When level repulsion fails: non-normality and chaos in open quantum systems

Abstract

For Hamiltonian systems, level statistics provide a faithful diagnostic of quantum chaos. By analogy, the statistics of the Lindbladian spectrum are often used in open quantum systems, and the Grobe-Haake-Sommers conjecture proposes that systems with chaotic classical counterparts should exhibit level repulsion in the Lindbladian spectrum. Here we point out an important flaw in this analogy: Hamiltonian and Lindbladian spectra behave differently and have distinct physical interpretations, and one should therefore not expect the latter to provide a reliable diagnostic. For Lindbladians, the late-time dynamics are not determined by the bulk of the eigenvalues but only by those eigenvalues -- and their corresponding eigenvectors -- with small real parts. Combined with the strong non-normality typical of Lindbladians, this allows situations in which the level statistics can be tuned almost arbitrarily without affecting the dynamics on either short or long time scales. We explicitly demonstrate this phenomenon and provide examples in which Ginibre level repulsion arises while the system dynamics at no time show signatures of chaos. We further relate this mechanism to the emergence of a non-Hermitian skin effect in Liouville space, linking boundary-induced eigenvector localization to the observed spectral instability. Our results show that level statistics cannot universally serve as a reliable diagnostic of quantum chaos in open quantum systems and highlight the need for alternative diagnostics that remain robust in strongly non-normal regimes.

Figures (3)

• Figure 1: Eigenvalue level statistics.a–b, Density plots of the complex spacing ratio values (\ref{['csr']}) for the open driven harmonic oscillator (\ref{['dho']}) and the open tight-binding model (\ref{['1dlat']}), shown in the left and right panels, respectively. As is standard in Hermitian spectral analyses, we focus on the central part of the spectrum. Both models display the characteristic “bitten-donut’’ shape predicted by random matrix theory for chaotic non-Hermitian spectra. For the open driven oscillator we obtain the averaged values $\langle r\rangle \approx 0.7483$ and $-\langle \cos\theta\rangle \approx 0.3013$, while for the open tight-binding model we find $\langle r\rangle \approx 0.7403$ and $-\langle \cos\theta\rangle \approx 0.2438$. These values are close to the predictions of the Ginibre unitary ensemble, $\langle r\rangle \approx 0.73810$ and $-\langle \cos\theta\rangle \approx 0.24051$. In the Supplementary Material SuppMat we further show that, upon increasing the Hilbert-space dimension, the averages of both quantities converge to these predicted values for both models. c–d, Unfolded spacing-distribution histograms for the same models, compared with the two-dimensional Poisson distribution (\ref{['eq:2D-poi']}) (green dashed line) and the Ginibre unitary ensemble prediction (\ref{['eq:gin_dist']}) (red dash-dotted line). Both models exhibit clear level repulsion and closely follow the Ginibre statistics. Spectral unfolding was performed following the procedure described in the Methods section. In a the parameters used are $\omega = 0$, $\eta = 1/(2\sqrt{2})$, $\gamma_2 = \gamma_1 - 5\times10^{-5} = 0.05$, with $N_{\mathrm{tr}} = 135$, while in b we have used $J = 1$, $\gamma_1 = 2$, $\gamma_2 = 0.1$, and $L = 135$.
• Figure 2: Uhlmann fidelity. Evolution of the Uhlmann fidelity (\ref{['eq:fidelity']}) for the oscillator model (\ref{['dho']}) in a, and for the tight-binding model (\ref{['1dlat']}) in b. The different curves (blue, black, red, and green) correspond to $N_{\mathrm{tr}} = 81,\,61,\,41$, and $21$ in a, and to $L = 81,\,61,\,41$, and $21$ in b. As is evident, the larger the system size considered, the longer the fidelity remains close to unity; only once the state approaches the imposed boundary (set by $N_{\mathrm{tr}}$ and $L$) does the fidelity begin to decrease. By increasing the system size, the evolution can therefore be made consistent with that of the corresponding integrable models for arbitrarily long times. The remaining parameters are the same as in Fig. \ref{['fig:levelstat']}, and as initial states we take the vacuum in a and the particle at the center site in b.
• Figure 3: Eigenvalue condition number and spectra.a,c Eigenvalue condition numbers (which squared give what is known as Petermann factors petermann2003calculated) of the steady-state eigenvalue (blue circles) and a representative bulk eigenvalue (red crosses) as a function of the Hilbert-space dimension: truncation $N_\mathrm{tr}$ for the open driven harmonic oscillator (a) and system size $L$ for the open tight-binding model (c). In both cases the condition number of the steady state remains approximately constant, whereas for bulk eigenvalues it grows exponentially with system size. The knee around $N_\mathrm{tr},L\approx60$ originates from finite numerical precision, as the numerics cannot reliably resolve exponentially small denominators. The same scaling is observed for the condition number $\kappa_{\hat{M}}(\hat{V})$ (\ref{['eq:degofnn']}), which also grows exponentially with system size, indicating that the Lindbladian becomes highly non-normal for sufficiently large systems. b,d Numerically obtained spectra $\mu_n$ together with a colormap indicating the logarithm of the corresponding eigenvalue condition numbers for the open driven oscillator (b) and the open tight-binding model (d), both shown for system sizes $N_\mathrm{tr},L=100$. Bulk eigenvalues exhibit extremely large condition numbers, whereas eigenvalues close to the steady state display much smaller values. The separation between moderate and large condition numbers coincides with the transition from regularly distributed to strongly repelling eigenvalues. Remaining parameters are the same as in Fig. \ref{['fig:levelstat']}.