Anderson Localization in Disordered Quantum Walks
(**Volkher Scholz**, Albert Werner, and Andre Ahlbrecht)
We study a Spin-$\frac{1}{2}$-particle moving in a one dimensional
lattice subjected to disorder induced by a random space dependent
coin. The discrete time evolution is given by a family of random
unitary quantum walk operators, where the shift operation is assumed
to be non-random. Each coin is an independent identically distributed
random variable with values in the group of two dimensional unitary
matrices. We find that if the probability distribution of the coins
is absolutely continuous with respect to the Haar measure, then the
system exhibits localization. That is, every initially localized
particle remains on average and up to exponential corrections in a
finite region of space for all times.