  We study generic open quantum systems with Markovian dissipation, focusing on
a class of stochastic Liouvillian operators of Lindblad form with independent
random dissipation channels (jump operators) and a random Hamiltonian. We
perform a thorough numerical study, focusing on global spectral features, the
spectral gap, and the steady-state purity and statistics. We establish that all
properties follow three different regimes as a function of the dissipation
strength, whose boundaries depend on the particular observable. Within each
regime, we determine the scaling exponents with the dissipation strength and
system size. On the analytical side, we compute the average spectral gap at
arbitrary dissipation and provide simple closed-form expressions for the
asymptotic values at strong and weak dissipation strength. We also consider
spectral correlations in generic complex spectra, such as that of the
Liouvillian. We introduce and study, both analytically and numerically, the
complex ratio of the nearest-neighbor spacing by next-to-nearest-neighbor
spacing. Besides the usual level repulsion, we find that complex spacing ratios
have a nontrivial angular dependence which offers a distinct signature of
quantum chaos. It distinguishes integrable systems, which support the flat
ratio distribution characteristic of Poisson statistics, and chaotic
dissipative quantum systems, which conform to Random Matrix Theory statistics.
We apply this new signature of quantum chaos to random Liouvillians and
boundary-driven dissipative spin-chains. The results of this thesis can help
understand the long-time dynamics, steady-state properties, and spectral
correlations of generic dissipative systems.
