  A thermodynamic theory is developed to describe the behavior of the
entanglement between the coin and position degrees of freedom of the quantum
walk on the line. This theory shows that, in spite of the unitary evolution, a
steady state is established after a Markovian transient stage. This study
suggests that if a quantum dynamics is developed in a composite Hilbert space
(i.e. the tensor product of several sub-spaces) then the behavior of an
operator that only belongs to one of the sub-spaces may camouflage the unitary
character of the global evolution.
