Optimal Thermalization under Indefinite Causal Order with Identical and Asymmetric Baths

Abstract: Indefinite causal order (ICO), in which the order of quantum operations is placed in a coherent superposition, has been demonstrated to enhance various information-processing tasks. Here, we investigate its impact on the thermodynamic processes generated by thermalizing quantum channels. We consider a two-level system interacting with two thermal baths under a quantum SWITCH, with the channel order controlled coherently by an ancillary qubit. We derive closed-form expressions for the effective inverse temperature $β_f$ of the postselected system state for both identical and distinct bath temperatures, and identify the control-qubit parameters that maximize heating or cooling. Our analysis reveals how the diagonal and coherent components of the control-qubit state contribute separately to the temperature shift, and how their interplay enables departures from the thermal response attainable under any fixed causal order. Bath asymmetry enhances these effects, while reduced purity of the control qubit state suppresses them. These results provide a systematic framework for assessing the thermodynamic capabilities of ICO and clarify the role of quantum coherence as a tunable thermodynamic resource.