-first-order-saturation entropy law for strips (Eq. 6). Other assumptions are the operator growth speed v_B acting as an effective light cone, the inequality v_E <= v_B, and the Hayden-Preskill recovery criterion. In the holographic and wormhole sections, standard AdS/CFT dictionary and large-N eikonal approximations are additional domain assumptions. No free parameters are fitted to force the final relation; f is an independent variable of the claim, not a fitted constant.

assumptions (6)

domain assumption Entanglement growth law for strips: S[A, psi(t)] = min{f s |A| + v_E s |partial A| t, s |A|}; linear growth until first-order saturation.
Equation (6). Needed to get tsat = R(1-f)/v_E and hence v_I = v_E/(1-f). Not proven; fails for spherical regions.
domain assumption Operator growth effective light cone: W(t) is approximately supported on an interval of length 2 v_B t on states of fixed energy density.
Section II A, Eq. (7). Defines v_B and is used for the f=1 argument and the upper bound.
domain assumption v_E(f) <= v_B for all f, with v_B as the effective maximum speed.
Section II A. Needed to use Eq. (1) without the min; not rigorously proven, though true in known cases.
domain assumption Hayden-Preskill recovery criterion: a region can recover the reference entanglement if it contains a small part of the scrambled output together with enough entanglement; identified with regions whose entanglement has not saturated.
Section II B. This identification converts saturation size into information speed; it is argued, not proven.
domain assumption AdS/CFT dictionary: HRT surfaces compute entanglement entropies and entanglement wedges recover bulk particles; valid for the Vaidya geometries.
Section IV A. Maps the boundary information-speed question to extremal surface phase diagrams.
domain assumption Large-N factorization and eikonal shockwave scattering in the traversable wormhole computation.
Appendix B. Used for Eq. (57); authors note the elastic eikonal approximation becomes unreliable at very large T.

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Cite this review

Pith. "Pith review of The Speed of Quantum Information Spreading in Chaotic Systems." pith.science (2026). https://pith.science/paper/PKDYT3IB

@misc{pith2026190806993,
  author       = {Pith},
  title        = {Pith review of: The Speed of Quantum Information Spreading in Chaotic Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PKDYT3IB}},
  note         = {Machine review of arXiv:1908.06993}
}

read the original abstract

We present a general theory of quantum information propagation in chaotic quantum many-body systems. The generic expectation in such systems is that quantum information does not propagate in localized form; instead, it tends to spread out and scramble into a form that is inaccessible to local measurements. To characterize this spreading, we define an information speed via a quench-type experiment and derive a general formula for it as a function of the entanglement density of the initial state. As the entanglement density varies from zero to one, the information speed varies from the entanglement speed to the butterfly speed. We verify that the formula holds both for a quantum chaotic spin chain and in field theories with an AdS/CFT gravity dual. For the second case, we study in detail the dynamics of entanglement in two-sided Vaidya-AdS-Reissner-Nordstrom black branes. We also show that, with an appropriate decoding process, quantum information can be construed as moving at the information speed, and, in the case of AdS/CFT, we show that a locally detectable signal propagates at the information speed in a spatially local variant of the traversable wormhole setup.

Figures

Figures reproduced from arXiv: 1908.06993 by the authors.

**Figure 2.** FIG. 2: (a) Comparing the information expansion between a product state and a maximally entangled state. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3: Penrose diagram of the planar, 1-sided AdS-RN-Vaidya spacetime. The infalling shell is in blue. The [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4: Phase diagram with the parameters [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5: Left panel: Red and green dots are [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: FIG. 7: Setup of the traversable wormhole calculation. The coupling between the two sides gives rise to a [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: FIG. 8: Plot of the imaginary part of [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]

**Figure 9.** Figure 9: FIG. 9: Upper left: Plot of [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]

**Figure 10.** Figure 10: FIG. 10: Plot of the area of the HRT surface versus boundary time, with the half-width kept fixed at the same value [PITH_FULL_IMAGE:figures/full_fig_p031_10.png]

