"Measurement-based Formulation of Quantum Heat Engine and Optimal Efficiency with Finite-Size Heat Baths"


Hiroyasu Tajima (Riken)


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]. On the other hand, it is also true that there are three unsolved problems for constructing the thermodynamics of small-size systems. (P1: Disunity in the formulations of work extraction) The above researches formulate the quantum heat engine in various ways, and the relationship among the formulations has not been sufficiently discussed. (P2: Problem about treating finite-size heat baths) 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 such baths. How small are the systems and the baths such that the upper bounds lose the optimality? Based on the above situation, we firstly propose a general measurement-based formation of the quantum heat engines. Our formulation describes an arbitrary quantum heat engine that has some equipment to assess the amount of the extracted work. With using our formulation, we derive two trade-off relations that clarify a problem of a widely-used formulation of quantum heat engine, in which the time evolution of the internal system (working body and heat baths) is formulated as a unitary transformation. The trade-off relations clarify that we can hardly know the amount of the extracted work when the time evolution of the internal system is close to unitary. Second, we derive the optimal efficiency of quantum (or classical) heat engines whose heat baths are n-particle systems. We give a concrete work extracting process which attains the optimal efficiency 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. We also give the stability condition of the heat baths; at the limit of n-> ∞, we can make the initial and final states of the heat baths are the same iff Qn=o(√n), where Qn is the endothermic amount from the hot bath. The details of the contents in this talk are in the articlesarXiv:1405.6457 and arXiv:1504.06150, which are in collaboration with Prof. Masahito Hayashi at Nagoya University.
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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).
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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).

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