  A suitable way of quantifying work for microscopic quantum systems has been
constantly debated in the field of quantum thermodynamics. One natural approach
is to measure the average increase in energy of an ancillary system, called the
battery, after a work extraction protocol. The quality of energy extracted is
usually argued to be good by quantifying higher moments of the energy
distribution, or by restricting the amount of entropy to be low. This limits
the amount of heat contribution to the energy extracted, but does not
completely prevent it. We show that the definition of "work" is crucial. If one
allows for a definition of work that tolerates a non-negligible entropy
increase in the battery, then a small scale heat engine can possibly exceed the
Carnot efficiency. This can be achieved even when one of the heat baths is
finite in size.
