  The asymptotic behavior of the quantum walk on the line is investigated
focusing on the probability distribution of chirality independently of
position. The long-time limit of this distribution is shown to exist and to
depend on the initial conditions, and it also determines the asymptotic value
of the entanglement between the coin and the position. It is shown that for
given asymptotic values of both the entanglement and the chirality distribution
it is possible to find the corresponding initial conditions within a particular
class of spatially extended Gaussian distributions. Moreover it is shown that
the entanglement also measures the degree of Markovian randomness of the
distribution of chirality.
