Abstract
We study discrete-time quantum walks on d-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter q. We solve the model analytically and observe that for prime d there exists a strong complementarity property between the eigenvectors of two quantum walk evolution operators that act in the 2d-dimensional Hilbert space. Namely, if d is prime the corresponding eigenvectors of the evolution operators obey | v_q|v'_q' | ≤ 1/√d for q≠ q' and for all |v_q and |v'_q'. We also discuss dynamical consequences of this complementarity. Finally, we show that the complementarity is still present in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.