We generalise the two-sided Bogoliubov inequality for classical particles (Delle Site et al. in J Stat Mech Theory 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.
