Quantum Chaos Diagnostics for Open Quantum Systems from Bi-Lanczos Krylov Dynamics

Abstract: In Hermitian systems, Krylov complexity has emerged as a powerful diagnostic of quantum dynamics, capable of distinguishing chaotic from integrable phases, in agreement with established probes such as spectral statistics and out-of-time-order correlators. By contrast, its role in non-Hermitian settings, relevant for modeling open quantum systems, remains less understood due to the challenges posed by complex eigenvalues and the limitations of standard approaches such as singular value decomposition. Here, we demonstrate that Krylov complexity, computed via the bi-Lanczos algorithm, effectively identifies chaotic and integrable phases in open quantum systems. The results align with complex spectral statistics and complex spacing ratios, highlighting the robustness of this approach. The universality of our findings is further supported through studies of both the non-Hermitian Sachdev-Ye-Kitaev model and non-Hermitian random matrix ensembles.