Petz-Rényi Relative Entropy of Thermal States and their Displacements

Abstract: In this article, we obtain the precise range of the values of the parameter $\alpha$ such that Petz-R\'enyi $\alpha$-relative entropy $D_{\alpha}(\rho||\sigma)$ of two displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states $\rho$ and $\sigma$ with inverse temperature parameters $r_1, r_2,\dots, r_n$ and $s_1,s_2, \dots, s_n$, respectively, we have [ 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}, ] where we adopt the convention that the minimum of an empty set is equal to infinity. Along the way, we prove a special case of a conjecture of Seshdreesan, Lami and Wilde (J. Math. Phys. 59, 072204 (2018)).