  Thermodynamics at the nanoscale is known to differ significantly from its
familiar macroscopic counterpart: the possibility of state transitions is not
determined by free energy alone, but by an infinite family of free-energy-like
quantities; strong fluctuations (possibly of quantum origin) allow to extract
less work reliably than what is expected from computing the free energy
difference. However, these known results rely crucially on the assumption that
the thermal machine is not only exactly preserved in every cycle, but also kept
uncorrelated from the quantum systems on which it acts. Here we lift this
restriction: we allow the machine to become correlated with the microscopic
systems on which it acts, while still exactly preserving its own state.
Surprisingly, we show that this restores the second law in its original form:
free energy alone determines the possible state transitions, and the
corresponding amount of work can be invested or extracted from single systems
exactly and without any fluctuations. At the same time, the work reservoir
remains uncorrelated from all other systems and parts of the machine. Thus,
microscopic machines can increase their efficiency via clever "correlation
engineering" in a perfectly cyclic manner, which is achieved by a catalytic
system that can sometimes be as small as a single qubit (though some setups
require very large catalysts). Our results also solve some open mathematical
problems on majorization which may lead to further applications in entanglement
theory.
