Krylov Complexity in Open Quantum Systems

• Krylov complexity is a measure of operator growth that extends to open quantum systems by incorporating dissipation and decoherence via a biorthogonal Krylov basis.
• Mapping operator evolution to a non-Hermitian tight-binding model reveals suppressed complexity growth and localized dynamics at strong dissipation.
• Numerical methods such as moments techniques and bi-Lanczos algorithms enable robust diagnostics to distinguish chaotic from integrable behavior in realistic quantum systems.

Krylov complexity is a quantitative diagnostic of operator growth in quantum systems, originally formulated for closed (unitary) systems but now systematically generalized to open quantum systems under non-unitary Lindblad or non-Hermitian dynamics. In the open-systems context, Krylov complexity retains its core structure—capturing the “spread” of an initially local operator over a dynamically constructed optimal basis—but introduces distinct signatures associated with dissipation, decoherence, and the interplay between integrability and chaos. The combination of moments methods, biorthogonal Krylov basis construction, and tight-binding analogies has established Krylov complexity as a universal tool for probing quantum information dynamics, chaos, and operator mixing in realistic, environmentally coupled quantum systems.

1. Krylov Complexity: Definition and Mathematical Framework

Krylov complexity measures the degree of operator growth relative to a tailor-made Krylov basis constructed via repeated action of a dynamical generator—Hamiltonian for closed, Lindbladian for open systems. For an operator $\mathcal{O}_0$ evolving in time, the Krylov basis $\{\mathcal{O}_n\}$ is built recursively: $$\mathcal{O}_1 = [H,\mathcal{O}_0], \quad \mathcal{O}_2 = [H,\mathcal{O}_1], \ldots$$ (in the closed case; for open systems, $$\mathcal{L}(\bullet) = [H, \bullet] + \mathrm{dissipator}$$).

Orthogonalization via (bi-)Lanczos or Arnoldi algorithms yields an (bi-)orthonormal basis. The time-evolved operator is expanded as

$$|\mathcal{O}(t)) = \sum_{n=0}^{D_K-1} i^n \varphi_n(t) |\mathcal{O}_n)$$

and Krylov complexity is defined as

$K(t) = \sum_{n=0}^{D_K-1} n |\varphi_n(t)|^2\,,$

where $|\varphi_n(t)|^2$ quantifies the “weight” of the operator at the $n$-th Krylov site.

In open quantum systems where the Lindbladian $\mathcal{L}$ is non-Hermitian, the expansion generalizes to

$$|O(t)\rangle = \sum_{n} \phi_n(t) |p_n\rangle,\quad \langle O(t)| = \sum_n \phi^*_n(t) \langle q_n|$$

with biorthogonality condition $\langle q_m|p_n\rangle = \delta_{mn}$. Krylov complexity remains

$$C_K(t) = \frac{\sum_n n |\phi_n(t)|^2}{\sum_n |\phi_n(t)|^2}\,,$$

monitoring operator delocalization in Krylov space even under dissipative evolution (Bhattacharya et al., 2023, Rabinovici et al., 8 Jul 2025).

2. Krylov Complexity and Lindbladian Evolution: Mapping to Non-Hermitian Tight-Binding

The introduction of dissipation—governed by Lindblad master equations or effective non-Hermitian generators—alters the structure of operator evolution in Krylov space:

$$\frac{d}{dt} O(t) = i[H, O] + \sum_i \left(L_i^\dagger O L_i - \frac{1}{2}\{L_i^\dagger L_i, O\}\right)\,.$$

Upon projection into the Krylov basis, the amplitudes $\phi_n(t)$ satisfy a generalized non-Hermitian tight-binding equation:

$$i\partial_t \phi_n = -b_{n+1}\phi_{n+1} - b_n \phi_{n-1} - i\gamma d_n \phi_n$$

where $b_n$ are Lanczos (off-diagonal) coefficients, and $d_n$ arises from the dissipator projection—typically found to grow linearly with $n$ up to a cutoff, reflecting increasing “complexity-dependent loss” (2207.13603).

This structure yields an effective description of operator dynamics as a particle on a semi-infinite chain with position-dependent loss, mapping dissipation to a non-Hermitian potential.

