Dissipation-Induced Threshold on Integrability Footprints

Abstract

The presence of a dissipative environment substantially disrupts unitary evolution. Nevertheless, key features of the evolution, such as its integrable or chaotic nature, are not immediately erased by dissipation. To understand this, we model environment-induced dissipation as a convex combination of unitary evolution and a random Kraus map, and examine how signatures of integrability gradually fade with increasing dissipation strength. Our analysis shows that in the weakly dissipative regime, the complex eigenvalue spectrum organizes into well-defined, high-density clusters. We estimate the critical dissipation threshold beyond which these clusters disappear, rendering the dynamics indistinguishable from chaotic evolution. This threshold depends only on the number of spectral clusters and the rank of the Kraus map. To characterize this transition, we introduce the eigenvalue angular velocity as a diagnostic of integrability loss. We illustrate our findings using several integrable quantum circuits, including the dissipative quantum Fourier transform. Our results provide a quantitative picture of how noise gradually erases the footprints of unitary integrability in open quantum systems.