Higher-order gap ratios of singular values in open quantum systems

Understanding open quantum systems using information encoded in its complex eigenvalues has been a subject of growing interest. In this paper, we study higher-order gap ratios of the singular values of generic open quantum systems. We show that $k$-th order gap ratio of the singular values of an open quantum system can be connected to the nearest-neighbor spacing ratio of positions of classical particles of a harmonically confined log-gas with inverse temperature $\beta'(k)$ where $\beta'(k)$ is an analytical function that depends on $k$ and the Dyson's index $\beta=1,2,$ and $4$ that characterizes the properties of the associated Hermitized matrix. Our findings are crucial not only for understanding long-range correlations between the eigenvalues but also provide an excellent way of distinguishing different symmetry classes in an open quantum system. To highlight the universality of our findings, we demonstrate the higher-order gap ratios using different platforms such as non-Hermitian random matrices, random dissipative Liouvillians, Hamiltonians coupled to a Markovian bath, and Hamiltonians with in-built non-Hermiticity.