3. Dissipation, Decoherence, and Saturation of Complexity

The primary impact of open-system effects is the suppression and eventual saturation of Krylov complexity:

• In closed (unitary) systems with linear Lanczos coefficient growth ($b_n \sim \alpha n$), $K(t)$ can grow indefinitely (or saturate at $\sim K/2$ in finite systems).
• In open systems, the linear growth of $d_n$ (the dissipative term) induces position-dependent decay, suppressing large-$n$ amplitudes, leading to the emergence of localized edge modes in the effective tight-binding model.
• At strong dissipation ($\gamma > \gamma_c$), the spectrum of the non-Hermitian Krylov Hamiltonian develops an imaginary gap, and long-time dynamics become dominated by slowly decaying edge-localized modes. The result is a much lower late-time value of $K(t)$ than in closed systems—operator growth is arrested by dissipation (2207.13603, Bhattacharya et al., 2023).

Numerical results for the Sachdev-Ye-Kitaev model and interacting fermion chains support this picture, with consistent observation of suppressed long-time Krylov complexity as dissipation increases, regardless of the underlying chaoticity of the system.

Moreover, Krylov complexity exhibits limited sensitivity to decoherence in certain regimes. For example, in the Caldeira-Leggett model, Krylov complexity reflects primarily dissipative effects; decoherence produces oscillatory features in the Krylov basis but does not yield a distinct signature for decoherence-onset because the Krylov basis is dynamically constructed and generally misaligned with the “pointer basis” in which decoherence is manifest (Bhattacharyya et al., 18 Sep 2025).

4. Numerical Implementation: Moments Methods and Bi-Lanczos Algorithms

The computational protocol for Krylov complexity in open systems involves three intertwined elements:

Construction of the Biorthogonal Krylov Basis

For non-Hermitian Lindbladians $\mathcal{L}_o$, the bi-Lanczos algorithm constructs two biorthogonal sets $\{|p_n\rangle\rangle\}, \{|q_n\rangle\rangle\}$ with recurrence relations: $$c_{j+1}|p_{j+1}\rangle\rangle = \mathcal{L}_o|p_j\rangle\rangle - a_j|p_j\rangle\rangle - b_j|p_{j-1}\rangle\rangle$$ and the dual,

$$b_{j+1}^*|q_{j+1}\rangle\rangle = \mathcal{L}_o^\dagger|q_j\rangle\rangle - a_j^*|q_j\rangle\rangle - c_{j-1}^*|q_{j-1}\rangle\rangle$$

with biorthogonality $\langle\langle q_m|p_n\rangle\rangle = \delta_{mn}$ (Bhattacharya et al., 2023, Rabinovici et al., 8 Jul 2025, Baggioli et al., 19 Aug 2025).

Moments Method

Given a two-point function $C(t) = \langle O(t)O(0)\rangle$, moments $\mu_n = \partial_t^n C(t)|_{t=0}$ generate the Krylov chain. For systems with non-Hermitian Lindbladians, after constructing the biorthogonal Krylov basis, moments are used to build a tridiagonal "Hamiltonian" (or Lindbladian) whose time evolution in Krylov space is solved either directly or numerically (Bhattacharyya et al., 18 Sep 2025).

Computation of Krylov Complexity

With the expansion coefficients $\phi_n(t)$ (which may require normalization at each $t$ due to non-unitarity), complexity is computed as

$$C_K(t) = \frac{\sum_n n |\phi_n(t)|^2}{\sum_n |\phi_n(t)|^2}\,.$$

This protocol is robust across both weak and strong dissipation, integrable and chaotic underlying models, and encompasses models such as the damped harmonic oscillator, the Caldeira-Leggett model, the open SYK model, and open spin chain systems.

5. Diagnostic Power: Interplay with Integrability and Chaos

In the closed system limit, Krylov complexity’s late-time plateau differentiates chaotic ($K \sim K/2$) from integrable ($K \ll K/2$) models, as higher disorder in the Lanczos sequence (i.e., the variance of $b_n$) is associated with suppressed operator delocalization—a manifestation of “localization” in Krylov space (Rabinovici et al., 2022).

In open systems, the following is observed:

Regime	Early-time $K(t)$	Late-time $K(t)$ saturation	Sensitivity to chaos
Closed	Linear/exponential growth	High (chaotic), low (integrable)	Pronounced
Open, weak dissipation	Similar to closed	Merged across integrable and chaotic	Diminished
Open, strong dissipation	Suppressed growth/saturation	Universally low	Vanishes

The initial (small $n$) behavior of off-diagonal bi-Lanczos coefficients $b_n$ remains a sensitive probe of chaos for weakly open systems—the early-time spread of complexity grows faster for systems with nearly linear $b_n$ (chaotic) than for sublinear or irregular $b_n$ (integrable). However, dissipation-induced fluctuations in higher-order $b_n$ coefficients equalize long-time saturated complexity across integrable and chaotic models (Bhattacharya et al., 2023).

