Krylov Complexity in Open Quantum Systems

TL;DR

The paper generalizes Krylov complexity from closed to open quantum systems by formulating operator growth under Lindblad dynamics and mapping it to a non-Hermitian tight-binding model on a half-infinite chain. The non-Hermitian terms, whose diagonal components grow linearly with the Krylov index, generate edge-localized modes that dominate long-time behavior, leading to a dissipation-induced saturation of Krylov complexity well below the closed-system scrambling value. Numerical studies on models such as the SYK and interacting fermions demonstrate the universality of these dynamics and illuminate how dissipation constrains chaos and complexity in open quantum settings. The work provides a framework for discussing complexity, chaos, and holography in open systems, with potential implications for holographic dualities in dissipative environments.

Abstract

Krylov complexity is a novel measure of operator complexity that exhibits universal behavior and bounds a large class of other measures. In this letter, we generalize Krylov complexity from a closed system to an open system coupled to a Markovian bath, where Lindbladian evolution replaces Hamiltonian evolution. We show that Krylov complexity in open systems can be mapped to a non-hermitian tight-binding model in a half-infinite chain. We discuss the properties of the non-hermitian terms and show that the strengths of the non-hermitian terms increase linearly with the increase of the Krylov basis index $n$. Such a non-hermitian tight-binding model can exhibit localized edge modes that determine the long-time behavior of Krylov complexity. Hence, the growth of Krylov complexity is suppressed by dissipation, and at long-time, Krylov complexity saturates at a finite value much smaller than that of a closed system with the same Hamitonian. Our conclusions are supported by numerical results on several models, such as the Sachdev-Ye-Kitaev model and the interacting fermion model. Our work provides insights for discussing complexity, chaos, and holography for open quantum systems.

Figures (5)

• Figure 1: (a) Schematic the mapping between Krylov complexity in open systems and a non-hermitian tight-binding model in a half-infinite chain. (b) Krylov complexity $\mathcal{K}(t)$ in open systems (red solid line), compared with $\mathcal{K}(t)$ in a closed system with the same Hamiltonian (blue dashed line). Krylov complexity is suppressed by dissipation.
• Figure 2: Coefficients of the non-hermitian terms $|d_{nm}|$ for the SYK model (a) and for the spinless fermion model (b). For (a), we have chosen $\hat{M}_i=\hat{\psi}_i$ and $\hat{O}=i\hat{\psi}_1\hat{\psi}_2$, and we have set $J=1$. The system contains total number of Majorana fermion $N=24$. For (b), we have chosen $\hat{M}_i=\hat{c}^\dag_i+\hat{c}_i$ and $\hat{O}=\hat{n}_{1}$, and we have set $J_1=1$, $J_2=0.2$, $V_1=0.6$ and $V_2=0.1$. The model contains total number of sites $N=13$.
• Figure 3: Coefficients $d_{n}$ for the diagonal components of the non-hermitian terms for the SYK model (a) and for the spinless fermion model (b). For (a), four different curves cover four different cases: $\hat{O}$ and $\hat{M}_i$ are either fermionic operator $\hat{\psi}_i$ or bosonic operator $i\hat{\psi}_i\hat{\psi}_j$. We have set $J=1$. The system contains total number of Majorana fermion $N=24$. For (b), four different curves cover four different cases: $\hat{O}$ and $\hat{M}_i$ are either fermionic operator $\hat{\psi}_i$ or bosonic operator $\hat{n}_i$, where $\hat{\psi}_i$ denotes $\hat{c}^\dag_i+\hat{c}_i$. We have set $J_1=1$, $J_2=0.2$, $V_1=0.6$ and $V_2=0.1$. The model contains total number of sites $N=13$. $n_\text{s}$ marks the places where $d_n$ saturates.
• Figure 4: Spectrum for the non-hermitian tight-binding model. (a-c) We choose $\beta_d=2.80$, $\alpha_d=0.35$, $\beta_b=0.66$, $\alpha_b=0.34$ and $n_\text{s}=1000$. (a) Eigen-energy $\epsilon=\epsilon^\prime-i\epsilon^{\prime\prime}$. $(\epsilon^\prime, \epsilon^{\prime\prime})$ for $\gamma=0.007<\gamma_\text{c}$ (b) $(\epsilon^\prime, \epsilon^{\prime\prime})$ for $\gamma=0.04>\gamma_\text{c}$. Here we shift the imaginary part $\epsilon^{\prime\prime}$ by a constant such that the smallest $\epsilon^{\prime\prime}$ is zero. (c) The wave functions for two modes below the gap in (b). (d) $\gamma_\text{c}$ versus $n_\text{s}$.
• Figure 5: A quantum circuit illustration of evaluation of $\text{Tr}_B[\hat{\rho}_{B}\hat{\xi}_\text{I}(t) \hat{M}^\dagger_\text{I}(t) \hat{O}^\text{S}_\text{I}(t) \hat{M}_\text{I}(t')\hat{\xi}_\text{I}^\dagger(t') ]$. (a) When $\hat{\xi}$ and $\hat{\xi}^{\dagger}$ are fermionic, they are supported on the total Hilbert space and the partial-trace's permutation is incorrect. (b) When $\hat{\xi}$ and $\hat{\xi}^{\dagger}$ are bosonic, they are only supported on the Hilbert space of bath and the partial-trace's permutation is correct.