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arXiv · 2205.05445

Complementarity in quantum walks

Andrzej Grudka·Pawel Kurzynski·Tomasz P. Polak·Adam S. Sajna·Jan Wojcik·Antoni Wojcik
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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.

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Andrzej Grudka, Pawel Kurzynski, Tomasz P. Polak, Adam S. Sajna, Jan Wojcik, Antoni Wojcik. 2022-05-11. Complementarity in quantum walks. https://doi.org/10.1088/1751-8121/acdcd0

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