{"id":"81fafcd4-2a8c-47ff-acef-2479f92403b3","arxiv_id":"1908.06993","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For chaotic systems with initial entanglement fraction f, quantum information spreads at speed v_E(f)/(1-f), interpolating between the entanglement speed at f=0 and the butterfly speed at f=1.","lead":"Chaotic quantum systems spread information at a speed that depends on how entangled the initial state is. The paper proposes a formula connecting this information speed to the entanglement speed and the butterfly speed, and tests it in spin chains and holographic models.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The formula v_I=v_E(f)/(1-f) hinges on the unproven strip entanglement-growth law in Eq. (6); if saturation is not linear and first-order, the central relation lacks a regime of validity.","rationale":"The reader's weakest_assumption correctly identifies Eq. (6) as the load-bearing condition: all of the paper's information-theoretic derivation reduces to the saturation time t_sat = R(1-f)/v_E, and that relation is exactly the content of Eq. (6). My independent read confirms this is the point where the argument is least secure. The paper is honest about the status of Eq. (6), flagging it as an assumption rather than a theorem, admitting failure for spherical regions, and acknowledging the large-f spin-chain data are unreliable. The holographic section does provide genuine evidence: the numerical phase diagrams and the area-vs-time plots in Appendix A demonstrate first-order linear growth for strips in the studied Vaidya-RN backgrounds, and the analytic f->1 limit connecting v_E(f) to v_B(1-f) is a clean parameter-free result. The spin-chain data at small f also give real, if system-size-limited, support. So the central claim is not internally inconsistent or unsupported; it is a conjecture whose validity is conditional on an unproven dynamical law for strip entanglement. That is exactly a 'conditional accept' situation, matching the reader's verdict. I would not change the verdict: the concern is real but does not by itself refute the formula, and the proposed random-circuit test would either confirm universality or expose the missing criterion for when Eq. (6) holds. I agree with the reader's identification rather than raising a separate objection.","tokens_in":35064,"tokens_out":8003,"duration_ms":87011,"concrete_test":"Simulate a one-dimensional random unitary circuit (or a spin chain with L >= 30) initialized with entanglement fraction f as in Eq. (16), and for strip intervals of several half-widths R measure (a) S_A(t) directly to test whether it obeys Eq. (6) with a single f-dependent slope v_E(f) and a first-order saturation at t_sat = R(1-f)/v_E, and (b) the mutual information wavefront I(A:REF) to extract v_I independently. If Eq. (6) is violated for any f < 1, or if v_I deviates from v_E(f)/(1-f) beyond the numerical uncertainty, the central formula fails in that model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Eq. (1) follows from Eq. (6), which assumes that for strip regions the entanglement entropy grows linearly at speed v_E(f) until a sharp first-order saturation, giving t_sat=R(1-f)/v_E and hence v_I=v_E/(1-f). This is the single most load-bearing step: if Eq. (6) fails, the saturation-time relation and the information-speed formula fail with it. The paper states Eq. (6) as a general expectation rather than a theorem, and it explicitly notes the law is false for spherical regions (Sec. IV D), so even the authors restrict the formula to strips. The only clean support across all f comes from the holographic near-extremal RN-Vaidya calculation, where the linear-growth-plus-first-order-saturation behavior is verified numerically, but that is one family of models and the analytic derivation in Appendix A is built on the same ansatz. The spin-chain test is strong only at small f; at large f the extraction of v_E is finite-size limited, and Fig. 2(b) even shows v_E/(1-f) exceeding v_B, forcing the ad hoc switch to the min formula Eq. (2). A second, related soft spot is the Hayden-Preskill identification of the just-saturated region as the minimal recoverable region, but Eq. (6) is the more concrete and falsifiable hinge. If some strip-like chaotic system exhibits nonlinear growth near saturation or a continuous saturation transition, the formula Eq. (1) simply does not apply; the paper does not supply a criterion beyond 'strips' to tell when it should hold.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":35332,"tokens_out":4233,"duration_ms":50112,"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":[{"comment":"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.","section":"Sec. II, Eq. (6)"},{"comment":"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.","section":"Sec. II.B, Eqs. (9)-(11)"},{"comment":"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.","section":"Sec. III, Fig. 2(b)"}],"minor_comments":[{"comment":"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.","section":"Sec. III, Fig. 2"},{"comment":"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.","section":"Eq. (29)"},{"comment":"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).","section":"Abstract and Sec. I"},{"comment":"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.","section":"Sec. V.B"},{"comment":"The text uses 'compliment' in 'the compliment of A'; this should be 'complement'.","section":"Sec. III, Eq. (14)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a strong contribution with a transparent central argument and substantial numerical and holographic support. My recommendation of major revision is driven by the conditional character of the central formula: the result is only as general as the unproved Eq. (6), and the spin-chain evidence weakens exactly where the formula is most distinctive. These issues are fixable within the manuscript's scope by adding a more explicit statement of the assumption's status, a finite-size scaling analysis, and a more rigorous treatment of the Hayden-Preskill recovery step. I do not see grounds for rejection, but I would not accept the paper in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThere 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.