  Sadi Carnot's theorem regarding the maximum efficiency of heat engines is
considered to be of fundamental importance in thermodynamics. This theorem
famously states that the maximum efficiency depends only on the temperature of
the heat baths used by the engine, but not on the specific structure of baths.
Here, we show that when the heat baths are finite in size, and when the engine
operates in the quantum nanoregime, a revision to this statement is required.
We show that one may still achieve the Carnot efficiency, when certain
conditions on the bath structure are satisfied; however if that is not the
case, then the maximum achievable efficiency can reduce to a value which is
strictly less than Carnot. We derive the maximum efficiency for the case when
one of the baths is composed of qubits. Furthermore, we show that the maximum
efficiency is determined by either the standard second law of thermodynamics,
analogously to the macroscopic case, or by the non increase of the max relative
entropy, which is a quantity previously associated with the single shot regime
in many quantum protocols. This relative entropic quantity emerges as a
consequence of additional constraints, called generalized free energies, that
govern thermodynamical transitions in the nanoregime. Our findings imply that
in order to maximize efficiency, further considerations in choosing bath
Hamiltonians should be made, when explicitly constructing quantum heat engines
in the future. This understanding of thermodynamics has implications for
nanoscale engineering aiming to construct small thermal machines.
