We explore the fundamental limits on thermodynamic irreversibility when cooling a quantum system in the presence of a finite-size reservoir. First, we prove that, for any noninteracting $n$-particle reservoir, the entropy production $\mathrm{\Sigma }$ decays at most linearly with $n$. Instead, we derive a cooling protocol in which $\mathrm{\Sigma }\propto 1/n^2$, which is, in fact, the best possible scaling. This becomes possible due to the presence of interactions in the finite-size reservoir, which must be prepared at the verge of a phase transition. We further show numerically that an intermediate scaling of $\mathrm{\Sigma }\propto 1/n^{\delta }$ with $\delta \in (1,2)$ can be achieved using a star-network configuration for the reservoir. Our results open the possibility of cooling with a higher energetic efficiency via interacting reservoirs.

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• Quantum information theory
• Quantum thermodynamics
• Stochastic thermodynamics