Convexity inequalities for estimating free energy and relative entropy

For quantum systems, it is shown that the relative entropy S(P,Q)=-Tr P log P+Tr P log Q of two positive semi-definite operators P and Q satisfies gamma ^-1 Tr(P-P^{1+ gamma} Q^{- gamma} )<or=S(P,Q)<or= gamma ^-1 Tr(P^{1- gamma} Q^gamma -P) for 0( gamma <or=1, and that these bounds become exact in the limit gamma to Phi . Analogous inequalities hold, for states in classical statistical mechanics or information theory, with trace replaced by integration or summation. Furthermore, the average of these bounds is, in general a better approximation to S(P,Q) than either bound alone, and the average is amenable to further improvement via repeated Richardson extrapolation. IF P and Q are Gibbs equilibrium states, then these inequalities can also be used to obtain bounds on the free energy of a perturbed system in terms of the free energy of the unperturbed system and expectations of the perturbation.