  We investigate the time evolution of the chirality reduced density matrix for
a discrete-time quantum walk on a one-dimensional lattice, which is obtained by
tracing out the spatial degree of freedom. We analyze the standard case,
without decoherence, and the situation where decoherence appears in the form of
broken links in the lattice. By examining the trace distance for possible pairs
of initial states as a function of time, we conclude that the evolution of the
reduced density matrix is non-Markovian, in the sense defined in [H. P. Breuer,
E. M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)]. As the level
of noise increases, the dynamics approaches a Markovian process. The highest
non-Markovianity corresponds to the case without decoherence. The reduced
density matrix tends always to a well-defined limit that we calculate, but only
in the decoherence-free case this limit is non-trivial.
