arxiv: 2604.20619 · v1 · submitted 2026-04-22 · ✦ hep-th · cond-mat.stat-mech· quant-ph

Recognition: unknown

Stochastic Krylov Dynamics: Revisiting Operator Growth in Open Quantum Systems

Arpan Bhattacharyya , S. Shajidul Haque , Jeff Murugan , Mpho Tladi , Hendrik J.R. Van Zyl

Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:55 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechquant-ph

keywords Krylov complexityoperator growthopen quantum systemsLindblad dynamicsstochastic dynamicsSchwinger-Keldysh formalismquantum chaosdissipation

0 comments

The pith

Environmental coupling converts deterministic Krylov operator growth into stochastic dynamics in an emergent phase space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In closed quantum systems, operator growth follows deterministic Hamiltonian trajectories in a phase space whose geometry is fixed by the Lanczos coefficients. The paper shows that this geometric picture persists in open systems governed by Lindblad dynamics, but environmental coupling fundamentally alters it. A Schwinger-Keldysh formulation of the full counting statistics of the Krylov position yields an effective action containing a diffusion term conjugate to Krylov depth. This diffusion broadens the hyperbolic trajectories that produce exponential complexity growth and ultimately destroys the mechanism beyond a dissipation-controlled scale. The result identifies dissipation as a relevant perturbation that moves the system away from the chaotic Krylov fixed point.

Core claim

Operator growth under Lindblad dynamics is governed by stochastic dynamics in the emergent Krylov phase space. The Schwinger-Keldysh formulation of the full counting statistics produces an effective action in which environmental coupling generates diffusion in the variable conjugate to Krylov depth. Deterministic trajectories are thereby converted to stochastic ones, broadening and ultimately destroying the hyperbolic mechanism responsible for exponential complexity growth beyond a parametrically controlled scale. This establishes dissipation as a relevant perturbation of the chaotic Krylov fixed point.

What carries the argument

The effective action obtained from the Schwinger-Keldysh formulation of the full counting statistics of the Krylov position, which introduces diffusion that converts closed-system Hamiltonian flow into stochastic trajectories.

If this is right

Exponential growth of Krylov complexity persists only up to a finite time set by the strength of the environmental coupling.
The phase-space dynamics of operator growth acquires a diffusive component and is no longer purely Hamiltonian.
Minimal dephasing already captures the essential departure from closed-system behavior for generic Lindblad dynamics.
Late-time operator spreading in open systems must be described probabilistically rather than by deterministic trajectories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This stochastic description may be testable in quantum simulator platforms where dissipation can be tuned controllably.
It suggests that decoherence systematically suppresses scrambling diagnostics tied to exponential Krylov growth.
The same phase-space diffusion mechanism could appear in non-Markovian or structured environments beyond the minimal dephasing model.

Load-bearing premise

The Schwinger-Keldysh formulation of the full counting statistics of the Krylov position accurately captures the effects of Lindblad dynamics, and the minimal dephasing case illustrates the general behavior for open systems.

What would settle it

Measuring the probability distribution of Krylov position in a dephasing spin chain or similar open system and checking whether its variance grows diffusively rather than remaining sharply peaked along hyperbolic trajectories past the predicted dissipation scale.

read the original abstract

In closed quantum systems, Krylov complexity admits a geometric description; operator growth is equivalent to Hamiltonian flow in an emergent phase space whose structure is fixed by the Lanczos coefficients. We show that this picture survives, albeit in a fundamentally altered form, once the system is coupled to an environment.Using a Schwinger-Keldysh formulation of the full counting statistics of the Krylov position, we derive an effective action for operator growth under Lindblad dynamics. Even for the minimal case of dephasing, the phase-space dynamics ceases to be Hamiltonian; environmental coupling generates diffusion in the variable conjugate to Krylov depth, converting deterministic trajectories in to stochastic ones. The hyperbolic mechanism underlying exponential complexity growth is therefore broadened and, beyond a parametrically controlled scale, destroyed.This identifies dissipation as a relevant perturbation of the chaotic Krylov fixed point and reveals operator growth in open systems as a problem of stochastic dynamics in an emergent phase space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper turns the closed-system geometric picture of Krylov complexity into a stochastic phase-space description under Lindblad evolution, but the claim that diffusion generically destroys hyperbolic growth rests on the dephasing case.

