Petz-Rényi Relative Entropy of Thermal States and their Displacements

• URL: http://arxiv.org/abs/2303.03380v2
• Date: Wed, 17 Apr 2024 17:15:44 GMT
• Title: Petz-Rényi Relative Entropy of Thermal States and their Displacements
• Authors: George Androulakis, Tiju Cherian John,
• Abstract summary: Petz-R'enyi $alpha$-relative entropy $D_alpha(rho||sigma)$ of two displaced thermal states is finite. We adopt the convention that the minimum of an empty set is equal infinity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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)).

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