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Operator growth in open quantum systems: lessons from the dissipative SYK
Budhaditya Bhattacharjee, Xiangyu Cao, Pratik Nandy, Tanay Pathak
TL;DR
The paper asks how environmental dissipation changes operator growth and Krylov complexity in open quantum systems. It studies dissipative q-body SYK using operator-size concentration, analytical moment methods, and numerical Arnoldi iteration. The coefficients grow asymptotically linearly, while complexity shows suppressed exponential growth followed by saturation at a time logarithmic in inverse dissipation strength.
Problem
Operator growth and Krylov complexity lack a generally satisfactory formulation for open Lindbladian systems, motivating a study of universal behavior under dissipation.
Source: §1 · p. 2, §2.3 · p. 8Method
The paper studies dissipative SYK using operator-size concentration, diagrammatic and combinatorial analysis, the moment method, finite-q large-N checks, and finite-q finite-N Arnoldi iteration.
Source: §1 · p. 4, §4 · p. 9, §4.1 · p. 12Results
Krylov complexity exhibits suppressed exponential growth followed by saturation, with the saturation time growing logarithmically with inverse dissipation strength.
Source: §1 · p. 4Takeaways & Limitations
The results support an open-system operator-growth hypothesis in which Krylov complexity upper-bounds normalized OTOC growth and may generalize the chaos bound.
Source: §1 · p. 4, §7 · pp. 24–25The proposed ideal scenario is not known to hold generally away from large N, although numerical evidence supports it in some quantum spin chains.
Source: §6 · p. 19
Abstract
from arXiv · showhide
We study the operator growth in open quantum systems with dephasing dissipation terms, extending the Krylov complexity formalism of Phys. Rev. X 9, 041017. Our results are based on the study of the dissipative $q$-body Sachdev-Ye-Kitaev (SYK$_q$) model, governed by the Markovian dynamics. We introduce a notion of ''operator size concentration'' which allows a diagrammatic and combinatorial proof of the asymptotic linear behavior of the two sets of Lanczos coefficients ($a_n$ and $b_n$) in the large $q$ limit. Our results corroborate with the semi-analytics in finite $q$ in the large $N$ limit, and the numerical Arnoldi iteration in finite $q$ and finite $N$ limit. As a result, Krylov complexity exhibits exponential growth following a saturation at a time that grows logarithmically with the inverse dissipation strength. The growth of complexity is suppressed compared to the closed system results, yet it upper bounds the growth of the normalized out-of-time-ordered correlator (OTOC). We provide a plausible explanation of the results from the dual gravitational side.
1 Introduction
The paper studies operator growth under Lindbladian dissipation using Krylov complexity, focusing on dissipative SYK and seeking analytical results beyond finite-size numerics. It finds suppressed exponential complexity growth followed by logarithmically delayed saturation, with implications for open-system scrambling.
Motivation
- The paper targets analytical operator-growth results for dissipative SYK in large N and large q, complementing finite-q and finite-N numerical studies.The model is analytically tractable in these limits, while prior dissipative calculations used finite-dimensional Arnoldi iteration.Source: §1 · p. 3
Approach
- Three approaches—diagrammatic and combinatorial analysis, finite-q large-N semi-analytics, and finite-q finite-N Arnoldi iteration—are used to study the same Krylov-complexity behavior.The large-q dissipative SYK model is identified as a setting where these approaches are equally viable.Source: §1 · pp. 3–4
Main result
- Krylov complexity grows exponentially before saturating, with a saturation time that increases logarithmically with inverse dissipation strength.The dissipative growth is suppressed relative to the closed-system result.Source: §1 · p. 4
Implications
- The proposed open-system picture argues that Krylov-complexity growth upper-bounds normalized OTOC growth and may generalize the chaos bound.The paper presents this as an argument and gives a plausible holographic interpretation from the dual gravitational side.Source: §1 · p. 4
2 Krylov complexity in open systems: general approaches
The section compares three extensions of Krylov complexity to Lindbladian dynamics: a closed-system Krylov basis, Arnoldi iteration, and a moment method. Each resolves some difficulties while introducing others, motivating a special setting where they coincide.
2 Krylov complexity in open systems: general approaches
- Open-system Krylov complexity is formulated for Lindblad dynamics with a maximally mixed stationary state, using operators evolved by the Heisenberg-picture Lindbladian.The Lindbladian combines a unitary Liouvillian with a dissipation contribution and acts on operator space.Source: §2 · pp. 4–5
2.1 Using the closed Krylov basis
- The closed Krylov-basis approach generates an orthonormal basis from repeated action of the Hamiltonian Liouvillian, but the dissipative part becomes a general matrix.The dissipative matrix elements lack a special structure in general and require additional conjectures.Source: §2.1 · p. 5
2.2 Arnoldi iteration
- Arnoldi iteration generates the full open-system Krylov basis and represents a general Lindbladian by an upper Hessenberg matrix.Only for a Hermitian superoperator does this reduce to the real symmetric tridiagonal Lanczos form.Source: §2.2 · p. 6
2.3 The moment method
- The moment method recursively transforms autocorrelation moments into two coefficient sequences, but its resulting basis is non-orthonormal and can be singular under the relevant inner product.Consequently, defining Krylov complexity from that basis is problematic.Source: §2.3 · p. 7
- None of the three approaches is generally satisfactory because each resolves some problems while worsening others.The paper’s central claim is that a suitable system can realize all the desired properties simultaneously.Source: §2.3 · p. 8
3 The dissipative SYK
The dissipative SYK model adds Hermitian Majorana jump operators to the q-body SYK Hamiltonian and is analyzed in large N and large q. Dissipation acts on Majorana strings according to their size, competing with Hamiltonian-driven operator growth.