\n\nWho 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.","headline":"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.","tokens_in":35924,"tokens_out":2187,"would_cite":true,"duration_ms":26227,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that one information speed, $v_I = v_E/(1-f)$, unifies the entanglement and butterfly speeds in chaotic many-body systems.","keywords":["quantum information scrambling","entanglement speed","butterfly speed","Hayden-Preskill protocol","entanglement growth","holographic entanglement entropy","AdS/CFT","chaotic many-body systems"],"falsifier":"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.","tokens_in":34838,"feed_emoji":"⚛️","tokens_out":6654,"duration_ms":64609,"temperature":0.7,"pith_summary":"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.","feed_headline":"One formula unifies the speeds of quantum scrambling","feed_subtitle":"Entanglement speed and butterfly speed appear as the two limits of a single information speed.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the Hayden-Preskill decoding criterion that the protocol here generalizes to a spatially local setting with dynamically generated memory.","marker":"[20]"},{"why":"Derives the entanglement speed $v_E$ for holographic thermalization from AdS-Vaidya, providing the $f=0$ baseline that this paper extends.","marker":"[47]"},{"why":"Supplies the extremal-surface methods for entanglement growth during holographic thermalization that the appendix adapts to finite $f$.","marker":"[65]"},{"why":"Provides the butterfly speed $v_B$ from Lieb-Robinson-type bounds and OTOC analysis, which sets the $f\\to1$ limit of the information speed.","marker":"[49]"},{"why":"Membrane model that independently predicts $v_E(f)\\sim v_B(1-f)$ near $f=1$, matching one of the paper's limits.","marker":"[50]"},{"why":"Coarse-grained entanglement growth model used for the random circuit comparison $v_E(f)=1-f$, giving $v_I=v_B$.","marker":"[51]"},{"why":"Introduces traversable wormholes via a double-trace deformation, the setup that the paper generalizes to spatially local signal propagation.","marker":"[52]"},{"why":"Supplies the traversable-wormhole correlator formalism and shockwave phase used in the local signal computation.","marker":"[53]"},{"why":"Proves $v_E\\le c$ in Lorentz-invariant theories, the bound that Appendix C sharpens to $v_E(f)\\le (1-f)v_B$.","marker":"[60]"}],"fun_headline_variants":["Quantum info speed unifies entanglement and butterfly limits","One formula for quantum scrambling: from entanglement to butterfly speed","Quantum chaos: single information speed from entanglement to butterfly","One speed governs quantum information spreading in chaos","Information speed spans from entanglement to butterfly speeds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quantum info speed unifies entanglement and butterfly limits","One formula for quantum scrambling: from entanglement to butterfly speed","Quantum chaos: single information speed from entanglement to butterfly","One speed governs quantum information spreading in chaos","Information speed spans from entanglement to butterfly speeds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000769,"raw_usage":{"total_tokens":3437,"prompt_tokens":1007,"completion_tokens":2430,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":623,"completion_tokens_details":{"reasoning_tokens":2358}},"tokens_in":623,"tokens_out":2430,"duration_ms":17191,"temperature":1.0,"reasoning_tokens":2358,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:29:40.817332+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"[60], there are several technical details that are not properly addressed","cited_arxiv_id":null,"evidence_quote":"Establishes the Hayden-Preskill decoding criterion that the protocol here generalizes to a spatially local setting with dynamically generated memory."},{"cited_title":"Kitaev, A simple model of quantum holography, in KITP Progr","cited_arxiv_id":null,"evidence_quote":"Derives the entanglement speed $v_E$ for holographic thermalization from AdS-Vaidya, providing the $f=0$ baseline that this paper extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the extremal-surface methods for entanglement growth during holographic thermalization that the appendix adapts to finite $f$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the butterfly speed $v_B$ from Lieb-Robinson-type bounds and OTOC analysis, which sets the $f\\to1$ limit of the information speed."},{"cited_title":"Hosur, X.-L","cited_arxiv_id":null,"evidence_quote":"Membrane model that independently predicts $v_E(f)\\sim v_B(1-f)$ near $f=1$, matching one of the paper's limits."},{"cited_title":"The Lyapunov Spectrum of Quantum Thermalisation","cited_arxiv_id":"1806.05204","evidence_quote":"Introduces traversable wormholes via a double-trace deformation, the setup that the paper generalizes to spatially local signal propagation."},{"cited_title":"Bohrdt, C","cited_arxiv_id":null,"evidence_quote":"Proves $v_E\\le c$ in Lorentz-invariant theories, the bound that Appendix C sharpens to $v_E(f)\\le (1-f)v_B$."}],"review_version":1}