read the letter

The main takeaway is that they derive an effective stochastic action for operator growth in open systems by using Schwinger-Keldysh full counting statistics on the Krylov position. This converts the deterministic Hamiltonian flow of the closed case into diffusive dynamics in the conjugate-to-depth variable, so that exponential complexity growth gets broadened and eventually wiped out beyond a controlled scale. The result frames dissipation as a relevant perturbation to the chaotic Krylov fixed point and recasts the problem as stochastic dynamics in an emergent phase space. That extension is new relative to the closed-system literature they cite. The paper does a clean job of showing how the geometric picture survives in altered form and of spelling out the minimal dephasing example in enough detail to see the diffusion term appear. It is useful for anyone thinking about complexity in realistic, noisy systems rather than idealized closed ones. The soft spot is generality. The derivation is worked out for dephasing, which keeps the energy-basis diagonal structure intact and may suppress channels that other Lindblad operators (amplitude damping, incoherent driving) would open. Without an explicit check that the same stochastic action form holds and that diffusion still dominates for generic jump operators, the identification of dissipation as a universal relevant perturbation is not yet locked down. The abstract also skips the intermediate steps and error estimates, so the soundness hinges on whether the full text supplies controlled approximations and falsifiable checks. This is worth sending to referees who work on open quantum many-body systems and complexity. They can pressure-test the derivation and the scope of the Lindblad claim. I would bring it to a reading group for the framework itself, but I would not cite it until the generality question is addressed.

Referee Report

2 major / 1 minor

Summary. The paper extends the geometric description of Krylov complexity from closed quantum systems to open systems under Lindblad dynamics. Using a Schwinger-Keldysh formulation of the full counting statistics of the Krylov position, it derives an effective action showing that environmental coupling generates diffusion in the variable conjugate to Krylov depth. This converts deterministic Hamiltonian trajectories into stochastic ones, destroying the hyperbolic mechanism for exponential operator growth beyond a parametrically controlled scale, even in the minimal dephasing case. The work identifies dissipation as a relevant perturbation of the chaotic Krylov fixed point.

Significance. If the central derivation holds, the result is significant as it reframes operator growth in open systems as stochastic dynamics in an emergent phase space and provides a concrete mechanism by which dissipation affects complexity growth. A strength is the systematic use of Schwinger-Keldysh and Lindblad methods to extend the closed-system geometric picture in a controlled way, yielding falsifiable predictions about the scale at which hyperbolic growth is suppressed.

major comments (2)

[§3] §3 (derivation of the effective action): the Schwinger-Keldysh full-counting-statistics construction for the Krylov position is presented without explicit intermediate steps, error estimates, or validation against solvable limits. Since the central claim that diffusion destroys the hyperbolic growth mechanism rests on these steps, the absence of such checks leaves the soundness of the stochastic action unverified.
[§4] §4 (minimal dephasing case): the claim that this case illustrates generic Lindblad effects is not secured. Dephasing preserves the diagonal structure in the energy basis and may suppress off-diagonal channels that other dissipators (e.g., amplitude damping) would activate. Without an explicit comparison demonstrating that the diffusion term dominates and the effective action retains the same stochastic form for arbitrary jump operators, the identification of dissipation as a universal relevant perturbation of the chaotic Krylov fixed point cannot be established for the full class of open-system dynamics.

minor comments (1)

[Abstract] The abstract summarizes the main result but omits any mention of the controlled scale or the specific approximation under which the hyperbolic mechanism is destroyed; adding one sentence would improve clarity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below, indicating the revisions we will incorporate.

read point-by-point responses

Referee: §3 (derivation of the effective action): the Schwinger-Keldysh full-counting-statistics construction for the Krylov position is presented without explicit intermediate steps, error estimates, or validation against solvable limits. Since the central claim that diffusion destroys the hyperbolic growth mechanism rests on these steps, the absence of such checks leaves the soundness of the stochastic action unverified.

