Conditional limit measure of a one-dimensional quantum walk with an absorbing sink
Mohamed Sabri, Etsuo Segawa, M. Štefaňák · Physical Review A · 2018
We consider a two-state quantum walk on a line where after the first step an absorbing sink is placed at the origin. The probability of finding the walker at position $j$, conditioned on that it has not returned to the origin, is investigated in the asymptotic limit. We prove a limit theorem for the conditional probability distribution and show that it is given by the Konno's density function modified by a prefactor ensuring that the distribution vanishes at the origin. In addition, we discuss the relation to the problem of recurrence of a quantum walk and determine the P\'olya number. Our approach is based on path counting and the stationary phase approximation.