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

TL;DR

This paper challenges the use of Lindbladian spectral statistics as a universal diagnostic for quantum chaos in open systems, highlighting fundamental differences from Hamiltonian systems and the influence of non-normality.

Contribution

It reveals that Lindbladian spectra can be arbitrarily tuned without affecting system dynamics, undermining their reliability as chaos indicators, and connects this to non-Hermitian skin effects.

Findings

Lindbladian spectra differ fundamentally from Hamiltonian spectra.

Level statistics can be manipulated independently of system dynamics.

Non-Hermitian skin effects relate to spectral instability and eigenvector localization.

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.