  In the case of the discrete time coined quantum walk the reduced dynamics of
the coin shows non-Markovian recurrence features due to information back-flow
from the position degree of freedom. Here we study how this non-Markovian
behavior is modified in the presence of open system dynamics. In the process,
we obtain useful insights into the nature of non-Markovian physics. In
particular, we show that in the case of (non-Markovian) random telegraph noise
(RTN), a further discernbile recurrence feature is present in the dynamics.
Moreover, this feature is correlated with the localization of the walker. On
the other hand, no additional recurruence feature appears for other
non-Markovian types of noise (Ornstein-Uhlenbeck and Power Law noise). We
propose a power spectral method for comparing the relative strengths of the
non-Markovian component due to the external noise and that due to the internal
position degree of freedom.
