Complementarity in quantum walks

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