  We propose to quantify the complexity of non-equilibrium steady state density
operators, as well as of long-lived Liouvillian decay modes, in terms of level
spacing distribution of their spectra. Based on extensive numerical studies in
a variety of models, some solvable and some unsolved, we conjecture that
integrability of density operators (e.g., existence of an algebraic procedure
for their construction in finitely many steps) is signaled by a Poissonian
level statistics, whereas in the generic non-integrable cases one finds level
statistics of a Gaussian unitary ensemble of random matrices. Eigenvalue
statistics can therefore be used as an efficient tool to identify integrable
quantum non-equilibrium systems.
