Open system dynamics in local Lindbladians with chaotic spectra

TL;DR

This work shows that open quantum systems governed by local Lindbladians with Ginibre-type spectra exhibit quasi-universal early-time dynamics for nonlinear observables, driven by bulk eigenmodes whose associated eigenoperators are highly nonlocal in Pauli space. Locality imposes a strong size–decay correlation: decay rates scale with operator size, so local operators couple mainly to slow, large-weight modes outside the bulk, while bulk modes govern nonlinear quantities like purity through universal short-time behavior. The study combines two Ginibre-like models (Ising with dissipation and a random Lindblad model), validates complex spacing ratio statistics, and analyzes eigenoperator size distributions, overlaps, and IPR scrambling to explain operator growth, decoherence, and the conditions under which anomalous large operators emerge. Overall, the results illuminate how random-matrix-like bulk spectra interact with locality to shape open-system dynamics, with implications for dissipative quantum chaos and the design of noisy quantum devices. The findings provide a framework to predict early-time universal decoherence and to understand when operator growth is suppressed or anomalously enhanced, depending on the balance between single-site and two-site dissipation. The work also highlights directions for extending RMT diagnostics to open systems beyond Ginibre statistics and for exploring the entanglement structure of Lindblad eigenoperators.

Abstract

We investigate the physical consequences of having a spectrum that satisfies random matrix theory (RMT) for generic Lindbladians, and compare its implications for spatially local and completely random Lindblad dynamics in one spatial dimension. We find that Lindbladians whose spectrum is described by RMT exhibit quasi-universal early-time dynamics for quantities non-linear in the density matrix, in the sense that for generic, highly entangled initial states, the early time evolution is independent of the choice of initial state. We numerically investigate how locality generically imposes constraints on the size-dependence of Lindblad eigenoperators. This size dependence implies that linear observables, such as expectation values of local operators, are highly sensitive to eigenmodes outside the bulk of the spectrum in the thermodynamic limit, and plays a central role in limiting operator growth in the presence of dissipation. We find that when single-site dissipation dominates, an operator's decoherence scales approximately linearly with its Pauli weight, even in the presence of 2-site jump operators. When two-site only dissipation dominates, however, this generic trend in operator size can be violated for numerically accessible system sizes, leading to long-lived high Pauli-weight operators.

Figures (32)

