**Abstract**
Today, theoretic analysis about quantum-scale heat engines is achieving a splendid success. They clarify that the average performance of these small-size heat engines obeys the second law of the macroscopic thermodynamics[1,2], and that the single-shot performance of the heat engines obeys different rules[3,4]. They also clarifies the thermodynamic laws for information processing [5,6].

In spite of these success, there are still two unsolved problems for constructing the thermodynamics of small-size systems.First, the above researches formulate the quantum heat engine in various ways, and the relationship among the formulations has not been sufficiently discussed. Second, The above statistical mechanical approches have never treated the finite-size heat baths. On the other hand, the macroscopic thermodynamics can treat macroscopic finite-size baths. It gives the optimal bounds of the performance of the heat engines with arbitrary-size baths whenever the baths are macroscopic. However, what is the quantitative definition of "macroscopic system"?

In this talk, we give two results which shed a light on those two problems. Firstly, we classify the previous formulations of quantum heat engines, and derive a trade-off relation that clarifies a problem of a widely-used formulation of quantum heat engine controlled by a classical controller, in which the time evolution of the internal system (working body and heat baths) is formulated as a unitary transformation. In order to dissolve this problem, we remodel a quantum heat engine controlled by a classical controller as a general measurement process. Our measurement-based formulation is consistent with the Fully Quantum formulation, which is another widely-used formulation of quantum heat engine.

Second, we derive the optimal efficiency of quantum (or classical) heat engines whose heat baths are $n$-particle systems. We give the optimal work extraction process as an energy-preserving unitary time evolution among the heat baths and the work storage. During the unitary, the entropy gain of the work storage is so negligibly small as compared with the energy gain of the work storage, i.e., we can interpret the energy gain as the extracted work. With using our results, we evaluate the accuracy of the macroscopic thermodynamics for the heat engines with finite-size heat baths from the statistical mechanical viewpoint.

The details of the contents in this talk are in the articles arXiv:1504.06150.v2 and arXiv:1405.6457.v2, which are collaborations with Prof. Masahito Hayashi at Nagoya University.

References:

[1] H. Tasaki, arXiv:cond-mat/0009244 (2000).

[2] P. Skrzypczyk, A. J. Short and P. Sandu, Nature Communications 5, 4185, (2014)

[3] M. Horodecki and J. Oppenheim, Nat. Commun. 4, 2059 (2013).

[4] F. G. S. L. Brandao, M. Horodeck, N. H. Y. Ng, J. Oppenheim, and S. Wehner, PNAS, 112,3215(2015).

[5] T. Sagawa, M. Ueda, Phys. Rev. Lett. 102 250602 (2009).

[6] L. Rio, J. Aberg, R. Renner, O. Dahlsten, and V. Vedral, Nature,474, 61, (2011).