  According to Thermodynamics, the efficiency of a heat engine is upper bounded
by Carnot efficiency. For macroscopic systems, the Carnot efficiency is,
however, achieved only for quasi static processes. And, considerable attention
has been paid to provide general evaluation of the efficiency at a finite
speed. Recently, several upper bounds of the efficiency have been derived in
the context of the trade-off among the efficiency, power, and other quantities
such as the fluctuation of power.
  Here, we show universal lower and upper bounds of the efficiency from the
thermodynamic uncertainty relations for the entropy production and for the heat
transfers. The lower bound is characterized by the ratio between the
fluctuation of the irreversible entropy production and mean output work. The
upper bound of the efficiency is described by a generalized precision of the
heat transfers among the working substance, and hot and cold reservoirs. We
explicitly derive necessary and sufficient conditions of both the lower and
upper bounds in a unified manner in terms of fluctuation theorem. Hence, our
result provides an operating principle of the heat engine.
