Relations between different quantum Rényi divergences

Abstract: Quantum generalizations of R\'enyi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum R\'enyi divergence $\bar{D}{\alpha}$ and the minimal quantum R\'enyi divergence $\widetilde{D}{\alpha}$. Moreover, the maximum quantum R\'enyi divergence $\widehat{D}{\alpha}$ is of particular mathematical interest. In this Master thesis, we investigate relations between these divergences and their applications in quantum information theory. Our main result is a reverse Araki-Lieb-Thirring inequality that implies a new relation between the minimal and the Petz divergence, namely that $\alpha \bar{D}{\alpha}(\rho | \sigma) \leqslant \widetilde{D}{\alpha}(\rho | \sigma)$ for $\alpha \in [0,1]$ and where $\rho$ and $\sigma$ are density operators. This bound suggests defining a "pretty good fidelity", whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction. In addition, we provide a new proof of the inequality $\widetilde{D}{1}(\rho | \sigma) \leqslant \widehat{D}_{1}(\rho | \sigma)\, ,$ based on the Araki-Lieb-Thirring inequality. This leads to an elegant proof of the logarithmic form of the reverse Golden-Thompson inequality.