  Open quantum walks often lead to a classical asymptotic behavior. Here, we
look for a simple open quantum walk whose asymptotic behavior can be
non-classical. We consider a discrete-time quantum walk on n-cycle subject to a
random coin-dependent phase shift at a single position. This finite system,
whose evolution is described by only two Kraus operators, can exhibit all kinds
of asymptotic behavior observable in quantum Markov chains: it either evolves
towards a maximally mixed state, or partially mixed state, or tends to an
oscillatory motion on an asymptotic orbit. We find that the asymptotic orbits
do not have a product structure, therefore the corresponding states can
manifest entanglement between the position and the coin degrees of freedom,
even if the system started in a product state.