**Figure 11.** Figure 11: FIG. 11: Left panel: Phase diagram for the case [PITH_FULL_IMAGE:figures/full_fig_p031_11.png]

**Figure 12.** Figure 12: FIG. 12: Schematic illustrating the decomposition of the time-evolution operator, [PITH_FULL_IMAGE:figures/full_fig_p034_12.png]

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Forward citations

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Reference graph

Works this paper leans on

99 extracted references · 26 canonical work pages · cited by 2 Pith papers

[1]

Consider a generic eternal black hole

A few generalities This subsection reviews the formalism of traversable wormhole as presented in [53]. Consider a generic eternal black hole. In Kruskal coordinates, the metric takes the form: ds2 =−a(UV )dUdV +r2(UV )dy2 (32) We deﬁne a0 = a(0) to be the value on the horizon of the functions a, and denote by r+ the horizon radius. We take the time coordi...
[2]

the sweet spot

For fc <f < 1, we have vI =vB. This is a qualitative diﬀerence from the strip case (ﬁgure 5): in the strip case, recall that vI <v B for all f <1. Whenf <fc, the smallest entanglement wedge containing the infalling particle is at a discontinuous transition. On the other hand, when fc <f < 1, that smallest entanglement wedge is at a continuous transition. ...
[3]

sweet spot

Example of planar BTZ black hole We now specialize to the planar BTZ black hole in order to have analytic tractability. The disadvantage of specializing to this solution is that vB is just the speed of light in 3 dimensions. Nevertheless, this will be suﬃcient for our purposes. The bulk-to-boundary propagator for the planar BTZ is given in Kruskal coordin...
[4]

sweet spot

The light-cone Equation (57) gives us C as a function of xL, xR, tL and tR. Note that the dependence on those four boundary spacetime coordinates is entirely through A and B. In order to see the “sweet spot” of traversability from (57), we now set t0 = 0,−tL =tR≡T and ∆xL = ∆xR =X, and study the function C(T,X ). C(T,X ) = αψ 2r+ eiπ 2 ∆ψ× ∫ ∞ 0 dv v2∆ψ−1...
[5]

initial” and “ﬁnal

Metric and notation We consider a special case of AdS-Vaidya style metrics, given by ds2 = L2 z2 ( −h(v,z )dv2− 2dvdz +d⃗ x·d⃗ x ) h(v,z ) =hi(z) + Θ(v)(hf(z)−hi(z)) (A1) where Θ(v) is a Heaviside step function, so hi(z) and hf(z) describe the black hole geometry in the initial ( v <0) and ﬁnal (v >0) regions, respectively, separated by the shock at v = 0...
[6]

The case immediately of interest to us is n =d− 1, and the corresponding boundary observable is an entanglement entropy

Boundary strips The simplest boundary regions to consider are generalized “strips” of ﬁnite width in the x1 direction (-R to R), inﬁnite extent in the x2 through xn direction, and xn+1,...,x d−1 set to zero. The case immediately of interest to us is n =d− 1, and the corresponding boundary observable is an entanglement entropy. The formulas reported here d...
[7]

tip”, or the deepest point of the extremal surface: z = zt and v = vt (we also use a subscript “t

Symmetries and Conserved Quantities The lack of explicit x dependence results in a constant of motion on extremal surfaces: J =zn√ Q =zn√ 1−h(v,z )(v′)2− 2v′z′ (A3) This allows us to simplify both equations of motion: ∂zQ− 2nz−1Q =∂x∂z′Q (A4) ∂vQ =∂x∂v′Q (A5) In both regions v >0 and v <0, the metric (A1) is independent of v. Setting ∂vhi,f = 0 in the v e...
[8]

To do this we examine the eﬀect of the shock itself on the extremal surface

Junction conditions It still remains to determine the value of Ef for a given zt,vt. To do this we examine the eﬀect of the shock itself on the extremal surface. The function h(v,z ) is not independent of v at the shock, so we must return to equations (A4) and (A5). Let the subscript “ c” denote quantities evaluated at the position of shock crossing, and ...

