  Interactions of quantum systems with their environment play a crucial role in
resource-theoretic approaches to thermodynamics in the microscopic regime.
Here, we analyze the possible state transitions in the presence of "small" heat
baths of bounded dimension and energy. We show that for operations on quantum
systems with fully degenerate Hamiltonian (noisy operations), all possible
state transitions can be realized exactly with a bath that is of the same size
as the system or smaller, which proves a quantum version of Horn's lemma as
conjectured by Bengtsson and Zyczkowski. On the other hand, if the system's
Hamiltonian is not fully degenerate (thermal operations), we show that some
possible transitions can only be performed with a heat bath that is unbounded
in size and energy, which is an instance of the third law of thermodynamics. In
both cases, we prove that quantum operations yield an advantage over classical
ones for any given finite heat bath, by allowing a larger and more physically
realistic set of state transitions.
