Nonclassical oscillations in pre- and post-selected quantum walks

Quantum walks are counterparts of classical random walks. They spread faster, which can be exploited in information processing tasks, and constitute a versatile simulation platform for many quantum systems. Yet, some of their properties can be emulated with classical light. This raises a question: which aspects of the model are truly nonclassical? We address it by carrying out a photonic experiment based on a pre- and post-selection paradox. The paradox implies that if somebody could choose to ask either if the particle is at position $x=0$ at even time steps or at position $x=d$ $(dg1)$ at odd time steps, the answer would be positive, no matter the question asked. Therefore, the particle seems to undergo long distance oscillations despite the fact that the model allows it to jump one position at a time. We translate this paradox into a Bell-like inequality and then into a contextuality witness. Finally, we experimentally verify this witness up to eight standard deviations.