  The classicalization of a decoherent discrete-time quantum walk on a line or
an n-cycle can be demonstrated in various ways that do not necessarily provide
a geometry-independent description. For example, the position probability
distribution becomes increasingly Gaussian, with a concomitant fall in the
standard deviation, in the former case, but not in the latter. As another
example, each step of the quantum walk on a line may be subjected to an
arbitrary phase gate, without affecting the position probability distribution,
no matter whether the walk is noiseless or noisy. This symmetry, which is
absent in the case of noiseless cyclic walk, but is restored in the presence of
sufficient noise, serves as an indicator of classicalization, but only in the
cyclic case. Here we show that the degree of quantum correlations between the
coin and position degrees of freedom, quantified by a measure based on the
disturbance induced by local measurements (Luo, Phys. Rev. A 77, 022301
(2008)), provides a suitable measure of classicalization across both type of
walks. Applying this measure to compare the two walks, we find that cyclic
quantum walks tend to classicalize faster than quantum walks on a line because
of more efficient phase randomization due to the self-interference of the two
counter-rotating waves. We model noise as acting on the coin, and given by the
squeezed generalized amplitude damping (SGAD) channel, which generalizes the
generalized amplitude damping channel.
