Letters in Mathematical Physics ( IF 1.2 ) Pub Date : 2024-04-17 , DOI: 10.1007/s11005-024-01805-z George Androulakis , Tiju Cherian John
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).
中文翻译：
Petz-Rényi 热态相对熵及其位移
在这封信中，我们获得了参数\(\alpha \)值的精确范围，使得 Petz–Rényi \(\alpha \) -相对熵\(D_{\alpha }(\rho ||\sigma ) \)两个忠实的位移热状态是有限的。更准确地说，我们证明，给定两个位移热状态\(\rho \)和\(\sigma \)以及逆温度参数\(r_1, r_2,\ldots , r_n\)和\(s_1,s_2, \ldots , s_n\)分别为\(0<r_j,s_j<\infty \)，对于所有j，我们有
$$\begin{对齐} D_{\alpha }(\rho ||\sigma )<\infty \Leftrightarrow \alpha< \min \left\{ \frac{s_j}{s_j-r_j}: j \in \{ 1, \ldots , n \} \text { 使得 } r_j<s_j \right\} , \end{对齐}$$我们采用空集的最小值等于无穷大的约定。考虑到\(\alpha >1 \)体系中Petz–Rényi \(\alpha \)相对熵的操作解释，该结果特别有用。在此过程中，我们还证明了 Seshadreesan 等人猜想的一个特例。 （《数学物理杂志》59（7）：072204，2018。https://doi.org/10.1063/1.5007167）。
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