Optimal Thermalization under Indefinite Causal Order with Identical and Asymmetric Baths

TL;DR

Indefinite causal order implemented via a quantum SWITCH can modify the thermodynamic response of a two-level system coupled to baths. The authors derive closed-form expressions for the postselected final inverse temperature $β_f$ in both identical and distinct bath scenarios and identify optimal control-qubit parameters that maximize cooling or heating, revealing how diagonal control and coherence terms independently shape the temperature shift. Bath asymmetry enhances ICO-induced deviations while reduced control-qubit purity suppresses them, establishing coherence as a tunable thermodynamic resource. These results provide a systematic framework for ICO-enabled thermodynamics and motivate future ICO-based refrigeration and work-extraction protocols.

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.

Figures (5)

• Figure 1: Schematic of the ICO thermalization protocol. A control qubit, prepared in an initial state $\rho_c$, coherently determines whether the system undergoes the sequence of thermal channels $\mathcal{E}_1 \circ\mathcal{E}_2$ or $\mathcal{E}_2 \circ\mathcal{E}_1$. After the joint evolution, the control is measured in a given basis, and the system is postselected on the chosen outcome. The resulting system state $\rho_f$ is diagonal in the energy basis and can be described by an effective inverse temperature $\beta_f$, which may differ from the temperatures of the baths.
• Figure 2: Optimal measurement polar angle $\Theta$ as a function of the initial control angle $\theta$ for a pure control qubit $(r=1)$. The three curves correspond to bath asymmetry parameters $n=\beta_{T2}/\beta_{T1}=0.5,~1.0,~2.0$. The optimal values identify the measurement directions that maximize or minimize the final inverse temperature.
• Figure 3: Heat maps of the temperature shift $\Delta\beta(\Theta,\theta)=\beta_f-\beta_{T1}$ for optimal phase differences $\Delta\Phi=\Phi-\phi$. Top row: $\Delta\Phi=0$ (phase aligned, constructive interference); bottom row: $\Delta\Phi=\pi$ (phase anti-aligned, destructive interference). Columns (left to right) correspond to bath-asymmetry ratios $n=\beta_{T2}/\beta_{T1}=0.5,\;1.0,\;2.0$. For all panels, we use $\beta_{T1}\Delta=1.0$, $\beta_i=\beta_{T1}$, and the control qubit in a pure state $(r=1)$. Color indicates the deviation of the ICO output inverse temperature from the reference bath temperature.
• Figure 4: Globally optimized output inverse temperatures $\beta_f^{\max}$ and $\beta_f^{\min}$, normalized to $\beta_{T1}$, as functions of bath-asymmetry parameter $n=\beta_{T2}/\beta_{T1}$. Solid lines show results for a pure control qubit $(r=1.0)$, and dashed lines correspond to a control qubit initialized in a mixed state $(r=0.5)$. For this figure, we use $\beta_{T1}\Delta=1.0$ and $\beta_i=\beta_{T1}$.
• Figure 5: Heat maps of $\Delta\beta(\Theta,\theta)=\beta_f-\beta_{T1}$ for the case of control qubit initialized in the mixed state with the length of the Bloch vector $r=0.5$. Layout and parameter choices are identical to Fig. \ref{['fig-3:dbeta_vs_theta_m_vs_theta_c']} from the main text. The reduced purity suppresses coherence-driven contributions and narrows the range of temperatures attainable through ICO.