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

Abstract

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 are affected by the appearance of bound states and their reflection symmetry. Here, 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 symmetries 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.

Disorder-free localization in quantum walks

“…For example, the two completely X-polarized spin configurations, where all s n are either + or −, yield the standard quantum walk with all of its eigenstates extended. On the other hand, the existence of a single spin-flip disorder leads some localized eigenstates |ψ m,sX around this impurity 19,20 .…”

“…Having been put forward almost three decades ago as a quantum counterpart of random walks 12 , quantum walks proved themselves so far to be a versatile model for quantum com-putation, not only due to their role in the development of new quantum algorithms 13 but also for providing a concrete framework for universality 14 . Despite their superiority over random walks in spreading rates 15,16 leading to faster computational algorithms 13 , they have also attracted considerable attention in terms of their non-diffusibility features in the presence of disorder such as quantum-to-classical transition 17 , dynamical localization 18 , and the emergence of bound states 19,20 . While rolling back to the classical behavior is attributed to wiping out coherence in the system due to dynamical disorder, it is the static disorder that leads to dynamical localization in quantum walks.…”

Strongly trapped space-inhomogeneous quantum walks in one dimension

“…Therefore, we are interested in the asymptotic behaviors and the class of strong trapping for two-phase QWs with one defect. In this paper, we introduce methods for the eigenvalue problem to reveal time-averaged limit distribution and the class of strong trapping by using the transfer matrix [5,15,16,18].…”

Localization of space-inhomogeneous three-state quantum walks

“…Solving eigenvalue problem via the transfer matrix was constructed for two-phase twostate QWs with one defect in [26] and for more general space-inhomogeneous QWs in [32]. The transfer matrix is also used in [33,34,35]. In this paper, we extend the transfer matrix method to n-state QWs with n − 2 self-loops.…”