论文标题
不同的量子差异之间的关系
Relations between different quantum Rényi divergences
论文作者
论文摘要
Rényi熵的量子概括是描述量子信息处理中各种操作任务的有用工具。这种概括的两个家族事实特别有用：petz量子rényi发散$ \ bar {d}_α$和最小量子rényidivergence $ \ widetilde {d}_α$。此外，最大的量子rényi差异$ \ widehat {d}_α$特别是数学上的兴趣。在本硕士论文中，我们研究了这些差异及其在量子信息理论中的应用之间的关系。我们的主要结果是反向Araki-lieb-thir-thir的不平等，暗示着最小和彼得斯之间存在新的关系，即$α\ bar {d}_α（ρ\ | c）\ leqslant \ leqslant \ wideTilde {d} _ $ $ p $ p $ coun [ $σ$是密度运算符。这一界限暗示了定义“相当不错的忠诚”，其与通常的保真度的关系意味着最佳和相当良好的测量以及最佳和相当不错的单线分数之间的已知关系。此外，我们还提供了不等式的新证明$ \ widetilde {d} _ {1}（ρ\ |σ）\ leqslant \ wideHat {d} _ {1} _ {1}（ρ\ |σ）\，，，基于Araki-lieb-lieb-thir-thir-thir-thir-thir-equequality。这导致了反向金汤姆森不平等的对数形式的优雅证明。
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.
