Shifting position decoherence in one-dimensional discrete time quantum walk

We have investigated the shifting position decoherence in one-dimensional discrete time quantum walk (DTQW) which arise from the tunneling effect in the experimental realization of quantum walk. We proved that the quantum behavior of 1DQW in the presence of tunneling effect doesn't fade, unlike the case in which coin subspace is subject to decoherence where in this case, system transition to classic even for weak strength of noise. We show that quadratic dependency of variance on time, which is pure quantum property of coherent QW, remain quadratic in the presence of tunneling decoherence and coin-position entanglement (CPE), which is another pure quantum property and converge to significant value for any strength of noise. Furthermore we show that this type of decoherence smooth the probability distribution and we represent the compact formula for probability distribution in terms of coherent one-dimensional quantum walk (1DQW).