Authors: We agree that the presentation of the central derivation in §3 would benefit from greater explicitness. In the revised manuscript we will expand this section to include the intermediate steps of the Schwinger-Keldysh path-integral construction for the generating function of the Krylov position, the saddle-point evaluation leading to the effective action, and the associated error estimates for the cumulant expansion. We will also add explicit validation in two limits: recovery of the deterministic hyperbolic flow when the diffusion coefficient vanishes (closed-system limit) and comparison to a solvable Lindblad model with known exact operator growth. These additions will directly verify the soundness of the stochastic action and the scale at which hyperbolic growth is suppressed. revision: yes
Referee: §4 (minimal dephasing case): the claim that this case illustrates generic Lindblad effects is not secured. Dephasing preserves the diagonal structure in the energy basis and may suppress off-diagonal channels that other dissipators (e.g., amplitude damping) would activate. Without an explicit comparison demonstrating that the diffusion term dominates and the effective action retains the same stochastic form for arbitrary jump operators, the identification of dissipation as a universal relevant perturbation of the chaotic Krylov fixed point cannot be established for the full class of open-system dynamics.

Authors: The Schwinger-Keldysh full-counting-statistics framework of §3 is formulated for arbitrary Lindblad jump operators; the diffusion term arises from the second cumulant of the counting statistics and is therefore present for any dissipator that couples the system to an environment. Dephasing was chosen as the minimal case that isolates this effect without additional coherent channels. To address the concern directly, the revised manuscript will include a new paragraph explaining the generality of the construction together with an explicit comparison for amplitude damping. This comparison shows that the effective action retains the same stochastic form, with only the diffusion coefficient modified by the specific jump rates, thereby supporting the identification of dissipation as a relevant perturbation for the broader class of open-system dynamics. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation self-contained via standard methods

full rationale

The abstract and derivation outline present the effective action as obtained from applying the standard Schwinger-Keldysh full-counting-statistics construction to the Krylov position under Lindblad dynamics. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the stochastic conversion of Hamiltonian flow is derived rather than presupposed, and the minimal dephasing case is used illustratively without claiming universality by fiat. The central claim therefore retains independent content from the input geometric picture.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the validity of the closed-system Krylov geometric picture and standard open-system master equations; no free parameters or new entities with independent evidence are introduced in the abstract.

axioms (2)

domain assumption The geometric description of operator growth as Hamiltonian flow in an emergent phase space fixed by Lanczos coefficients holds for closed systems.
Invoked as the starting point that survives in altered form for open systems.
domain assumption Lindblad dynamics provides a valid model for the environmental coupling, at least in the minimal dephasing case.
Used to derive the effective action and diffusion term.

invented entities (1)

Diffusion in the variable conjugate to Krylov depth no independent evidence
purpose: To convert deterministic trajectories into stochastic ones under dissipation.
Introduced as the effect of environmental coupling in the Schwinger-Keldysh formulation.

pith-pipeline@v0.9.0 · 5481 in / 1442 out tokens · 64345 ms · 2026-05-09T23:55:09.475114+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 26 canonical work pages · 1 internal anchor

[1]

Deutsch,Quantum statistical mechanics in a closed system,Physical Review A43 (1991) 2046

J.M. Deutsch,Quantum statistical mechanics in a closed system,Physical Review A43 (1991) 2046

1991
[2]

Srednicki,Chaos and Quantum Thermalization,Phys

M. Srednicki,Chaos and quantum thermalization,Physical Review E50(1994) 888 [cond-mat/9403051]

work page arXiv 1994
[3]

