Quantum Fisher information matrices from Rényi relative entropies

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, I establish formulas for their $\alpha$-$z$ information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.