Petz–Rényi relative entropy of thermal states and their displacements

Abstract

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).