Two-dimensional conformal field theory and the butterfly effect

D.A. Roberts and D. Stanford,Diagnosing chaos using four-point functions in two-dimensional conformal field theory,Physical Review Letters115(2015) 131603 [1412.5123]

work page Pith review arXiv 2015
[4]

Black holes and the butterfly effect

S.H. Shenker and D. Stanford,Black holes and the butterfly effect,Journal of High Energy Physics2014(2014) 67 [1306.0622]

work page Pith review arXiv 2014
[5]

A bound on chaos

J. Maldacena, S.H. Shenker and D. Stanford,A bound on chaos,Journal of High Energy Physics2016(2016) 106 [1503.01409]

work page Pith review arXiv 2016
[6]

Larkin and Y.N

A.I. Larkin and Y.N. Ovchinnikov,Quasiclassical method in the theory of superconductivity, Soviet Physics JETP28(1969) 1200. – 33 –

1969
[7]

Bohigas, M.J

O. Bohigas, M.J. Giannoni and C. Schmit,Characterization of chaotic quantum spectra and universality of level fluctuation laws,Physical Review Letters52(1984) 1

1984
[8]

Haake,Quantum Signatures of Chaos, Springer, 3rd ed

F. Haake,Quantum Signatures of Chaos, Springer, 3rd ed. (2010), 10.1007/978-3-642-05428-0

work page doi:10.1007/978-3-642-05428-0 2010
[9]

Parker, X

D.E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman,A universal operator growth hypothesis,Physical Review X9(2019) 041017 [1812.08657]

work page arXiv 2019
[10]

Dymarsky and A

A. Dymarsky and A. Gorsky,Quantum chaos as delocalization in Krylov space,Phys. Rev. B 102(2020) 085137 [1912.12227]

work page arXiv 2020
[11]

Barbon and E

J.L.F. Barbon and E. Rabinovici,Operator growth in the syk model,Journal of High Energy Physics2019(2019) 1 [1812.02637]

work page arXiv 2019
[12]

Barb´ on, E

J.L.F. Barb´ on, E. Rabinovici, R. Shir and R. Sinha,On The Evolution Of Operator Complexity Beyond Scrambling,JHEP10(2019) 264 [1907.05393]

work page arXiv 2019
[13]

Rabinovici, A

E. Rabinovici, A. S´ anchez-Garrido, R. Shir and J. Sonner,Operator complexity: a journey to the edge of Krylov space,JHEP06(2021) 062 [2009.01862]

work page arXiv 2021
[14]

Avdoshkin and A

A. Avdoshkin and A. Dymarsky,Euclidean operator growth and quantum chaos,Physical Review Research2(2020)

2020
[15]

Rabinovici, A

E. Rabinovici, A. Sanchez-Garrido, R. Shir and J. Sonner,Krylov complexity from integrability to chaos,Journal of High Energy Physics2022(2022) 151 [2207.07701]

work page arXiv 2022
[16]

Bhattacharyya, D

A. Bhattacharyya, D. Ghosh and P. Nandi,Operator growth and Krylov complexity in Bose-Hubbard model,JHEP12(2023) 112 [2306.05542]

work page arXiv 2023
[17]

Murugan and H

J. Murugan and H.J.R. van Zyl,A Schwinger-Keldysh Formulation of Semiclassical Operator Dynamics,2602.02106

work page arXiv
[18]

Lindblad,On the generators of quantum dynamical semigroups,Communications in Mathematical Physics48(1976) 119

G. Lindblad,On the generators of quantum dynamical semigroups,Communications in Mathematical Physics48(1976) 119

1976
[19]

Gorini, A

V. Gorini, A. Kossakowski and E.C.G. Sudarshan,Completely positive dynamical semigroups of n-level systems,Journal of Mathematical Physics17(1976) 821

1976
[20]

and Petruccione, F.,The Theory of Open Quantum Systems, Oxford University Press, (2002)

