  According to the second law, the efficiency of cyclic heat engines is limited
by the Carnot bound that is attained by engines that operate between two
thermal baths under the reversibility condition whereby the total entropy does
not increase. Quantum engines operating between a thermal and a
squeezed-thermal bath have been shown to surpass this bound. Yet, their maximum
efficiency cannot be determined by the reversibility condition, which may yield
an unachievable efficiency bound above unity. Here we identify the fraction of
the exchanged energy between a quantum system and a bath that necessarily
causes an entropy change and derive an inequality for this change. This
inequality reveals an efficiency bound for quantum engines energised by a
non-thermal bath. This bound does not imply reversibility, unless the two baths
are thermal. It cannot be solely deduced from the laws of thermodynamics.