• Figure 1: The eigenvalue spectrum of dissipative Ising model: The spectral data is shown for $J=1,h_x=1.3,h_z=1.2$, and $\gamma=0.8$ in the weak reflection symmetry sector labeled by $\mathcal{R}=1$. (a) The scatter plot of complex eigenvalues $\lambda_j$ is shown for an $N=8$ site one-dimensional spin chain. The green colored patch shows the eigenvalues used to compute CSR statistics in Fig. \ref{['fig:zzPro']}. (b) The values of the largest absolute real part $(X_m)$, the largest imaginary part $(Y_m)$, and the average value of the real parts $(\overline{X})$ are plotted as a function of the system size $N$. They approximately fall on the trend-line $\sim N$ indicated by dashed black lines. The spread of the bulk of the spectrum along real(imaginary) direction computed in terms of $\sigma_X$ ($\sigma_Y$) increases as a function of system size and is proportional to $\sqrt{N}$ indicated by magenta colored lines (see \ref{['eq:xm-ym', 'eq:xbar', 'eq:sigmaXY']} for details). (c) The marginal distribution of the eigenvalue, $\rho_X = \int [\text{d Im}(\lambda)]\ \rho(\lambda)$, is shown as a function of the real part of the eigenvalues for system size $N=8$. The black line shows the Gaussian distribution with mean $\overline{X}$ and standard deviation $\sigma_X$. (d) The marginal distribution $\rho_Y=\int [\text{d Re}(\lambda)]\ \rho(\lambda)$ is plotted as a function of the imaginary part of the eigenvalues. The overlaying curve shows the Gaussian distribution with zero mean and $\sigma_Y$ standard deviation. The distributions are normalized such that $\int \text{dx} \rho(\text{x})=1$. The inset shows the number of real eigenvalues ($n_{\text{real}}$) as a function of the dimension of the operator space $d_L$.
• Figure 2: Statistics of complex spacing ratios (CSR) in dissipative Ising model ($J=1,h_x=1.2,h_z=1.3,$ and $\gamma=0.8$): (a) The CSR $z_j=|z_j|e^{i\theta_j}$ is defined for each eigenvalue $\lambda_j$ according to Eq. \ref{['eq:zz']}. The 2d distribution of the $z_j$ in the complex plane is shown for an $N=8$ size system. The data is shown for the eigenvalues in the upper-half of the complex plane away from the real axis ($|\text{Re}(\lambda)-\overline{X}|\le2\sigma_X$, and $\sigma_Y/4\le\text{Im}(\lambda)\le3\sigma_Y/2$ --- see Fig. \ref{['fig:ee4']}(a)). The marginal probability distribution of (b) the absolute value $|z|$ and (c) the angular variable $\theta$ are shown in blue colored histograms. The orange curves in each panel show the corresponding distribution for the non-Hermitian Ginibre ensemble. The Ginibre results are numerically computed by diagonalizing 10 independent samples of $10^4\times 10^4$ sized non-Hermitian random matrix with complex-valued Gaussian matrix elements (GinUE ensemble). The distributions are normalized as $\int dx \rho(x)=1$. (d) The mean of the absolute values $|z|$ is shown as a function of system size $N$. It approaches $|z|_{\text{GinUE}}$ (black dashed line) with increasing system size. (e) The angular average $\langle\cos{\theta}\rangle$ is plotted for different systems sizes. The angular repulsion is captured by the negative value of $\cos(\theta)$ (i.e., large angular separation between neighbors), where the average value approaches the Ginibre ensemble value with increasing system size.
• Figure 3: Eigenoperator overlaps in dissipative Ising model ($J=1,h_x=1.2,h_z=1.3,$ and $\gamma=0.8$): (a) The probability distribution function of overlaps $|\alpha_j|/\text{mean}(|\alpha|)$, where $\alpha_j=\frac{1}{\langle \langle l_j|r_j\rangle \rangle }$ and $\text{mean}(|\alpha|)$ is average of their absolute value (only includes the eigenmodes inside $\lambda\in \Omega$). The blue histogram shows data for the dissipative Ising model, and the red line shows the complex Ginibre ensemble. The data is gathered for the eigenmodes in the window $\Omega=\{\lambda_j|\ |\text{Re}(\lambda_j)-\overline{X}|\le \sigma_X, \frac{\sigma_Y}{4}\le\text{Im}(\lambda_j)\le\sigma_Y\}$, where $\overline{X}$ is the center and $\sigma_{X(Y)}$ are standard deviations of the spectrum in $X(Y)$ direction. (b) The mean and standard deviation evaluated for the eigenmodes inside the window $\Omega$ are plotted as a function of increasing dimension of the operator space $d_L\approx 2^{2N-1}$. The dashed lines show the best fits to a power law function $a d_L^b$. The resulting fits to mean and standard deviation are shown in the respective colors. The yellow line depicts the trivial scaling $\propto\sqrt{d_L}$ for reference. (c-d) The scatter plot of $|\alpha_j|/\sqrt{d_L}$ is shown as a function of real (panel (c)) and imaginary part (panel(d)) of the eigenvalues $\lambda_j/N$ where $N$ is the total number of sites in the system. The dashed black lines show the boundary of the window $\Omega$ used in panel (a--b) for $N=8$.
• Figure 4: The complex eigenvalue ($\lambda$) spectrum for the dissipative Ising model is shown for $N=8$ system size ($J=1,h_x=1.2,h_z=1.3,$ and $\gamma=0.8$). Each eigenvalue $\lambda$ is colored according to the $\log_{10}$ of size distribution (a) $p_{s=8}(\hat{r})$ of its right, and (b) $p_{s=8}(\hat{l})$ of its left eigenoperator as defined in Eq. \ref{['eq:evec-size-dist']}.
• Figure 5: The coarse-grained weights of eigenoperators on basis-operators of size $s$ (see Eq. \ref{['eq:ps_tilde']}) are plotted as a function of the real part of eigenvalues $\lambda$. The data is shown for the (a) right and (b) left eigenoperators of $N=8$ size dissipative Ising model with parameters $J=1,h_x=1.3,h_z=1.2$, and $\gamma=0.8$. The coarse-graining scale is set to $\Delta=0.5$. The weight on $s=0$ basis operator is identically zero for the right eigenoperators with $\lambda\neq 0$ (panel (a)), since they are traceless due to the biorthogonality relation. The error bars show the variation of the weights of eigenmodes in the window $\Omega(x_0):=\{\lambda_j:|\text{Re}(\lambda_j)-x_0|\leq\Delta/2\}$ about its mean value $\tilde{p}_{s}(x_0)$. The value of top error bar is given by $e_t(x_0)=\text{min}(\text{max}_{j\in \Omega(x_0)}p_s(x_j),\sigma_{j\in \Omega(x_0)}p_s(x_j))$ and the bottom error bar is given by $e_b(x_0)=\text{min}(\text{min}_{j\in \Omega(x_0)}p_s(x_j),\sigma_{j\in \Omega(x_0)}p_s(x_j))$ where $\sigma$ is standard deviation of values within given window.