The quantum switch implements indefinite causal order via a coherent control of the orderings of multiple quantum operations, leading to various advantages and applications, one of which being in work extraction where enhancements to daemonic ergotropy is possible. Motivated by recent developments in the connections between non-Markovianity and the quantum switch, we construct a non-Markovian process that reduces to the two-party quantum switch in the fully non-Markovian limit. By controlling the amount of non-Markovianity in the process, we identified two operational regimes with differing behaviors. One has its daemonic ergotropy dependent on the presence and amount of non-Markovianity, achieving the maximum in the quantum switch case of full non-Markovianity. The other regime, however, has no advantages from non-Markovianity. We compare this non-Markovian process with the case of a superposition of independent channels, where two channels are placed in a coherent superposition without indefinite causal order, uncovering the advantages of non-Markovianity. Finally, the conditions required for the production of positive daemonic ergotropy are also derived for the case of fully non-Markovian and fully Markovian limits, where we compare against the conditions required for the superposition of independent channels.

• Received 6 April 2023
• Accepted 13 June 2023

DOI:https://doi.org/10.1103/PhysRevA.108.012201