Non-Markovian decoherent quantum walks

Quantum walk acts obviously different from its classical counterpart, but the decoherence will lessen and close the gap between them. To understand this process, it is necessary to investigate the evolution of the quantum walk under different situations of decoherence. In this article, we study a non-Markovian decoherent quantum walk on a line. In the short time regime, the behavior of the walk deviates from both idea quantum walks and classical random walks. The position variance as a measure of the quantum-walk collapses and revivals for a short time and tends to have a linear relation with time, that is the walker's behavior shows a diffusive spread in the long time limit, which is caused by the non-Markovian dephasing affecting on quantum correlations between the quantum walker and his coin. We also study both quantum discord and measurement-induced disturbance as measures of quantum correlations and observe that both collapse and revival in the short time regime and tend to be zero in the long time limit. Therefore, the quantum walk with a non-Markovian decoherence tends to diffusive spreading behavior in the long time limit, while in the short time regime, it oscillates between a ballistic and diffusive spreading behavior, and the quantum correlation collapses and revivals due to the memory effect.