  We show that the entanglement between the internal (spin) and external
(position) degrees of freedom of a qubit in a random (dynamically disordered)
one-dimensional discrete time quantum random walk (QRW) achieves its maximal
possible value asymptotically in the number of steps, outperforming the
entanglement attained by using ordered QRW. The disorder is modeled by
introducing an extra random aspect to QRW, a classical coin that randomly
dictates which quantum coin drives the system's time evolution. We also show
that maximal entanglement is achieved independently of the initial state of the
walker, study the number of steps the system must move to be within a small
fixed neighborhood of its asymptotic limit, and propose two experiments where
these ideas can be tested.
