Quantum walk on the line: Entanglement and nonlocal initial conditions

The conditional shift in the evolution operator of a quantum walk generates entanglement between the coin and position degrees of freedom. This entanglement can be quantified by the von Neumman entropy of the reduced density operator (entropy of entanglement). We show analytically that for a Hadamard walk with local initial conditions the asymptotic entanglement is 0. 872 for all initial coin states. When nonlocal initial conditions are considered, the asymptotic entanglement varies smoothly between almost complete entanglement and no entanglement (product state). An exact expression for the asymptotic (long-time) entanglement is obtained for initial conditions in the position subspace spanned by 1⟩.