Petz-R\'enyi Relative Entropy of Thermal States and their Displacements

TL;DR

This paper determines the exact conditions on the parameter alpha for which the Petz-Rényi alpha-relative entropy between two displaced thermal states remains finite, and proves a special case of a related conjecture.

Contribution

It precisely characterizes the finiteness range of Petz-Rényi relative entropy for displaced thermal states and proves a special case of a conjecture in quantum information theory.

Findings

Derived the exact alpha range for finiteness of D_alpha between displaced thermal states.

Proved a special case of a conjecture by Seshdreesan, Lami, and Wilde.

Established conditions based on inverse temperature parameters for finiteness.

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)).