  A long standing open problem whether a heat engine with finite power achieves
the Carnot efficiency is investigated. We rigorously prove a general trade-off
inequality on thermodynamic efficiency and time interval of a cyclic process
with quantum heat engines. In a first step, employing the Lieb-Robinson bound
we establish an inequality on the change in a local observable caused by an
operation far from support of the local observable. This inequality provides a
rigorous characterization of the following intuitive picture that most of the
energy released from the engine to the cold bath remains near the engine when
the cyclic process is finished. Using the above description, we finally prove
an upper bound on efficiency with the aid of quantum information geometry. In
addition, since our inequality falls down into the conventional second law of
thermodynamics in Markovian limit, we adopt a completely different treatment
which is developed in the context of classical stochastic processes. Our result
generally excludes the possibility of a process with finite speed at the Carnot
efficiency in quantum heat engines. In particular, the obtained constraint
covers engines evolving with non-Markovian dynamics, which almost all previous
studies on this topics fail to address.
