Limitations of discrete-time quantum walk on a one-dimensional infinite chain

How well can we manipulate the state of a particle via a discrete-time quantum walk? We show that the discrete-time quantum walk on a one-dimensional infinite chain with coin operators that are independent of the position can only realize product operators of the form eiξA⊗1p, which cannot change the position state of the walker. We present a scheme to construct all possible realizations of all the product operators of the form eiξA⊗1p. When the coin operators are dependent on the position, we show that the translation operators on the position can not be realized via a DTQW with coin operators that are either the identity operator 1 or the Pauli operator σx.