Relative Entropy via Distribution of Observables

Abstract: We obtain formulas for Petz-R\'enyi and Umegaki relative entropy from the idea of distribution of a positive selfadjoint operator. Classical results on R\'enyi and Kullback-Leibler divergences are applied to obtain new results and new proofs for some known results about Petz-R\'enyi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz-R\'enyi $\alpha$-relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.