Show all 99 references

[9]

Governing equations Everywhere oﬀ the shock itself, the governing equations which we will integrate to ﬁnd R, t, and A become: dx dz = ∓1√ H(z) dv dz = −1 h(z) ( 1± E√ H(z) ) dA dz =An−1(Lzt)n ∓1 z2n √ H(z) (A12) with H(z) =h(z) (z2n t z2n− 1 ) +E2 The top signs are used if z(...
[10]

With the boundary time set to track time elapsed after the shock, we let t =v on the boundary

Surfaces at late time t We are interested in areas of extremal surfaces associated with late times t and large boundary width R. With the boundary time set to track time elapsed after the shock, we let t =v on the boundary. The boundary time associated with an extremal surface...
[11]

linear growth regime

Limit f → 0 For pre-shock geometries with relatively small mass, zi,+ >> zf,+ (or in the limit of empty AdS), we can push zt >>z f,+,z∗ c . In this case, R receives a contribution linear inzt, while area and time receive negligible contributions. Thus by sending zt arbitrarily...
[12]

To reach an analogous large R limit, zt is pushed extremely close to the outermost pre-shock horizon

Finite f In the case that there is a horizon zi,+∼zf,+,zt cannot be pushed arbitrarily large. To reach an analogous large R limit, zt is pushed extremely close to the outermost pre-shock horizon. Let zt =zi,+(1−δ). As δ→ 0, the dominant pre-shock contributions to A and R then ...
[13]

Limit f → 1 In general spacetimes and dimensions, analytic expressions for zm are hard to come by, but considering the large f limit allows some insight. As the pre-shock outer horizon is pushed to the post-shock outer horizon (so the initial entanglement fractionf→ 1), the mi...
[14]

For largert type “E” surfaces, those which do not probe the pre-shock geometry, take over

Saturation time and vI Return now to equation (A28): ∆A =aeqAn−1 ( t×vn + zn f,+ zn i,+ R ) This tracks ∆A for a minimal extremal surface only up to the saturation time, when the area is equal to its equilibrium value. For largert type “E” surfaces, those which do not probe th...
[15]

numerical vE(f)

Numerical results for strips for f ⁄= 0: a closer look In this subsection, we present a few numerical results that serve as intermediate steps in the production of Figure 5 and provide some intuition for the behavior of the relevant surfaces. First, we plot the shape of a samp...
[16]

Derivation of ˜C First, we discuss the steps involved in the derivation of equations (38)-(47) for ˜C. Starting with the deﬁnition (37) of ˜C, let us expand the exponential in that deﬁnition in series: ˜C = ∑ n (ig)n n! ⟨ φR ( 1 K K∑ i=1 Oi ROi L )n φL ⟩ (B1) 32 We now focus o...
[17]

From ˜C to C Having computed ˜C in the previous subsection, we need relate this to the quantity C, the imaginary part of which indicates communication through the wormhole. Recalling that C = e−i⟨V⟩ ˜C, the two diﬀer by an overall phase ⟨V⟩ = g K ∑K i=1⟨Oi R(t0,x 0)Oi L(t0,x 0...
[18]

Simpliﬁed argument Let us assume there is a light-cone velocity vL, potentially depending on energy density ϵ, such that information does not signiﬁcantly propagate beyond the eﬀective light-cone. Here is a precise version: Given a strip A, we can widen A by an amount vL∆t on ...
[19]

This is done by directly constructing the approximation U ≈ UA′UAc

Light-cone velocity and butterﬂy velocity Here we argue that vL = vB. This is done by directly constructing the approximation U ≈ UA′UAc. The main idea is to break the Hamiltonian up into two pieces, one in ( A′)c and one in A′. Using the interaction picture, the evolution the...
[20]

