Operator convex functions defined on the positive half-line play a prominent role in the theory of quantum information, where they are used to define quantum $f$-divergences. Such functions admit integral representations in terms of rational functions. Obtaining high-quality rational approximants of operator convex functions is particularly useful for solving optimization problems involving quantum $f$-divergences using semidefinite programming. In this paper we study the quality of rational approximations of operator convex (and operator monotone) functions. Our main theoretical results are precise global bounds on the error of local Pad\'e-like approximants, as well as minimax approximants, with respect to different weight functions. While the error of Pad\'e-like approximants depends inverse polynomially on the degree of the approximant, the error of minimax approximants has root exponential dependence and we give detailed estimates of the exponents in both cases. We also explain how minimax approximants can be obtained in practice using the differential correction algorithm.

翻译：定义在半正半轴上的算子凸函数在量子信息理论中扮演着重要角色，用于定义量子$f$-散度。这类函数可借助有理函数表示为积分形式。获得算子凸函数的高质量有理逼近对解决涉及量子$f$-散度的半定规划优化问题尤为实用。本文研究了算子凸（及算子单调）函数的有理逼近质量。主要理论成果包括：针对局部类Padé逼近和极小化极大逼近，给出了在不同权重函数下的精确全局误差界。类Padé逼近的误差与逼近程度呈逆多项式关系，而极小化极大逼近的误差则具有指数根依赖关系，我们为两种情形提供了详细的指数估计，并阐述了如何通过微分校正算法在实际中获得极小化极大逼近。