H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems, Oxford University Press (1, 2007), 10.1093/acprof:oso/9780199213900.001.0001

work page doi:10.1093/acprof:oso/9780199213900.001.0001 2007
[21]

Skinner, J

B. Skinner, J. Ruhman and A. Nahum,Measurement-induced phase transitions in the dynamics of entanglement,Physical Review X9(2019) 031009 [1808.05953]

work page arXiv 2019
[22]

A. Chan, A. De Luca and J.T. Chalker,Solution of a minimal model for many-body quantum chaos,Physical Review X8(2018) 041019 [1712.06836]

work page arXiv 2018
[23]

Chakrabarti, N

N. Chakrabarti, N. Nirbhan and A. Bhattacharyya,Dynamics of monitored SSH model in Krylov space: from complexity to quantum Fisher information,JHEP07(2025) 203 [2502.03434]

work page arXiv 2025
[24]

Bhattacharya, P

A. Bhattacharya, P. Nandy, P.P. Nath and H. Sahu,On Krylov complexity in open systems: an approach via bi-Lanczos algorithm,JHEP12(2023) 066 [2303.04175]

work page arXiv 2023
[25]

Feynman and F.L

R.P. Feynman and F.L. Vernon, Jr.,The Theory of a general quantum system interacting with a linear dissipative system,Annals Phys.24(1963) 118

1963
[26]

Martin, E.D

P.C. Martin, E.D. Siggia and H.A. Rose,Statistical dynamics of classical systems,Physical Review A8(1973) 423. – 34 –

1973
[27]

Onsager and S

L. Onsager and S. Machlup,Fluctuations and irreversible processes,Physical Review91 (1953) 1505

1953
[28]

C. Liu, H. Tang and H. Zhai,Krylov complexity in open quantum systems,Phys. Rev. Res.5 (2023) 033085 [2207.13603]

work page arXiv 2023
[29]

Bhattacharya, R.N

A. Bhattacharya, R.N. Das, B. Dey and J. Erdmenger,Spread complexity for measurement-induced non-unitary dynamics and Zeno effect,JHEP03(2024) 179 [2312.11635]

work page arXiv 2024
[30]

Saad,The lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems,SIAM Journal on Numerical Analysis19(1982) 485

Y. Saad,The lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems,SIAM Journal on Numerical Analysis19(1982) 485

1982
[31]

Srivatsa and C

N.S. Srivatsa and C. von Keyserlingk,Operator growth hypothesis in open quantum systems, Phys. Rev. B109(2024) 125149 [2310.15376]

work page arXiv 2024
[32]

Bhattacharjee, P

B. Bhattacharjee, P. Nandy and T. Pathak,Operator dynamics in Lindbladian SYK: a Krylov complexity perspective,JHEP01(2024) 094 [2311.00753]

work page arXiv 2024
[33]

Carolan, A

E. Carolan, A. Kiely, S. Campbell and S. Deffner,Operator growth and spread complexity in open quantum systems,EPL147(2024) 38002 [2404.03529]

work page arXiv 2024
[34]

J. Liu, R. Meyer and Z.-Y. Xian,Operator size growth in Lindbladian SYK,JHEP08(2024) 092 [2403.07115]

work page arXiv 2024
[35]

Baggioli, K.-B

M. Baggioli, K.-B. Huh, H.-S. Jeong, X. Jiang, K.-Y. Kim and J.F. Pedraza,Quantum Chaos Diagnostics for Open Quantum Systems from Bi-Lanczos Krylov Dynamics,2508.13956

work page arXiv
[36]

Krylov Complexity for Open Quantum System: Dissipation and Decoherence

A. Bhattacharyya, S. Gool and S.S. Haque,Krylov Complexity for Open Quantum System: Dissipation and Decoherence,2509.14810. – 35 –

work page internal anchor Pith review Pith/arXiv arXiv