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–Rényi \(\alpha \)-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 \(r_1, r_2,\ldots , r_n\) and \(s_1,s_2, \ldots , s_n\), respectively, \(0<r_j,s_j<\infty \), for all j, we have

$$\begin{aligned} D_{\alpha }(\rho ||\sigma )<\infty \Leftrightarrow \alpha< \min \left\{ \frac{s_j}{s_j-r_j}: j \in \{ 1, \ldots , n \} \text { such that } r_j<s_j \right\} , \end{aligned}$$

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–Rényi \(\alpha \)-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).