Measurement-based Formulation of Quantum Heat Engine

There exist two formulations for quantum heat engine. One is semi-classical scenario, and the other is full quantum scenario. The former is formulated as a unitary evolution for the internal system, and is adopted by the community of statistical mechanics. In the latter, the whole process is formulated as unitary. It was adopted by the community of quantum information. However, their formulation does not consider measurement process. In particular, the former formulation does not work when the amount of extracted work is observed. In this paper, we formulate the quantum heat engine as the measurement process because the amount of extracted work should be observed in a practical situation. Then, we clarify the contradiction of the former formulation by using a novel trade-off relation. The trade-off relation clarifies the impossibility of proper work extraction by an internal unitary process.