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A long standing open problem whether a heat engine with finite power achievesthe Carnot efficiency is investigated. We rigorously prove a general trade-offinequality on thermodynamic efficiency and time interval of a cyclic processwith quantum heat engines. In a first step, employing the Lieb-Robinson boundwe establish an inequality on the change in a local observable caused by anoperation far from support of the local observable. This inequality provides arigorous characterization of the following intuitive picture that most of theenergy released from the engine to the cold bath remains near the engine whenthe cyclic process is finished. Using the above description, we finally provean upper bound on efficiency with the aid of quantum information geometry. Inaddition, since our inequality falls down into the conventional second law ofthermodynamics in Markovian limit, we adopt a completely different treatmentwhich is developed in the context of classical stochastic processes. Our resultgenerally excludes the possibility of a process with finite speed at the Carnotefficiency in quantum heat engines. In particular, the obtained constraintcovers engines evolving with non-Markovian dynamics, which almost all previousstudies on this topics fail to address.

Efficiency versus Speed in Quantum Heat Engines: Rigorous Constraint from Lieb-Robinson Bound