Complementarity in quantum walks

• URL: http://arxiv.org/abs/2205.05445v2
• Date: Wed, 22 Jun 2022 19:30:04 GMT
• Title: Complementarity in quantum walks
• Authors: Andrzej Grudka, Pawel Kurzynski, Tomasz P. Polak, Adam S. Sajna, Jan Wojcik, Antoni Wojcik
• Abstract summary: We study discrete-time quantum walks on $d$-cycles with a position and coin-dependent phase-shift. For prime $d$ there exists a strong complementarity property between the eigenvectors of two quantum walk evolution operators. We show that the complementarity is still present in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.
• Score: 0.08896991256227595
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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