Petz–Rényi relative entropy of thermal states and their displacements

IF 1.3 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
George Androulakis, Tiju Cherian John
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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).

热态及其位移的 Petz-Rényi 相对熵
在这封信中,我们得到了参数 \(\alpha \)的精确取值范围,使得两个忠实的位移热态的 Petz-Rényi \(\alpha \)-相对熵 \(D_{\alpha }(\rho ||\sigma )\) 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),它们分别具有反温度参数(r_1, r_2,\ldots , r_n\)和(s_1,s_2, \ldots , s_n\),对于所有的 j,我们有 $$\begin{aligned} (0<r_j,s_j<\infty \)。D_{alpha }(|||sigma )<\infty\Leftrightarrow \alpha< \min \left\{ \frac{s_j}{s_j-r_j}: j 在 \{ 1, \ldots , n \} 中\text { such that } r_j<s_j\right\}, end{aligned}$$其中我们采用了空集的最小值等于无穷大的约定。这个结果对于佩兹-雷尼(Petz-Rényi)熵((\alpha\)-relative entropy)在制度\(\alpha >1 \)中的运算解释特别有用。同时,我们还证明了 Seshadreesan 等人猜想的一个特例(J Math Phys 59(7):072204, 2018. https://doi.org/10.1063/1.5007167)。
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来源期刊
Letters in Mathematical Physics
Letters in Mathematical Physics 物理-物理:数学物理
CiteScore
2.40
自引率
8.30%
发文量
111
审稿时长
3 months
期刊介绍: The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.
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