Quantum Polya theorem

In this paper we discuss return probability properties of quantum random walk on the line. In the classical case this property is well known as the Polya theorem. We study in detail not only usually discussed “Hadamar walk”. In the general case quantum random walk depends on parameter $\theta$ ($0\le\theta\le\pi$). It was shown that in the most of cases when $0<\theta<\pi$ quantum random walk is weak localized and recurrent and the return probability tends to $0$ with the speed $1/t$. Other cases are also studied and described.