No-cooling theorem for the simultaneous-coupling RI qubit thermal machine

Prove that in the simultaneous-coupling repeated-interaction thermal machine—where a system qubit interacts simultaneously with hot and cold ancilla qubits prepared in Gibbs states at inverse temperatures β_H and β_C—the heat exchanged with the cold bath per collision in the limit-cycle state, Q_C^{(∞)}, is always nonnegative for arbitrary interaction parameters J_{xx}^H, J_{yy}^H, J_{xx}^C, J_{yy}^C and collision duration τ, thereby establishing the impossibility of refrigeration beyond the perturbative short-collision-time regime.

Background

The paper proves a rigorous no-go theorem for refrigeration in the alternating-coupling repeated-interaction (RI) model, showing Q_C^{(∞)} ≥ 0 for all parameters. In the simultaneous-coupling model, the authors provide a perturbative short-time proof and demonstrate numerical evidence suggesting the same constraint at strong coupling.

They explicitly conjecture that the impossibility of refrigeration extends to the simultaneous-coupling RI model beyond the Dyson-limit, making a full, nonperturbative proof a central open question.