Model
- The model contains N Majorana fermions with q-body random interactions and is studied in the analytically tractable limit N →∞ followed by q →∞.At large q, the coupling and dissipation strength are rescaled to keep the relevant quantities finite.Source: §3 · p. 8
Dissipation
- Openness is introduced through Hermitian fermionic jump operators Li = √µ ψi, with µ controlling the dissipation strength.Hermiticity ensures that the infinite-temperature state remains stationary.Source: §3 · p. 8
- The dissipative Lindbladian acts on any Majorana string of size s with a rate proportional to µs, while the Hamiltonian Liouvillian drives operator-size growth.This establishes the competition between dissipation and Hamiltonian operator growth.Source: §3 · p. 8
Doubled-space formulation
- Under the Choi-Jamiolkowski representation, the dissipative operator dynamics are equivalent to non-Hermitian dynamics generated by two coupled SYK systems.The infinite-temperature state maps to the identity state in the doubled Hilbert space.Source: §3 · p. 9
4 Large q exact result
In the large q dissipative SYK model, the closed- and open-system Krylov constructions coincide, yielding exact Lanczos-coefficient results. Operator size concentration provides the diagrammatic mechanism, while the coefficients determine moment and continued-fraction consequences.
4 Large q exact result
- The closed-system Krylov basis equals the open-system basis, and Arnoldi produces a symmetric tridiagonal matrix matching the moment-method Lanczos coefficients.These equivalences hold exactly for the dissipative SYK model in the large q limit with O proportional to ψ1.Source: §4 · p. 9
- The diagonal Lanczos coefficients are imaginary and grow linearly with n, with slope set by the dissipation strength, while b_n matches the closed-system result [1].The new dissipative effect appears in a_n; b_n remains identical to the closed-system coefficient.Source: §4 · p. 9
4.1 Operator size concentration
- In the large q limit of the closed SYK model, the n-th Krylov basis operator is a linear combination of Majorana strings of common length s = n(q −2) + 1.The identity is proved diagrammatically via a removal map establishing an n(n + 1)/2-to-one correspondence between marked ordered diagrams with n + 1 arcs and ordered diagrams with n arcs.Source: §4.1 · pp. 10, 12, 13
- This concentration makes the Liouvillian tridiagonal in the generated basis, which is orthogonal and therefore identical to the normalized Krylov basis.The same structure yields the announced b_n coefficients and completes the large-q derivation.Source: §4.1 · p. 13
4.2 Consequence on moments
- The moment method independently connects the dissipative autocorrelation function to the Lanczos coefficients and produces a continued-fraction formula for the moments.For μ = 0, the result reduces to the known closed-system function g(t) = 2 ln sech(J t) and the continued fraction governed by b_n [1].Source: §4.2 · pp. 13–14
- The large-q expansion used for these moment consequences applies only for times t much smaller than t_c, with t_c approximately q at large q.Beyond that timescale, the expansion breaks down and requires a different approximation [52].Source: §4.2 · p. 14
5 Finite q numerics
Finite-q numerics show that operator-size concentration remains a good approximation, producing Arnoldi matrices close to the ideal linear-growth scenario at large N and finite N before size-induced saturation.
5.1 Large N: operator size distribution
- Operator-size concentration remains an excellent approximation for finite q, with distributions peaked at maximal size and mean size growing linearly in n.For q = 4, lower-size weights decay exponentially, while the standard-deviation-to-mean ratio remains small; the property is exact only for the first five basis operators.Source: §5.1 · pp. 15–16
5.2 Large N: Arnoldi iteration
- For weaker dissipation, deviations from the ideal Arnoldi scenario scale as ε_n ∝ µ^n, remaining relevant only beyond the operator-dynamics region n ≲ 1/µ.The matrix is expected to remain close to ideal up to approximately n ≲ 10^2/µ.Source: §5.2 · pp. 17–18
- Finite-q large-N Arnoldi matrices are close to symmetric tridiagonal form, with imaginary diagonal and primary off-diagonal elements growing linearly in n.The diagonal growth is proportional to dissipation, while primary off-diagonal growth is nearly dissipation-independent; secondary elements are much smaller.Source: §5.2 · pp. 16–18
5.3 Arnoldi iteration at finite N
- For q = 4 and N = 18, finite-size Arnoldi matrices remain close to the ideal scenario, with slight primary-off-diagonal asymmetry, small other off-diagonal elements, and imaginary diagonals growing linearly until saturation.Source: Figure 2 · p. 17, §5.3 · p. 19
- At finite N, diagonal elements grow linearly with n until a saturation plateau, whose value increases linearly with system size N.The plateau arises because Majorana-string lengths are bounded above by N; primary off-diagonal entries show analogous finite-size behavior.Source: §5.3 · p. 19
6 Operator growth and Krylov complexity
The dissipative SYK model yields linear asymptotic Lanczos growth and a localized Krylov wavefunction, causing exponential K-complexity growth to saturate under dissipation. Exact solutions show the saturation scale and imply bounded normalized OTOC growth, while the holographic interpretation remains heuristic.
