  We prove that any asymptotics of a finite-dimensional quantum Markov
processes can be formulated in the form of a generalized Jaynes principle in
the discrete as well as in the continuous case. Surprisingly, we find that the
open system dynamics does not require maximization of von Neumannentropy. In
fact, the natural functional to be extremized is the quantum relative entropy
and the resulting asymptotic states or trajectories are always of the
exponential Gibbs-like form. Three versions of the principle are presented for
different settings, each treating different prior knowledge: for asymptotic
trajectories of fully known initial states, for asymptotic trajectories
incompletely determined by known expectation values of some constants of motion
and for stationary states incompletely determined by expectation values of some
integrals of motion. All versions are based on the knowledge of the underlying
dynamics. Hence our principle is primarily rooted in the inherent physics and
it is not solely an information construct. The found principle coincides with
the MaxEnt principle in the special case of unital quantum Markov processes. We
discuss how the generalized principle modifies fundamental relations of
statistical physics.
