Krylov Complexity for Open Quantum System: Dissipation and Decoherence

We investigate Krylov complexity in open quantum systems using Lindblad master equations for bosonic bath models, with particular emphasis on the Caldeira–Leggett model. Krylov complexity is computed from the moments of the two-point function within the standard master equation framework. For the damped harmonic oscillator, the results reveal clear dissipative features in Krylov complexity. In the Caldeira–Leggett model, in the high-temperature limit, we find that Krylov complexity saturates in the full system and reproduces the expected dissipative behavior when the decoherence term is suppressed in the master equation. Conversely, when the dissipative term is suppressed, the contribution from decoherence exhibits the familiar oscillatory dynamics of the coherent system, along with additional novel features. However, Krylov complexity appears insensitive to the onset of decoherence, as no clear distinctive signature is observed. We attribute this to the fact that Krylov complexity is defined in the Krylov basis, which does not coincide with the conventional basis typically used to study decoherence.

💡 Research Summary

This paper investigates the behavior of Krylov complexity (KC) in open quantum systems by employing Lindblad master equations for bosonic bath models, with a particular focus on the Caldeira–Leggett model. The authors first review the definition of KC as the expectation value of the position of an operator on a one‑dimensional Krylov chain, where the chain coefficients (Lanczos coefficients) are extracted from the moments of a two‑point correlation function. For closed (unitary) systems the diagonal Lanczos coefficients vanish, but for open (non‑unitary) dynamics the Lindbladian is non‑Hermitian, requiring a generalization of the standard Lanczos algorithm. The paper discusses three possible approaches—Arnoldi, closed‑Krylov, and bi‑Lanczos—and adopts the bi‑Lanczos scheme, which preserves a tridiagonal structure by constructing a bi‑orthonormal basis from the action of the Lindbladian and its adjoint. A similarity transformation is introduced to symmetrize the off‑diagonal elements, allowing the moments method to be applied with generalized coefficients (\tilde b_n = \sqrt{b_n c_n}).

The first concrete system studied is a damped harmonic oscillator coupled to a zero‑temperature electromagnetic field. The Lindblad jump operator is taken as (L = \sqrt{\gamma}, a), representing photon emission with rate (\gamma). By computing the Lanczos coefficients from the two‑point function of the chosen operator, the authors find that the coefficients truncate rapidly, leading to a KC that grows sharply at early times and then saturates to a constant value. This saturation reflects the irreversible loss of energy to the bath and demonstrates that KC can capture dissipative dynamics.

The second, more elaborate example is the Caldeira–Leggett model, which describes a quantum particle linearly coupled to an infinite set of harmonic oscillators (a thermal bath). In the high‑temperature limit the master equation separates into a dissipative term (proportional to (\gamma)) and a decoherence term (proportional to the diffusion constant (D)). The authors perform a systematic “term‑by‑term” analysis: they first switch off the decoherence term and keep only the dissipative contribution, then vice‑versa. When only the dissipative term is present, KC again shows rapid early‑time growth followed by saturation, mirroring the behavior of the damped oscillator. When only the decoherence term is retained, the KC curve displays only mild oscillatory modulations and its long‑time value remains essentially unchanged compared with the undriven case. In other words, the onset of decoherence does not produce a clear, distinctive signature in KC.

The authors attribute this insensitivity to the fact that KC is defined with respect to the Krylov basis generated by repeated action of the Lindbladian on the chosen initial operator. This basis does not coincide with the conventional position‑momentum basis typically used to diagnose decoherence, so the decoherence dynamics are “hidden” in the Krylov representation. Consequently, while KC is a useful probe of dissipative processes, it appears blind to pure decoherence in the models examined.

The paper concludes by emphasizing two main points: (i) Krylov complexity can reliably signal energy‑loss (dissipation) in open quantum systems, but (ii) it fails to provide a clear diagnostic for decoherence unless the Krylov basis is appropriately tailored. The authors suggest that future work should explore alternative operator choices (e.g., directly using the reduced density matrix) or non‑symmetric Krylov constructions that might be more sensitive to phase‑randomizing processes. The appendices contain explicit calculations of the Lanczos coefficients for the damped oscillator and details of the two‑point correlation functions used in the moments method.

Overall, this study provides a systematic framework for applying Krylov complexity to open quantum dynamics, demonstrates its strengths and limitations through concrete models, and points toward promising directions for refining complexity‑based diagnostics of dissipation and decoherence in quantum technologies and quantum thermodynamics.