[60], there are several technical details that are not properly addressed

Some technical points While the above discussion captures the conceptual essence of the argument, which is a minor variation of that in Ref. [60], there are several technical details that are not properly addressed. We next turn to these detailed points to make a more careful ...
[21]

Kitaev and J

A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006)

2006
[22]

Li and F

H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identiﬁcation of topological order in non-abelian fractional quantum hall eﬀect states, Phys. Rev. Lett. 101, 010504 (2008)

2008
[23]

Amico, R

L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008)

2008
[24]

Vidal, Entanglement renormalization, Phys

G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99, 220405 (2007)

2007
[25]

Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Ann

U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (N. Y). 326, 96 (2011)

2011
[26]

J. M. Maldacena, The Large N limit of superconformal ﬁeld theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999), [Adv. Theor. Math. Phys.2,231(1998)], arXiv:hep-th/9711200 [hep-th]

1999 arXiv
[27]

Witten, Anti-de Sitter space and holography, Adv

E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998), arXiv:hep-th/9802150 [hep-th]

1998 arXiv
[28]

S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B428, 105 (1998), arXiv:hep-th/9802109 [hep-th]

1998 arXiv
[29]

Ryu and T

S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006), arXiv:hep-th/0603001 [hep-th]

2006 arXiv
[30]

van Raamsdonk, Building up spacetime with quantum entanglement, General Relativity and Gravitation 42, 2323 (2010), arXiv:1005.3035 [hep-th]

M. van Raamsdonk, Building up spacetime with quantum entanglement, General Relativity and Gravitation 42, 2323 (2010), arXiv:1005.3035 [hep-th]

2010 arXiv
[31]

Susskind, Entanglement is not enough, Fortsch

L. Susskind, Entanglement is not enough, Fortsch. Phys. 64, 49 (2016), arXiv:1411.0690 [hep-th]

2016 arXiv
[32]

Stanford and L

D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90, 126007 (2014), arXiv:1406.2678 [hep-th]

2014 arXiv
[33]

A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Complexity, action, and black holes, Phys. Rev. D 93, 086006 (2016), arXiv:1512.04993 [hep-th]

2016 arXiv
[34]

A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116, 191301 (2016), arXiv:1509.07876 [hep-th]

2016 arXiv
[35]

Ryu and T

S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, J. High Energy Phys. 08, 045, arXiv:hep- th/0605073 [hep-th]
[36]

Faulkner, M

T. Faulkner, M. Guica, T. Hartman, R. C. Myers, and M. Van Raamsdonk, Gravitation from Entanglement in Holographic CFTs, J. High Energy Phys. 03, 051, arXiv:1312.7856 [hep-th]

arXiv
[37]

Faulkner, F

T. Faulkner, F. M. Haehl, E. Hijano, O. Parrikar, C. Rabideau, and M. Van Raamsdonk, Nonlinear Gravity from Entan- glement in Conformal Field Theories, J. High Energy Phys. 08, 057, arXiv:1705.03026 [hep-th]

arXiv
[38]

Almheiri, X

A. Almheiri, X. Dong, and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, J. High Energy Phys. 04, 163, arXiv:1411.7041 [hep-th]

arXiv
[39]

Harlow, The Ryu–Takayanagi Formula from Quantum Error Correction, Commun

D. Harlow, The Ryu–Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354, 865 (2017), arXiv:1607.03901 [hep-th]

2017 arXiv
[40]

Hayden and J

P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, J. High Energy Phys. 09, 120, arXiv:0708.4025 [hep-th]

arXiv
[41]

Sekino and L

Y. Sekino and L. Susskind, Fast scramblers, JHEP 2008 (10), 065, arXiv:0808.2096 [hep-th]

2008 arXiv
[42]

Brown and O

W. Brown and O. Fawzi, Scrambling speed of random quantum circuits, arXiv:1210.6644 [quant-ph] (2012)

2012 arXiv
[43]

