Quantum-classical dynamical distance for walks on graphs: Quantumness and decoherence.

Quantum walks (QWs), the quantum counterpart of classical random walks, are a powerful tool for many applications, from the modelling of quantum transport to universal quantum computation and quantum algorithms. We introduce a fidelity-based measure to quantify the differences in the dynamics of classical and quantum continuous-time walks over a graph. We provide graph-independent analytic expressions of this quantum-classical dynamical distance in the case of unitary quantum dynamics. We show that at short times it is proportional to the coherence of the walker, $i.e.$, a genuine quantum feature, whereas at long times it depends only on the size of the graph. We then address different models of decoherence in QWs and employ the quantum-classical dynamical distance as a figure of merit to assess whether, and to which extent, decoherence classicalizes the CTQW, $i.e.$, turns it into the analogue classical process. We show that while dephasing in the position basis asymptotically destroys the quantumness of the walker, making it equivalent to a classical random walk, decoherence in the energy basis does not fully classicalize the walker and partially preserves its quantum features.