We prove a large deviation principle for the expectation of macroscopic
observables in quantum (and classical) Gibbs states. Our proof is based on
Ruelle-Lanford functions and direct subadditivity arguments, as in the
classical case, instead of relying on G\"artner-Ellis theorem, and cluster
expansion or transfer operators as done in the quantum case. In this approach
we recover, expand, and unify quantum (and classical) large deviation results
for lattice Gibbs states. In the companion paper \cite{OR} we discuss the
characterization of rate functions in terms of relative entropies.Comment: 22 page