A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical Method in the Theory of Superconductivity, Sov. Phys. JETP 28, 1200 (1969)

1969
[44]

S. H. Shenker and D. Stanford, Black holes and the butterﬂy eﬀect, JHEP 2014 (3), 67. 39

2014
[45]

S. H. Shenker and D. Stanford, Stringy eﬀects in scrambling, J. High Energy Phys. 05, 132, arXiv:1412.6087 [hep-th]

arXiv
[46]

Sachdev, Bekenstein-hawking entropy and strange metals, Phys

S. Sachdev, Bekenstein-hawking entropy and strange metals, Phys. Rev. X 5, 041025 (2015)

2015
[47]

Kitaev, A simple model of quantum holography, in KITP Progr

A. Kitaev, A simple model of quantum holography, in KITP Progr. Entanglement Strongly-Correlated Quantum Matter (2015)

2015
[48]

Maldacena, S

J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, JHEP 2016 (8), 106

2016
[49]

I. L. Aleiner, L. Faoro, and L. B. Ioﬀe, Microscopic model of quantum butterﬂy eﬀect: Out-of-time-order correlators and traveling combustion waves, Ann. Phys. (N. Y). 375, 378 (2016)

2016
[50]

Hosur, X.-L

P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, JHEP 2016 (2), 4

2016
[51]

Leviatan, F

E. Leviatan, F. Pollmann, J. H. Bardarson, D. A. Huse, and E. Altman, Quantum thermalization dynamics with Matrix- Product States, arXiv:1702.08894 (2017)

2017 arXiv
[52]

Hallam, J

A. Hallam, J. Morley, and A. G. Green, The Lyapunov spectra of quantum thermalisation, Nature Commun. 10, 2708 (2019), arXiv:1806.05204 [cond-mat.str-el]

2019 arXiv
[53]

C. D. White, M. Zaletel, R. S. K. Mong, and G. Refael, Quantum dynamics of thermalizing systems, Phys. Rev. B 97, 035127 (2018)

2018
[54]

Gu, X.-L

Y. Gu, X.-L. Qi, and D. Stanford, Local criticality, diﬀusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 2017 (5), 125

2017
[55]

Zhou and D

T. Zhou and D. J. Luitz, Operator entanglement entropy of the time evolution operator in chaotic systems, Phys. Rev. B 95, 094206 (2017)

2017
[56]

D. J. Luitz and Y. Bar Lev, Information propagation in isolated quantum systems, Phys. Rev. B 96, 020406(R) (2017)

2017
[57]

A. A. Patel, D. Chowdhury, S. Sachdev, and B. Swingle, Quantum Butterﬂy Eﬀect in Weakly Interacting Diﬀusive Metals, Phys. Rev. X 7, 031047 (2017)

2017
[58]

Nahum, S

A. Nahum, S. Vijay, and J. Haah, Operator Spreading in Random Unitary Circuits, Phys. Rev. X 8, 021014 (2018)

2018
[59]

C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator Hydrodynamics, OTOCs, and Entangle- ment Growth in Systems without Conservation Laws, Phys. Rev. X 8, 021013 (2018)

2018
[60]

Bohrdt, C

A. Bohrdt, C. B. Mendl, M. Endres, and M. Knap, Scrambling and thermalization in a diﬀusive quantum many-body system, New J. Phys. 19, 063001 (2016)

2016
[61]

Xu and B

S. Xu and B. Swingle, Accessing scrambling using matrix product operators, Nature Phys.16, 199 (2019), arXiv:1802.00801 [quant-ph]

2019 arXiv
[62]

S. Sahu, S. Xu, and B. Swingle, Scrambling dynamics across a thermalization-localization quantum phase transition, Phys. Rev. Lett. 123, 165902 (2019), arXiv:1807.06086 [cond-mat.str-el]

