  Optimal (reversible) processes in thermodynamics can be modelled as
step-by-step processes, where the system is successively thermalized with
respect to different Hamiltonians by an external thermal bath. However, in
practice interactions between system and thermal bath will take finite time,
and precise control of their interaction is usually out of reach. Motivated by
this observation, we consider finite-time and uncontrolled operations between
system and bath, which result in thermalizations that are only partial in each
step. We show that optimal processes can still be achieved for any non-trivial
partial thermalizations at the price of increasing the number of operations,
and characterise the corresponding tradeoff. We focus on work extraction
protocols and show our results in two different frameworks: A collision model
and a model where the Hamiltonian of the working system is controlled over time
and the system can be brought into contact with a heat bath. Our results show
that optimal processes are robust to noise and imperfections in small quantum
systems, and can be achieved by a large set of interactions between system and
bath.
