Non-Markovianity and bound states in quantum walks with a phase impurity

We study the discrete-time quantum walk on the line with a single phase impurity. The spread and localisation properties of discrete-time walks initialized at the impurity site arc affected by the appearance of bound states and their reflection symmetry. Mere, we measure localisation by means of an effective localisation length and an effective participation ratio, which are obtained by averaging over all eigenstates and over all initial states, respectively. We observe that the reduced coin system dynamics undergoes oscillations in the long-time limit with the frequencies determined by the sublattice operator and the bound state quasi-energy differences. The oscillations give rise to non-Markovian evolution, which we quantify using the trace distance and entanglement based measures of non-Markovianity. Indeed, we reveal that the degree of the non-Markovian behaviour is closely related to the emergence of bound states due to the phase impurity. We also show that the considered measures give qualitatively different results depending on the number and synunetries of supported bound states. Finally, comparing localisation and non-Markovianity measures, we demonstrate that the degree of non-Markovianity becomes maximum when the walker is most localised in position space.