Krylov complexity and oscillatory quantum dynamics in two-field inflation

Hai-Bing Fu; Lei-Hua Liu; Tao Li

REVIEW

Krylov complexity and oscillatory quantum dynamics in two-field inflation

Tao Li, Lei-Hua Liu, Hai-Bing Fu

High Energy Physics - Theory · Submitted 15 Oct 2026

Pith's verdict: Extends Krylov complexity to two-field inflation with a clear two-channel structure, but the central factorization assumption is stated, not verified, leaving the quantitative results unsupported.

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arXiv ID2610.21574 (version 3)

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CitationHai-Bing Fu, Lei-Hua Liu, Tao Li. "Krylov complexity and oscillatory quantum dynamics in two-field inflation." arXiv:2610.21574 (2026). https://pith.science/paper/EWWTDEX5

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Editor's note

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This paper takes the next logical step: Krylov complexity in two-field inflation with coupled curvature and entropy perturbations. The main new result is that the two squeezing eigenchannels show different onset times and oscillatory dynamics after horizon exit. The derivation from the quadratic action to the squeezing equations is clean, and the single-field limit checks out. The paper is well-structured and cites the relevant literature.

The load-bearing issue is the off-diagonal pair-production coefficient A_12. The authors state explicitly that they focus on diagonal channels and assume subdominant off-diagonal contributions (Sec. II, b119–b121), but they never compute A_12 or bound its size for their chosen parameters. Without that check, the factorized squeezed state (Eq. 26) and the resulting two-channel evolution equations (30–33) rest on an unverified premise. If A_12 is comparable to the diagonal coefficients, the whole two-channel picture collapses, and the quantitative claims about distinct onset times, oscillatory patterns, and channel hierarchy could be artifacts. The paper acknowledges the limitation, but acknowledging it does not substitute for checking it.

Other soft spots: no error estimates or convergence checks on the numerical solutions; the parameter choices (epsilon=0.002, mu=0.5H, lambda=0.1) are arbitrary, and their effect on the oscillatory structures is unexplored. These are fixable but need attention.

Who is this for? Researchers working on quantum information measures in cosmology. The paper extends a known framework to a slightly richer setting; the qualitative claim that two channels behave differently is plausible, but the quantitative details require the missing verification.

Recommendation: I would send this to peer review. The idea is solid and the derivation is sound, but the authors must compute or bound A_12 and demonstrate that the factorized description is valid for their parameter range. Without that, the paper is incomplete.

Referee report

Significance

If the factorized two-channel description is valid for the chosen parameters, the paper provides a clean and useful extension of the single-field Krylov-complexity formalism to a multi-field setting. The derivation from the quadratic action through the mode functions to the squeezing equations is transparent, and the closed-form results K_k = sinh^2 r_1 + sinh^2 r_2 and S_K = sum_i [(1+K_ik) ln(1+K_ik) - K_ik ln K_ik] are simple and testable. The authors also correctly note that their two squeezing eigenchannels are not separately the curvature and entropy fields, which avoids a potentially serious misinterpretation. However, the central factorization ansatz (Eq. 26) is adopted under an explicitly stated but unverified dominance assumption for the off-diagonal pair-production coefficient A_12, and the Lanczos-coefficient part of the paper lacks a derivation. For these reasons the qualitative claims about channel-dependent onset times and oscillatory Krylov behavior are currently conditional on an unquantified approximation.

