不同量子Rényi发散之间的关系
Relations between different quantum Rényi divergences
摘要：
Rényi熵的量子推广是描述量子信息处理中各种操作任务的有用工具。有两类这样的推广结果特别有用：佩茨量子雷尼发散$\bar{D}_α$和最小量子雷尼发散$\widetilde{D}_α$。此外，最大量子雷尼发散$\widehat{D}_α$在数学上特别有趣。在这篇硕士论文中，我们将探讨这些分歧之间的关系及其在量子信息理论中的应用。我们的主要结果是一个反向的Araki-Lieb-Thirring不等式，它暗示了极小值和Petz散度之间的一个新的关系，即$α\bar{D}_α(ρ\| σ) \leqslant \widetilde{D}_α(ρ\| σ)$对于$α\in [0,1]$，其中$ρ$和$σ$是密度算子.这个界限建议定义一个“相当好的保真度”，其与通常保真度的关系意味着最佳和相当好的测量之间的已知关系，以及最佳和相当好的单重态分数。此外，基于Araki-Lieb-Thirring不等式，给出了不等式$\widetilde{D}_{1}(ρ\| σ) \leqslant \widehat{D}_{1}(ρ\| σ)\, ,$的一个新的证明.这导致了一个优雅的证明对数形式的反向戈登-汤普森不等式。
英文摘要：
Quantum generalizations of Rényi'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 Rényi divergence $\bar{D}_α$ and the minimal quantum Rényi divergence $\widetilde{D}_α$. Moreover, the maximum quantum Rényi divergence $\widehat{D}_α$ 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 $α\bar{D}_α(ρ\| σ) \leqslant \widetilde{D}_α(ρ\| σ)$ for $α\in [0,1]$ and where $ρ$ and $σ$ 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 $\widetilde{D}_{1}(ρ\| σ) \leqslant \widehat{D}_{1}(ρ\| σ)\, ,$ based on the Araki-Lieb-Thirring inequality. This leads to an elegant proof of the logarithmic form of the reverse Golden-Thompson inequality.
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