Indefinite causal order is not always a resource for thermodynamic processes

Abstract

Indefinite causal order is a key feature involved in the study of quantum higher order transformations. Recently, intense research has been focused on possible advantages related to the lack of definite causal order of quantum processes. Quite often the quantum switch is claimed to provide advantages in information-theoretic and thermodynamic tasks. We address here the question whether indefinite causal order is a resource for quantum thermodynamics. Inspired by previous results in the literature taking free energy and ergotropy as the figures of merit, we propose a framework for properly comparing the thermodynamic value of processes by comparing higher order transformations of the same type, and show that for the tasks considered here indefinite causal order is not necessary for thermodynamic advantages. More specifically, we show that there is a non-Markovian process, a causally ordered higher-order transformation, outperforming the results obtained for the quantum switch. We also discuss a possible way to study the advantages that may arise from indefinite causal order in a general scenario.