6. Block Krylov Complexity and Operator-Averaging

Standard single-seed Krylov complexity may fail to robustly distinguish chaotic and integrable dynamics, especially in open systems where the Krylov sequence's details depend on the initial seed operator. The block-Lanczos (multiseed) method mitigates this by evolving a complete set of “simple” (few-body) operators and averaging the resulting complexity (Craps et al., 24 Sep 2024):

$$C_{\text{mult}}(t) = \frac{1}{m}\sum_{n=0}^{m-1} C_K^{(n)}(t)$$

where $C_K^{(n)}(t)$ is the complexity for seed operator $O_{0,n}$. Late-time average $\bar{C}_{\text{mult}}$ is a robust probe—yielding a sharp distinction between integrable and chaotic (and, by extension, noisy or dissipative) regimes even in presence of strong non-unitarity.

7. Broader Physical Consequences and Limitations

The generalization of Krylov complexity to open quantum systems reveals several important physical features:

• Suppression and Saturation: Dissipation universally suppresses operator growth; Krylov complexity halts at a low value determined by slow edge-localized modes in Krylov space, linking to concepts in Anderson localization and information “pinning” by dissipation (2207.13603, Bhattacharyya et al., 18 Sep 2025).
• Chaos Diagnostics: While early-time operator growth still encodes chaos signatures, prolonged dissipation obscures distinction between chaotic and integrable behavior at long times, aligning with observed “universality” of late-time complexity plateaus in open systems (Bhattacharya et al., 2023, Baggioli et al., 19 Aug 2025).
• Decoherence Insensitivity: Krylov complexity primarily responds to dissipative, not decoherent, timescales; onset of decoherence may not be visible in complexity constructed in the Krylov basis, consistent with basis misalignment regarding physically preferred (“pointer”) states (Bhattacharyya et al., 18 Sep 2025).
• Connections to Holography and Black Hole Dynamics: The correspondence between late-time Krylov complexity growth and the complexity–volume proposal in evaporating black holes ties operator growth in open quantum systems to gravitational dynamics (Mohan, 2023).

8. Summary Table: Krylov Complexity in Representative Open Quantum Models

Model (Reference)	Open-system ingredient	Krylov complexity feature	Diagnostic role
Lindbladian spin chains (Bhattacharya et al., 2023)	Bulk/boundary dissipation	Early discrimination of chaos, late universal saturation	Chaos diagnostics, dissipation paper
Damped oscillator (Bhattacharyya et al., 18 Sep 2025)	Markovian bosonic bath	Suppressed, oscillatory $K(t)$, low asymptotic plateau	Dissipative effects
Caldeira–Leggett (Bhattacharyya et al., 18 Sep 2025)	Dissipation vs decoherence	Selective suppression/oscillation in $K(t)$	Disentangle dissipation and decoherence
Open SYK (2207.13603, Bhattacharya et al., 2023)	Strong non-Hermiticity	Edge-localized Krylov modes, complexity saturation	Universality, non-Hermitian random matrices
Early universe (Zhai et al., 27 Nov 2024)	Cosmological open-system	Rapid decoherence-like suppression, establishment of low-complexity “order”	Quantum information in cosmology
Random matrix GinUE (Baggioli et al., 19 Aug 2025)	Non-Hermitian chaos	KC peak in chaotic (Ginibre) vs Poisson (integrable)	Generalized chaos probe

9. Outlook

Krylov complexity in open quantum systems furnishes a universal, basis-adapted quantifier of operator growth that is sensitive to dynamical regimes, dissipation, and system-environment coupling. The bi-Lanczos and moments methods enable practical computation of operator complexity under non-unitary evolution, and the saturation, suppression, and early-time dynamics of Krylov complexity provide deep connections to quantum chaos, localization, information scrambling, and even holographic duality. For distinguishing chaos from integrability in strongly open settings, block Krylov complexity and multiseed methods offer enhanced diagnostic robustness. The framework extends naturally to bosonic baths, spin chains, quantum gases, black hole analogs, and early-universe cosmology, establishing Krylov complexity as a powerful metric for operator spreading and complexity in quantum systems subject to realistic environmental interactions.