**7. Conclusions and Outlook**

In this paper, we have reviewed and extended the analysis of a QPC (or a QPC-like) device, with a transmission probability with a smoothed step-like energy-dependence, as a steady-state thermoelectric heat engine. The interest in a QPC for heat-to-power conversion derives from its optimal performance with respect to the output power, which goes along with rather large efficiencies. We have analyzed the influence of the barrier smoothness on this behavior and found that strong non-linear-response conditions are required to recover a comparable performance.

In addition to the typically studied performance quantifiers—output power and efficiency—we have broadened the analysis by adding the *power fluctuations* as an independent quantification of performance. The bound on the combination of these three quantities set by the recently identified thermodynamic uncertainty relation, suggests investigating this as a combined performance quantifier.

We have shown that the bound of the thermodynamic uncertainty relation is further restricted if one adds the practical constraint of finite (positive!) output power. In the linear response, we quantify this restriction by the figure of merit *ZT*. Interestingly, we have found that this combined performance quantifier maximized over the voltage has large values in those parameter regions in which the maximized efficiency is large, while regions of maximal output power are not distinguished. However, while efficiencies take their maximal value in regions close to the stopping voltage in which finite power is produced, accounting for fluctuations shifts the optimal performance value to the limit of zero voltage and zero power production.

Whether this result is unique to the QPC as steady-state heat engine or can be generalized for other thermoelectric devices is a topic of further studies. Our analysis also naturally raises the question of how to quantify the performance of the QPC when operated as a refrigerator [14,15]. Given that QPCs are standard components in many mesoscopic experiments and both the currents and noise are experimentally accessible, we anticipate that our results could be tested in experiments in the near future.

**Author Contributions:** The project was suggested and conceived by J.S. Numerical maximization of different quantities were mostly performed by S.K. and P.S., analytical estimates were mainly performed by N.D., M.M., and P.S. Thermodynamic interpretation of the TUR aspects was provided by P.P.P. All authors (S.K., N.D., M.M., P.P.P., J.S., P.S.) contributed to the discussion of results and the preparation of the manuscript.

**Funding:** This research was funded by the Knut and Alice Wallenberg Foundation via an Academy Fellowship (N.D., M.M., J.S.), by the Swedish VR (S.K., J.S., P.S.), and by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 796700 (P.P.P.).

**Acknowledgments:** We acknowledge fruitful discussions with Artis Svilans and Heiner Linke and helpful comments on the manuscript by Robert Whitney and David Sánchez.

**Conflicts of Interest:** The authors declare no interest in conflicts.
