  We characterize quantumness of the so-called quantum walks (whose dynamics is
governed by quantum mechanics) by introducing two computable measures which are
stronger than the variance of the walker's position probability distribution.
The first measure is based on comparing probability distributions of a quantum
walk and all classical random walks (through the classical relative entropy of
the distributions), and it quantifies non-Gaussianity of the probability
distribution of the walk. Next, after assigning a density matrix to classical
walks, we introduce a more powerful measure by employing quantum relative
entropy. We show that this measure exceeds the first one by the quantum
coherence of the walk. There are walks labeled classical by the variance
whereas our measures identify some quantumness therein. As an application, we
study a model of quantum (energy) transport on a simple lattice, and compare
the behavior of its relative transport efficiencies with that of the
quantumness. Our measures help partly explain why in some quantum transport
phenomena a considerably high efficiency may appear---this is where quantumness
is appreciable.