2019 arXiv
[63]

You and Y

Y.-Z. You and Y. Gu, Entanglement features of random Hamiltonian dynamics, Phys. Rev. B 98, 014309 (2018)

2018
[64]

Nahum, J

A. Nahum, J. Ruhman, and D. A. Huse, Dynamics of entanglement and transport in one-dimensional systems with quenched randomness, Phys. Rev. B 98, 035118 (2018)

2018
[65]

M. Heyl, F. Pollmann, and B. D´ ora, Detecting Equilibrium and Dynamical Quantum Phase Transitions in Ising Chains via Out-of-Time-Ordered Correlators, Phys. Rev. Lett. 121, 016801 (2018)

2018
[66]

C. B. Daˇ g and L. M. Duan, Detection of out-of-time-order correlators and information scrambling in cold atoms: Ladder-XX model, Phys. Rev. A 99, 052322 (2019), arXiv:1807.11085 [quant-ph]

2019 arXiv
[67]

Liu and S

H. Liu and S. J. Suh, Entanglement Tsunami: Universal Scaling in Holographic Thermalization, Phys. Rev. Lett. 112, 011601 (2014), arXiv:1305.7244 [hep-th]

2014 arXiv
[68]

D. A. Roberts, D. Stanford, and L. Susskind, Localized shocks, JHEP 2015, 51, arXiv:1409.8180 [hep-th]

2015 arXiv
[69]

D. A. Roberts and B. Swingle, Lieb-Robinson Bound and the Butterﬂy Eﬀect in Quantum Field Theories, Phys. Rev. Lett. 117, 091602 (2016), arXiv:1603.09298 [hep-th]

2016 arXiv
[70]

Mezei, Membrane theory of entanglement dynamics from holography, Phys

M. Mezei, Membrane theory of entanglement dynamics from holography, Phys. Rev. D 98, 106025 (2018)

2018
[71]

Jonay, D

C. Jonay, D. A. Huse, and A. Nahum, Coarse-grained dynamics of operator and state entanglement, arXiv:1803.00089 (2018)

2018 arXiv
[72]

P. Gao, D. L. Jaﬀeris, and A. Wall, Traversable Wormholes via a Double Trace Deformation, J. High Energy Phys. 12, 151, arXiv:1608.05687 [hep-th]

arXiv
[73]

Maldacena, D

J. Maldacena, D. Stanford, and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65, 1700034 (2017), arXiv:1704.05333 [hep-th]

2017 arXiv
[74]

V. E. Hubeny, M. Rangamani, and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 2007 (7), 062, arXiv:0705.0016 [hep-th]

2007 arXiv
[75]

Czech, J

B. Czech, J. L. Karczmarek, F. o. Nogueira, and M. Van Raamsdonk, The gravity dual of a density matrix, Classical and Quantum Gravity 29, 155009 (2012), arXiv:1204.1330 [hep-th]

2012 arXiv
[76]

Bousso, S

R. Bousso, S. Leichenauer, and V. Rosenhaus, Light-sheets and AdS/CFT, Phys. Rev. D86, 046009 (2012), arXiv:1203.6619 [hep-th]

2012 arXiv
[77]

X. Dong, D. Harlow, and A. C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, ”Phys. Rev. Lett.” 117, 021601 (2016), arXiv:1601.05416 [hep-th]

2016 arXiv
[78]

Cotler, P

J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle, and M. Walter, Entanglement Wedge Reconstruction via Universal Recovery Channels, Phys. Rev. X 9, 031011 (2019), arXiv:1704.05839 [hep-th]

2019 arXiv
[79]

Mezei and D

M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, JHEP 2017 (5), 65, arXiv:1608.05101 [hep-th]

2017 arXiv
[80]

Hartman and N

T. Hartman and N. Afkhami-Jeddi, Speed Limits for Entanglement, arXiv:1512.02695 [hep-th] (2015)

2015 arXiv
[81]

