Petz-Rényi Relative Entropy of Thermal States and their Displacements

In this article, we obtain the precise range of the values of the parameter $\alpha$ such that Petz-Rényi $\alpha$-relative entropy $D_{\alpha }(\rho \mathrm{\mid }\mathrm{\mid }\sigma )$ 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 $$D_{\alpha }(\rho \mathrm{\mid }\mathrm{\mid }\sigma )<\mathrm{\infty }\iff \alpha <\min \{\frac{s_j}{s_j-r_j}:j\in \{1,\dots ,n\}\text{ such that }{r}_j<s_j\},$$ 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)).