Petz-Rényi relative entropy of thermal states and their displacements

In this letter, we obtain the precise range of the values of the parameter alpha such that Petz-R & eacute;nyi alpha-relative entropy D alpha(rho||sigma)of two faithful displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states rho and sigma with inverse temperature parameters r1,r2,...,rnands1,s2,...,sn, respectively,0<rj,sj<infinity, for allj,we have D alpha(rho||sigma) <infinity double left right arrow alpha<min{sjsj-rj: j is an element of{1,...,n}such that rj<sj}, where we adopt the convention that the minimum of an empty set is equal to infinity. This result is particularly useful in the light of operational interpretations of the Petz-R & eacute;nyi alpha-relative entropy in the regime alpha>1. Along the way, we also prove a special case of a conjecture of Seshadreesan et al. (J Math Phys 59(7):072204, 2018.https://doi.org/10.1063/1.5007167).