Properties of open quantum walks on $\mathbb {Z}$
Ilya Sinayskiy, Francesco Petruccione · Physica Scripta · 2012
The connection between the asymptotic behavior of the open quantum walk and the spectrum of generalized quantum coins was studied. In the case of simultaneously diagonalizable transition operators, an exact expression for the probability distribution of the position of the walker for an arbitrary number of steps was found. For a large number of steps, the probability distribution consists of, maximally, two ‘soliton’-like solutions and a certain number of Gaussian distributions. The number of different contributions to the final probability distribution is equal to the number of distinct absolute values in the spectrum of the transition operators. The presence of the zeros in the spectrum is an indicator of the ‘soliton’-like solutions in the probability distribution.