6 Operator growth and Krylov complexity
- Operator size concentration makes the Lindbladian approximately symmetric tridiagonal, with both diagonal and off-diagonal Lanczos coefficients growing linearly.This holds exactly at large N and large q and approximately beyond that regime.Source: §6 · p. 19
6.1 General argument
- In weak dissipation, K-complexity initially grows as e^(2αt), then saturates at ξ proportional to α/µ when the wavefunction reaches the dissipation-controlled localization scale.The saturation time grows logarithmically with inverse dissipation; strong dissipation removes the exponential-growth regime.Source: §6.1 · p. 20
6.2 Exact solution
- The exact solvable coefficients reproduce an exponential stationary tail with inverse width ξ^-1 = χµ/(2α) and saturation time t = ln(1/u)/2 + O(1).The exact K-complexity is proportional to ξ ∼ 1/u at small u, with its proportionality constant controlled by η.Source: §6.2 · pp. 22–23
- The exact solution confirms closed-system growth K(t) ∼ e^(2αt) before dissipation corrections dominate and drive K-complexity to a constant.The normalized variance grows as ηe^(4t) in the growth regime and saturates at η/(4u^2).Source: §6.2 · p. 23
- For large-q SYK, the exact solvable coefficients apply up to subleading 1/q terms, while most of the wavefunction remains localized near the origin and K-complexity is proportional to 1/q.The exact coefficient structure also preserves an SL(2,R) form, but the imaginary diagonal coefficients suppress complexity.Source: §6.2 · pp. 22–23
6.3 Bound on chaos
- Any nonzero dissipation bounds normalized OTOC growth because large-size Majorana strings are eliminated at rates proportional to their length.Thus the dissipative SYK model has no nonzero critical dissipation threshold below which the OTOC grows indefinitely at N → ∞.Source: §6.3 · pp. 23–24
6.4 Holographic interpretation
- The holographic explanation is heuristic: dissipation may effectively increase Keldysh-wormhole coupling and shorten scrambling, but the geometry remains insufficiently understood.The authors leave broader gravitational and JT-complexity analyses for future work.Source: §6.4 · p. 24
7 Conclusion
The paper extends an operator-growth hypothesis to dissipative SYK through operator size concentration and finds consistent linear Lanczos growth across analytical and numerical approaches. Dissipation suppresses closed-system complexity growth, followed by saturation, while broader universality remains unresolved.
7 Conclusion
- Operator size concentration enables a diagrammatic and combinatorial derivation of two asymptotically linear Lanczos-coefficient sets in dissipative q-body SYK.One coefficient set is dissipation-sensitive, while the other tracks Hamiltonian chaotic growth.Source: §7 · p. 24
- The analytical large-q results agree with finite-q Schwinger-Dyson analysis and Arnoldi iteration at finite q and finite N.The three approaches are especially consistent in the large-q dissipative SYK setting.Source: §7 · pp. 24–25
- Dissipation suppresses exponential K-complexity growth relative to the closed system and is proposed to produce subsequent saturation at a scrambling-time scale.The authors suggest this may be universal for dissipative systems, but present it as a belief rather than an established theorem.Source: §7 · pp. 24–25
- The asymptotic growth of a_n in generic integrable systems, bi-Lanczos formulations, random jump operators, and complex-fermion dissipation remain open problems.These directions are identified as necessary for testing how broadly the proposed conclusions apply.Source: §7 · p. 25
A Appendix: Derivation of Eq. (3.4)
The appendix derives Eq. (3.4) by evaluating the Lindbladian action on fermionic and bosonic string operators. The derivation separates the two parity cases and combines their summations to obtain the stated result.
Case 1
- For fermionic string operators, the derivation uses the negative Lindbladian sign and requires the string to have even length.The first summation is split into parts organized by string length and evaluated through the required shifts.Source: A Appendix: Derivation of Eq. (3.4) · pp. 25–26
Conclusion of derivation
- Combining the evaluated sums with Eq. (A.2) yields Eq. (3.4), whose right-hand side is nonzero for even N and can have either sign.The sign depends on the number of fermions and the string length.Source: A Appendix: Derivation of Eq. (3.4) · p. 26
Case 2
- For bosonic string operators, the positive Lindbladian sign applies and the string must have even length; the preceding computation then proceeds analogously.The appendix states that the earlier calculations follow exactly before inserting the bosonic parity condition.Source: A Appendix: Derivation of Eq. (3.4) · p. 26