Estimation of quantum relative entropy and its R\'{e}nyi generalizations is a fundamental statistical task in quantum information theory, physics, and beyond. While several estimators of these divergences have been proposed in the literature along with their computational complexities explored, a limit distribution theory which characterizes the asymptotic fluctuations of the estimation error is still premature. As our main contribution, we characterize these asymptotic distributions in terms of Fr\'{e}chet derivatives of elementary operator-valued functions. We achieve this by leveraging an operator version of Taylor's theorem and identifying the regularity conditions needed. As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting asymptotic distribution is a centered normal, with its variance characterized in terms of the Pauli operators and states. We utilize the knowledge of the aforementioned limit distribution to obtain asymptotic performance guarantees for a multi-hypothesis testing problem.

翻译：量子相对熵及其Rényi推广的估计是量子信息论、物理学及其他领域中一项基础性统计任务。尽管文献中已提出多种针对这些散度的估计器并探讨了其计算复杂度，但表征估计误差渐近波动的极限分布理论仍不成熟。作为我们的主要贡献，我们通过初等算子值函数的Fr´echet导数刻画了这些渐近分布。我们利用算子的泰勒定理版本并识别所需的正则条件来实现这一目标。作为结果的应用，我们考虑了一种基于量子态Pauli层析成像的量子相对熵估计器，并证明其渐近分布为中心化正态分布，其方差由Pauli算子和状态表征。我们利用上述极限分布的知识，为多假设检验问题获得了渐近性能保证。