Complementarity in quantum walks

Abstract

The eigenbases of two quantum observables, { | a i ⟩ } i = 1 D and { | b j ⟩ } j = 1 D , form mutually unbiased bases (MUB) if | ⟨ a i | b j ⟩ | = 1 / D for all i and j. In realistic situations MUB are hard to obtain and one looks for approximate MUB (AMUB), in which case the corresponding eigenbases obey | ⟨ a i | b j ⟩ | ⩽ c / D , where c is some positive constant independent of D. In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks (QWs) 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 the eigenvectors of two QW evolution operators form AMUB. Namely, if d is prime the corresponding eigenvectors of the evolution operators, that act in the D-dimensional Hilbert space ( D = 2 d ), obey | ⟨ v q | v q ′ ′ ⟩ | ⩽ 2 / D for q ≠ q ′ and for all | v q ⟩ and | v q ′ ′ ⟩ . Finally, we show that the analogous AMUB relation still holds in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.