Abstract
We revisit the one-dimensional discrete time quantum walk with three states and the Grover coin, the simplest model that exhibits localization in a quantum walk. We derive analytic expressions for the localization and a long-time approximation for the entire probability density function (PDF). We find the possibility for asymmetric localization to the extreme that it vanishes completely on one site of the initial conditions. We also connect the time-averaged approximation of the PDF found by Inui et al. [Phys. Rev. E 72, 056112 (2005)] to a spatial average of the walk. We show that this smoothed approximation predicts moments of the real PDF accurately.
- Received 20 May 2014
DOI:https://doi.org/10.1103/PhysRevA.90.012307
©2014 American Physical Society