  We design a heat engine with multi-heat-reservoir, ancillary system and
quantum memory. We then derive an inequality related with the second law of
thermodynamics, and give a new limitation about the work gain from the engine
by analyzing the entropy change and quantum mutual information change during
the process. In addition and remarkably, by combination of two independent
engines and with the help of the entropic uncertainty relation with quantum
memory, we find that the total maximum work gained from those two heat engines
should be larger than a quantity related with quantum entanglement between the
ancillary state and the quantum memory. This result provides a lower bound for
the maximum work extracted, in contrast with the upper bound in the
conventional second law of thermodynamics. However, the validity of this
inequality depends on whether the maximum work can achieve the upper bound.
