Properties of Open Quantum Walks on $\mathbb{Z}$
  A connection between the asymptotic behavior of the open quantum walk and the spectrum of a generalized quantum coins is studied. For the case of simultaneously diagonalizable transition operators an exact expression for probability distribution of the position of the walker for arbitrary number of steps is found. For a large number of steps the probability distribution consist of maximally two "soliton"-like solution 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 spectrum is an indicator of the "soliton"-like solutions in the probability distribution. 
I. Sinayskiy and F. Petruccione
quant-ph