Quantum walks under superposition of causal order

TL;DR

This paper investigates how superpositions of causal order can be integrated into quantum walks, revealing that only certain types like periodic walks or those with disorder exhibit such superpositions, with implications for non-Markovian dynamics.

Contribution

It establishes criteria for incorporating superpositions of causal order into quantum walks, highlighting the special role of periodic and disordered walks and analyzing their causal asymmetries.

Findings

Superposition of causal order occurs only in periodic or disordered quantum walks.

Periodic quantum walks show causal asymmetry and non-Markovian behavior.

Non-Markovianity correlates with indefinite causal order in quantum walks.

Abstract

We set the criteria under which superposition of causal order can be incorporated in to quantum walks. In particular, we show that only periodic quantum walks or those with at least one disorder exhibit Superposition of causal order under the action of `quantum switch'. We exemplify our results with a simple example of two-period discrete-time quantum walks. In particular, we observe that periodic quantum walks exhibit causal asymmetry pertaining to the dynamics of the reduced coin state: the dynamics are more non-Markovian for one temporal order than the other. We also note that the non-Markovianity of the reduced coin state due to indefiniteness in causal order tends to match the dynamics of a particular temporal order of the coin state. We substantiate our results with numerical simulations.