Quantum generalizations of the Fisher information are important in quantum information science, with applications in high energy and condensed matter physics and in quantum estimation theory, machine learning, and optimization. One can derive a quantum generalization of the Fisher information matrix in a natural way as the Hessian matrix arising in a Taylor expansion of a smooth divergence. Such an approach is appealing for quantum information theorists, given the ubiquity of divergences in quantum information theory. In contrast to the classical case, there is not a unique quantum generalization of the Fisher information matrix, similar to how there is not a unique quantum generalization of the relative entropy or the R\'enyi relative entropy. In this paper, I derive information matrices arising from the log-Euclidean, $\alpha$-$z$, and geometric R\'enyi relative entropies, with the main technical tool for doing so being the method of divided differences for calculating matrix derivatives. Interestingly, for all non-negative values of the R\'enyi parameter $\alpha$, the log-Euclidean R\'enyi relative entropy leads to the Kubo-Mori information matrix, and the geometric R\'enyi relative entropy leads to the right-logarithmic derivative Fisher information matrix. Thus, the resulting information matrices obey the data-processing inequality for all non-negative values of the R\'enyi parameter $\alpha$ even though the original quantities do not. Additionally, I derive and establish basic properties of $\alpha$-$z$ information matrices resulting from the $\alpha$-$z$ R\'enyi relative entropies. For parameterized thermal states and time-evolved states, I establish formulas for their $\alpha$-$z$ information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.

翻译：Fisher信息的量子推广在量子信息科学中至关重要，其在高能物理、凝聚态物理、量子估计理论、机器学习及优化等领域均有应用。通过将光滑散度在泰勒展开中的Hessian矩阵自然导出，可获得Fisher信息矩阵的一种量子推广。鉴于散度在量子信息理论中的普遍性，这一方法对量子信息理论研究者颇具吸引力。与经典情形不同，Fisher信息矩阵的量子推广并非唯一，这类似于相对熵或Rényi相对熵的量子推广也非唯一。本文从对数欧几里得、α-z及几何Rényi相对熵出发推导了相应的信息矩阵，主要技术工具是用于计算矩阵导数的均差方法。有趣的是，对于所有非负的Rényi参数α值，对数欧几里得Rényi相对熵导出Kubo-Mori信息矩阵，而几何Rényi相对熵导出右对数导数Fisher信息矩阵。因此，所得信息矩阵对所有非负的Rényi参数α值均满足数据处理不等式，尽管原始量本身并不满足。此外，本文推导并建立了由α-z Rényi相对熵导出的α-z信息矩阵的基本性质。针对参数化热态与时间演化态，本文建立了其α-z信息矩阵的计算公式，并提出了用于估计这些矩阵的混合量子-经典算法，相关成果可应用于量子玻尔兹曼机器学习。