Incompatibility between Carnot efficiency and finite power in Markovian dynamics

In Markovian dynamics with the local detailed balance condition, we decompose the total entropy production rate into microscopic transitions. By applying this decomposition to the heat to work conversion process, we rigorously show that the Carnot efficiency implies zero power for any heat engine, even with broken time-reversal symmetry beyond the linear response regime. Moreover, we propose a trade-off relationship between the entropy production rate and the heat flow between the system and bath.