Major objections (3)

[Sec. II, Eqs. (14)-(20), and Sec. III, Eq. (26)] The factorized squeezed state (Eq. 26) and the resulting evolution equations (30)-(34) are valid only when the off-diagonal pair-production coefficient A_12 is subdominant. The authors state this condition (Sec. II: "The two-channel description is adopted when the off-diagonal contribution remains subdominant") and later state that the validity of the reduction "is checked by the relative size of the off-diagonal Hamiltonian coefficients," but no such check is reported anywhere in the manuscript. Because A_12 can be computed from the same mode functions via Eq. (15), this is a straightforward quantitative task. Please report |A_12| relative to sqrt(|A_11 A_22|), or a comparable measure, over the full integration range, and show explicitly that the off-diagonal channel is subdominant for the parameters H=1, epsilon=0.002, eta_SR=0.01, mu=0.5H, lambda=0.1, k=0.01. Without this check, the subsequent claims in Eqs. (53), (60), and (66) are supported only conditionally. The authors acknowledge this: “The two-channel description is adopted when the off-diagonal contribution remains subdominant.”
[Sec. V.B, Eq. (66)] The quantities b_n^(1) = |A_11| sqrt(n)/2 and b_m^(2) = |A_22| sqrt(m)/2 are introduced as "channel-resolved Lanczos coefficients," but no derivation from the Lanczos recursion is given. Moreover, the pair-operator matrix elements written in Eqs. (64)-(65) are n and n+1, not sqrt(n) and sqrt(n+1), so Eq. (66) does not follow from those matrix elements in an obvious way. If these coefficients are intended as hopping amplitudes in a two-dimensional occupation lattice, the definition should be stated explicitly and connected to the Hamiltonian in the occupied-sector basis; if they are intended as actual Lanczos coefficients, the Lanczos recursion should be applied to the two-channel state and compared with Eq. (66) in the one-channel limit. As written, the quantitative Lanczos results shown in Fig. 6 are unsupported.
[Sec. II, numerical setup] The numerical procedure is not specified sufficiently for reproducibility. The paper states that two linearly independent initial modes are considered with Bunch-Davies initial conditions, but it does not give the explicit initial conditions for the four mode functions, the form of a(tau) used in the quasi-de Sitter background, the differential equations actually solved, the numerical integrator, or error tolerances. The later statement that the angle equations are replaced by equivalent complex squeezing variables is also not accompanied by the explicit equations used. Since the central qualitative claims, especially the different onset times of r_1 and r_2, rest entirely on the numerical integration, please provide the complete numerical setup and a convergence or accuracy test.

Minor objections (4)

[Sec. II, around Eq. (14)] The discussion of A_12 contains a repetitive sentence that should be rewritten for clarity: "The off-diagonal coefficient A_12 characterizes pair production involving both channels and the present analysis focuses on..." should be split into a definition and a statement of the adopted approximation.
[Throughout] There are several typographical and formatting issues, including "HAMIL TONIAN" in the Sec. II heading, non-rendered characters in the Schrödinger equation and in several formulas, and inconsistent use of phi versus varphi for the squeezing angle. These should be corrected in the final version.
[Sec. IV.B, Eq. (56)] In the single-field consistency check, the expression for B_k in terms of the Mukhanov variable is stated without showing that it reduces to the standard Hamiltonian coefficient associated with the v action. A brief derivation or a comment on the absence of the z''/z term would make the consistency check more convincing.
[Sec. V.B] The notation b_n^(1) conflicts with the conventional Lanczos coefficient b_n used in Sec. IV. Please use a different symbol, such as h_n^(1), for the channel-resolved hopping amplitudes to avoid confusion.

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The paper

The paper claims

Two-field inflation reveals distinct Krylov complexity oscillations across two squeezing channels.

The reading

This paper extends Krylov complexity—a measure of how quantum states spread in an abstract operator space—from single-field inflation to two-field inflation, where curvature and entropy perturbations couple. The authors show that the four-mode squeezed state describing such a system contains two independent squeezing channels. Solving the coupled mode equations numerically, they find that the two channels develop different squeezing histories after horizon exit: the second channel grows strong earlier and dominates the intermediate evolution, while the first channel shows a delayed rise. The resulting Krylov complexity oscillates across horizon crossing, with oscillations originating from phase-sensitive squeezing dynamics and amplified by the nonlinear relation K = sinh²r. These results provide a quantum-information signature of genuine multi-field effects, going beyond what power spectra or bispectra alone can reveal.

