  We generalise the two-sided Bogoliubov inequality for classical particles
from [L. Delle Site et al., J.Stat.Mech.Th.Exp. 083201 (2017)] to systems of
quantum particles. As in the classical set-up, the inequality leads to upper
and lower bounds for the free energy difference associated with the
partitioning of a large system into smaller, independent subsystems. From a
thermodynamic modelling point of view, the free energy difference determines
the finite size correction needed to consistently treat a small system as a
representation of a large system. Applications of the bounds to quantify finite
size effects are ubiquitous in physics, chemistry, material science, or
biology, to name just a few; in particular it is relevant for molecular
dynamics simulations in which a small portion of a system is usually taken as
representative of the idealized large system.
