Petz-Rényi Relative Entropy of Thermal States and their Displacements

Abstract

In this article, 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 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 \[ D_α(ρ||σ)<\infty \Leftrightarrow α< \min \left\{ \frac{s_j}{s_j-r_j}: j \in \{ 1, \ldots , n \} \text{ such that } r_j<s_j \right\}, \] 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)).

Closer to the published version- better exposition resulting from the peer review process. Added extra text in the `Introduction' section on the applications and relevance of this work

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• The Chernoff lower bound for symmetric quantum hypothesis testing
• Hypothesis testing for Gaussian states on bosonic lattices
• Page curves and typical entanglement in linear optics
• Quantum $f$-divergences via Nussbaum-Szkoła Distributions and Applications to $f$-divergence Inequalities
• Relative Entropy via Distribution of Observables

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