What carries the argument

The central object is the four-mode squeezed state (Eq. 26), factorized into two independent two-mode squeeze operators each acting on a (k, −k) pair for one of the two field sectors. The squeezing parameters r₁ₖ and r₂ₖ evolve according to first-order equations (34) driven by the diagonal pair-production coefficients A₁₁ and A₂₂, which are explicit functions of the coupled curvature–entropy mode functions. The Krylov complexity then follows directly as Kₖ = sinh² r₁ₖ + sinh² r₂ₖ, and the Krylov entropy factorises similarly. The phase-sensitive source term sin(2φᵢₖ − arg Aᵢᵢ) is the mechanism that generates oscillatory modulations of the squeezing growth rates, which become strongly amplified in the superhorizon regime.

Extended reading notes

Core claim

In two-field inflation, the diagonal squeezing basis of the four-mode squeezed state yields two independent squeezing channels, each governed by its own Hamiltonian coefficients constructed from coupled curvature–entropy mode functions. The two channels exhibit different onset times for strong squeezing: the second channel develops rapidly after horizon exit and dominates the intermediate stage, while the first channel shows a delayed but steep enhancement. Both channels produce oscillatory Krylov complexity across horizon crossing and pronounced fluctuations in the superhorizon regime. The oscillations arise because the instantaneous squeezing rate is proportional to sin(2φᵢₖ − arg Aᵢᵢ), and the resulting modulations are strongly amplified through the nonlinear relation Kᵢₖ = sinh² rᵢₖ. The total Krylov complexity, Krylov entropy, and channel-resolved Lanczos coefficients all reflect this two-channel hierarchy, which is absent in single-field models.

Load-bearing premise

The analysis assumes that off-diagonal pair production between the two squeezing channels (coefficient A₁₂) is subdominant, so the two channels can be treated independently and the factorized squeezed state (Eq. 26) remains accurate.

Editorial extensions

If this is right

Krylov complexity can serve as a quantum-information diagnostic that distinguishes single-field from multi-field inflationary dynamics, even when isocurvature perturbations are observationally suppressed.
The oscillatory features in Krylov complexity offer a new theoretical handle on the role of phases during horizon crossing; these phases are not erased by the usual squeezing process in multi-field settings.
The two-channel hierarchy identified here implies that at intermediate times one mode carries most of the quantum information spreading, affecting predictions for entropy and Lanczos growth.
Because the Krylov complexity is directly tied to the squeezing strengths, future extensions to N-field inflation will produce N independent squeezing channels, leading to even richer lattice dynamics in Krylov space.

Reading between the lines

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

If the off-diagonal pair production coefficient A₁₂ is not subdominant for realistic parameter values (e.g., stronger curvature–entropy coupling λ > 0.1), the factorized squeezed state (Eq. 26) may be inaccurate, and one would need to treat the full Sp(4,R) structure. The oscillations and onset hierarchy reported here could then be modified or washed out by channel mixing.
The phase-sensitive oscillations in Kₖ might be imprinted on some observer-level correlator if the state is subsequently decohered or if particle production couples to a detector, potentially providing a testable signature beyond the power spectrum.
The two-channel Krylov construction could be mapped onto a two-dimensional Krylov lattice; the hopping amplitudes bₙ⁽¹⁾ and bₘ⁽²⁾ grow differently, suggesting that operator spreading becomes anisotropic in the two-mode occupation space after horizon crossing.

