  The study of dissipation and decoherence in generic open quantum systems
recently led to the investigation of spectral and steady-state properties of
random Lindbladian dynamics. A natural question is then how realistic and
universal those properties are. Here, we address these issues by considering a
different description of dissipative quantum systems, namely, the discrete-time
Kraus map representation of completely positive quantum dynamics. Through
random matrix theory (RMT) techniques and numerical exact diagonalization, we
study random Kraus maps, allowing for a varying dissipation strength, and their
local circuit counterpart. We find the spectrum of the random Kraus map to be
either an annulus or a disk inside the unit circle in the complex plane, with a
transition between the two cases taking place at a critical value of
dissipation strength. The eigenvalue distribution and the spectral transition
are well described by a simplified RMT model that we can solve exactly in the
thermodynamic limit, by means of non-Hermitian RMT and quaternionic free
probability. The steady state, on the contrary, is not affected by the spectral
transition. It has, however, a perturbative crossover regime at small
dissipation, inside which the steady state is characterized by uncorrelated
eigenvalues. At large dissipation (or for any dissipation for a large-enough
system), the steady state is well described by a random Wishart matrix. The
steady-state properties thus coincide with those already observed for random
Lindbladian dynamics, indicating their universality. Quite remarkably, the
statistical properties of the local Kraus circuit are qualitatively the same as
those of the nonlocal Kraus map, indicating that the latter, which is more
tractable, already captures the realistic and universal physical properties of
generic open quantum systems.
