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We investigate the time evolution of the chirality reduced density matrix fora discrete-time quantum walk on a one-dimensional lattice, which is obtained bytracing out the spatial degree of freedom. We analyze the standard case,without decoherence, and the situation where decoherence appears in the form ofbroken links in the lattice. By examining the trace distance for possible pairsof initial states as a function of time, we conclude that the evolution of thereduced 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 levelof noise increases, the dynamics approaches a Markovian process. The highestnon-Markovianity corresponds to the case without decoherence. The reduceddensity matrix tends always to a well-defined limit that we calculate, but onlyin the decoherence-free case this limit is non-trivial.

Chirality asymptotic behavior and non-Markovianity in quantum walks on a line