Original abstract

We investigate Krylov-space dynamics of primordial perturbations in two-field inflation. Starting from the quadratic action for coupled curvature and entropy perturbations, we solve the mode equations with Bunch-Davies initial conditions and construct the corresponding quadratic Hamiltonian. The Gaussian quantum state is described by a four-mode squeezed state with two squeezing channels. We derive the evolution of the squeezing strength and squeezing angle and use them to evaluate the Krylov complexity, Krylov entropy, and channel-resolved Lanczos coefficients. Our numerical results show that the two channels remain weakly squeezed inside the horizon but develop different squeezing histories after horizon exit. The second channel develops strong squeezing earlier and dominates the intermediate evolution, whereas the first channel shows a delayed enhancement. The corresponding Krylov complexities exhibit distinct oscillatory structures across horizon crossing and pronounced fluctuations in the strongly squeezed superhorizon regime. These features arise from the phase-sensitive squeezing dynamics and reflect the different responses of the two channels to curvature-entropy coupling.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-16: 1 request, 6,764 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We investigate Krylov-space dynamics of primordial perturbations in two-field inflation. Starting from the quadratic action for coupled curvature and entropy perturbations, we solve the mode equations with Bunch-Davies initial conditions and construct the corresponding quadratic Hamiltonian. The Gaussian quantum state is described by a four-mode squeezed state with two squeezing channels. We derive the evolution of the squeezing strength and squeezing angle and use them to evaluate the Krylov complexity, Krylov entropy, and channel-resolved Lanczos coefficients. Our numerical results show that the two channels remain weakly squeezed inside the horizon but develop different squeezing histories after horizon exit. The two channels exhibit different onset times and distinct oscillatory Krylov dynamics. The second channel develops strong squeezing earlier and dominates the intermediate evolution, whereas the first channel shows a delayed enhancement. The corresponding Krylov complexities exhibit oscillatory structures across horizon crossing and pronounced fluctuations in the strongly squeezed superhorizon regime. These oscillations originate from the relative phases entering the squeezing equations and are amplified through the nonlinear relation between Krylov complexity of different channels.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate Krylov-space dynamics of primordial perturbations in two-field inflation.	contribution 0.99	0.92	0.05	asserted 0.80
s1	Starting from the quadratic action for coupled curvature and entropy perturbations, we solve the mode equations with Bunch-Davies initial conditions and construct the corresponding quadratic Hamiltonian.	method 1.00	0.93	0.03	asserted 0.99
s2	The Gaussian quantum state is described by a four-mode squeezed state with two squeezing channels.	method 0.82	0.54	0.05	asserted 1.00
s3	We derive the evolution of the squeezing strength and squeezing angle and use them to evaluate the Krylov complexity, Krylov entropy, and channel-resolved Lanczos coefficients.	method 0.93	0.96	0.07	asserted 1.00
s4	Our numerical results show that the two channels remain weakly squeezed inside the horizon but develop different squeezing histories after horizon exit.	result 1.00	0.95	0.18	asserted 1.00
s5	The two channels exhibit different onset times and distinct oscillatory Krylov dynamics.	result 1.00	0.89	0.24	asserted 1.00
s6	The second channel develops strong squeezing earlier and dominates the intermediate evolution, whereas the first channel shows a delayed enhancement.	result 0.99	0.93	0.21	asserted 1.00
s7	The corresponding Krylov complexities exhibit oscillatory structures across horizon crossing and pronounced fluctuations in the strongly squeezed superhorizon regime.	result 1.00	0.95	0.30	asserted 1.00
s8	These oscillations originate from the relative phases entering the squeezing equations and are amplified through the nonlinear relation between Krylov complexity of different channels.	interpretation 0.99	0.86	0.25	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.68 · evidence from simulation or computation 0.94.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Evolution of the perturbation amplitudes, mode-function amplitudes, and phases during

**Figure 2.** FIG. 2. Evolution of the Hamiltonian coefficients

**Figure 3.** FIG. 3. Evolution of the squeezing strength

**Figure 4.** FIG. 4. shows the two channel contributions

**Figure 5.** FIG. 5. Evolution of the Krylov entropy during two-field inflation. The blue solid and red dashed

**Figure 6.** FIG. 6. Evolution of the channel-resolved Lanczos coefficients

Figures reproduced from arXiv:2610.21574 by the authors.

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Discussion

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