Casini, H

H. Casini, H. Liu, and M. Mezei, Spread of entanglement and causality, JHEP 2016 (7), 77, arXiv:1509.05044 [hep-th]

2016 arXiv
[82]

Nahum, J

A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum Entanglement Growth under Random Unitary Dynamics, Phys. Rev. X 7, 031016 (2017). 40

2017
[83]

Zhou and A

T. Zhou and A. Nahum, Emergent statistical mechanics of entanglement in random unitary circuits, Phys. Rev. B 99, 174205 (2019)

2019
[84]

Faulkner, A

T. Faulkner, A. Lewkowycz, and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 2013, 74, arXiv:1307.2892 [hep-th]

2013 arXiv
[85]

Liu and S

H. Liu and S. J. Suh, Entanglement growth during thermalization in holographic systems, Phys. Rev. D 89, 066012 (2014), arXiv:1311.1200 [hep-th]

2014 arXiv
[86]

Blake and A

M. Blake and A. Donos, Diﬀusion and Chaos from near AdS 2 horizons, J. High Energy Phys. 02, 013, arXiv:1611.09380 [hep-th]

arXiv
[87]

Mezei, On entanglement spreading from holography, J

M. Mezei, On entanglement spreading from holography, J. High Energy Phys. 05, 064, arXiv:1612.00082 [hep-th]

arXiv
[88]

Almheiri, A

A. Almheiri, A. Mousatov, and M. Shyani, Escaping the Interiors of Pure Boundary-State Black Holes, arXiv:1803.04434 [hep-th] (2018)

2018 arXiv
[89]

Caceres, A

E. Caceres, A. S. Misobuchi, and M.-L. Xiao, Rotating traversable wormholes in AdS, J. High Energy Phys. 12, 005, arXiv:1807.07239 [hep-th]

arXiv
[90]

Z. Fu, B. Grado-White, and D. Marolf, A perturbative perspective on self-supporting wormholes, Class. Quant. Grav. 36, 045006 (2019), arXiv:1807.07917 [hep-th]

2019 arXiv
[91]

Freivogel, V

B. Freivogel, V. Godet, E. Morvan, J. F. Pedraza, and A. Rotundo, Lessons on Eternal Traversable Wormholes in AdS, J. High Energy Phys. 07, 122, arXiv:1903.05732 [hep-th]

1903 arXiv
[92]

Z. Fu, B. Grado-White, and D. Marolf, Traversable Asymptotically Flat Wormholes with Short Transit Times, Class. Quant. Grav. 36, 245018 (2019), arXiv:1908.03273 [hep-th]

2019 arXiv
[93]

Marolf and S

D. Marolf and S. McBride, Simple Perturbatively Traversable Wormholes from Bulk Fermions, J. High Energy Phys. 11, 037, arXiv:1908.03998 [hep-th]

1908 arXiv
[94]

Liao and V

Y. Liao and V. Galitski, Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators, Phys. Rev. B 98, 205124 (2018), arXiv:1807.09799 [cond-mat.dis-nn]

2018 arXiv
[95]

Romero-Berm´ udez, K

A. Romero-Berm´ udez, K. Schalm, and V. Scopelliti, Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?, JHEP 07, 107, arXiv:1903.09595 [hep-th]

1903 arXiv
[96]

Xu and B

S. Xu and B. Swingle, Locality, Quantum Fluctuations, and Scrambling, Phys. Rev. X 9, 031048 (2019), arXiv:1805.05376 [cond-mat.str-el]

2019 arXiv
[97]

Caceres and A

E. Caceres and A. Kundu, Holographic Thermalization with Chemical Potential, J. High Energy Phys. 09, 055, arXiv:1205.2354 [hep-th]

arXiv
[98]

Kundu and J

S. Kundu and J. F. Pedraza, Spread of entanglement for small subsystems in holographic CFTs, Phys. Rev. D95, 086008 (2017), arXiv:1602.05934 [hep-th]

