We study the computation of the $α$-Rényi capacity of a classical-quantum (c-q) channel for $α\in(0,1)$. We propose an exponentiated-gradient (mirror descent) iteration that generalizes the Blahut-Arimoto algorithm. Our analysis establishes relative smoothness with respect to the entropy geometry, guaranteeing a global sublinear convergence of the objective values. Furthermore, under a natural tangent-space nondegeneracy condition (and a mild spectral lower bound in one regime), we prove local linear (geometric) convergence in Kullback-Leibler divergence on a truncated probability simplex, with an explicit contraction factor once the local curvature constants are bounded.

翻译：本文研究经典量子信道在$α\in(0,1)$范围内的$α$-Rényi容量计算问题。我们提出一种指数梯度（镜像下降）迭代算法，该算法推广了经典的Blahut-Arimoto算法。通过分析熵几何结构下的相对光滑性，我们证明了目标函数值具有全局次线性收敛性。此外，在自然的切空间非退化条件（及某一参数区间下的温和谱下界假设）下，我们证明了算法在截断概率单纯形上的Kullback-Leibler散度具有局部线性（几何）收敛性，并在局部曲率常数有界时给出了显式收缩因子。