Relations between different quantum R\'enyi divergences

TL;DR

This paper explores relationships between different quantum R\'enyi divergences, establishing new inequalities and bounds that deepen understanding of their mathematical properties and operational implications in quantum information theory.

Contribution

It introduces a reverse Araki-Lieb-Thirring inequality linking minimal and Petz divergences, and provides new proofs of key inequalities in quantum R\'enyi divergence theory.

Findings

Established a bound \(\alpha \bar{D}_{\alpha} \leq \widetilde{D}_{\alpha}\) for \(\alpha \in [0,1]\)

Proposed a 'pretty good fidelity' concept related to these divergences

Provided new proofs of known inequalities using the Araki-Lieb-Thirring inequality

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.