Complementarity in quantum walks

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 $|⟨v_q|v_{q^{\prime }}^{\prime }⟩|\le 1/\sqrt{d}$ for $q\ne q^{\prime }$ and for all $|v_q⟩$ and $|v_{q^{\prime }}^{\prime }⟩$. 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.

5+7 pages, 2 figures, comments welcome

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