Quantum Markov processes: From attractor structure to explicit forms of asymptotic states

Markov processes play an important role in physics and in particular in the theory of open systems. In this paper we study the asymptotic evolution of trace-nonincreasing homogeneous quantum Markov processes (both types, discrete quantum Markov chains and continuous quantum Markov dynamical semigroups) equipped with a subinvariant faithful state in the Schrödinger and the Heisenberg picture. We derive a fundamental theorem specifying the structure of the asymptotics and uncover a rich set of transformations between attractors of quantum Markov processes in both pictures. Moreover, we generalize the structure theorem derived earlier for quantum Markov chains to quantum Markov dynamical semigroups showing how the internal structure of generators of quantum Markov processes determines attractors in both pictures. Based on these results we provide two characterizations of all asymptotic and stationary states, both strongly reminding in form the well-known Gibbs states of statistical mechanics. We prove that the dynamics within the asymptotic space is of unitary type, i.e. quantum Markov processes preserve a certain scalar product of operators from the asymptotic space, but there is no corresponding unitary evolution on the original Hilbert space of pure states. Finally simple examples illustrating the derived theory are given.