2017 arXiv
[99]

P. W. Shor, Scrambling Time and Causal Structure of the Photon Sphere of a Schwarzschild Black Hole, arXiv:1807.04363 [gr-qc] (2018)

2018 arXiv

Pith tools

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

[1] [1]

Consider a generic eternal black hole

A few generalities This subsection reviews the formalism of traversable wormhole as presented in [53]. Consider a generic eternal black hole. In Kruskal coordinates, the metric takes the form: ds2 =−a(UV )dUdV +r2(UV )dy2 (32) We deﬁne a0 = a(0) to be the value on the horizon of the functions a, and denote by r+ the horizon radius. We take the time coordi...

[2] [2]

the sweet spot

For fc <f < 1, we have vI =vB. This is a qualitative diﬀerence from the strip case (ﬁgure 5): in the strip case, recall that vI <v B for all f <1. Whenf <fc, the smallest entanglement wedge containing the infalling particle is at a discontinuous transition. On the other hand, when fc <f < 1, that smallest entanglement wedge is at a continuous transition. ...

[3] [3]

sweet spot

Example of planar BTZ black hole We now specialize to the planar BTZ black hole in order to have analytic tractability. The disadvantage of specializing to this solution is that vB is just the speed of light in 3 dimensions. Nevertheless, this will be suﬃcient for our purposes. The bulk-to-boundary propagator for the planar BTZ is given in Kruskal coordin...

[4] [4]

sweet spot

The light-cone Equation (57) gives us C as a function of xL, xR, tL and tR. Note that the dependence on those four boundary spacetime coordinates is entirely through A and B. In order to see the “sweet spot” of traversability from (57), we now set t0 = 0,−tL =tR≡T and ∆xL = ∆xR =X, and study the function C(T,X ). C(T,X ) = αψ 2r+ eiπ 2 ∆ψ× ∫ ∞ 0 dv v2∆ψ−1...

[5] [5]

initial” and “ﬁnal

Metric and notation We consider a special case of AdS-Vaidya style metrics, given by ds2 = L2 z2 ( −h(v,z )dv2− 2dvdz +d⃗ x·d⃗ x ) h(v,z ) =hi(z) + Θ(v)(hf(z)−hi(z)) (A1) where Θ(v) is a Heaviside step function, so hi(z) and hf(z) describe the black hole geometry in the initial ( v <0) and ﬁnal (v >0) regions, respectively, separated by the shock at v = 0...

[6] [6]

The case immediately of interest to us is n =d− 1, and the corresponding boundary observable is an entanglement entropy

Boundary strips The simplest boundary regions to consider are generalized “strips” of ﬁnite width in the x1 direction (-R to R), inﬁnite extent in the x2 through xn direction, and xn+1,...,x d−1 set to zero. The case immediately of interest to us is n =d− 1, and the corresponding boundary observable is an entanglement entropy. The formulas reported here d...

[7] [7]

tip”, or the deepest point of the extremal surface: z = zt and v = vt (we also use a subscript “t

Symmetries and Conserved Quantities The lack of explicit x dependence results in a constant of motion on extremal surfaces: J =zn√ Q =zn√ 1−h(v,z )(v′)2− 2v′z′ (A3) This allows us to simplify both equations of motion: ∂zQ− 2nz−1Q =∂x∂z′Q (A4) ∂vQ =∂x∂v′Q (A5) In both regions v >0 and v <0, the metric (A1) is independent of v. Setting ∂vhi,f = 0 in the v e...

[8] [8]

To do this we examine the eﬀect of the shock itself on the extremal surface

Junction conditions It still remains to determine the value of Ef for a given zt,vt. To do this we examine the eﬀect of the shock itself on the extremal surface. The function h(v,z ) is not independent of v at the shock, so we must return to equations (A4) and (A5). Let the subscript “ c” denote quantities evaluated at the position of shock crossing, and ...