Complementarity in quantum walks

TL;DR

This paper analyzes discrete-time quantum walks on cycles with phase-dependent dynamics, revealing a complementarity property between eigenvectors for prime cycle lengths, with implications for quantum particle behavior.

Contribution

It introduces an analytically solvable model of quantum walks with phase shifts and uncovers a novel complementarity property for prime cycle lengths, extending to a continuous Dirac particle model.

Findings

Eigenvectors exhibit a bound of 1/√d for prime d

Complementarity property holds for both discrete and continuous models

Eigenvector overlaps are limited, indicating strong quantum complementarity

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 $|\langle v_q|v'_{q'} \rangle| \leq 1/\sqrt{d}$ for $q\neq q'$ and for all $|v_q\rangle$ and $|v'_{q'}\rangle$. 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.