Quantum information quantities play a substantial role in characterizing operational quantities in various quantum information-theoretic problems. We consider numerical computation of four quantum information quantities: Petz-Augustin information, sandwiched Augustin information, conditional sandwiched Renyi entropy and sandwiched Renyi information. To compute these quantities requires minimizing some order-$\alpha$ quantum Renyi divergences over the set of quantum states. Whereas the optimization problems are obviously convex, they violate standard bounded gradient/Hessian conditions in literature, so existing convex optimization methods and their convergence guarantees do not directly apply. In this paper, we propose a new class of convex optimization methods called mirror descent with the Polyak step size. We prove their convergence under a weak condition, showing that they provably converge for minimizing quantum Renyi divergences. Numerical experiment results show that entropic mirror descent with the Polyak step size converges fast in minimizing quantum Renyi divergences.

中文翻译：

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通过具有 Polyak 步长的镜像下降最小化量子 Renyi 散度

量子信息量在表征各种量子信息理论问题中的操作量方面发挥着重要作用。我们考虑四种量子信息量的数值计算：Petz-Augustin 信息、夹心奥古斯丁信息、条件夹心仁义熵和夹心仁义信息。要计算这些数量，需要最小化量子态集合上的一些阶数-$\alpha$ 量子 Renyi 散度。而优化问题显然是凸的，它们违反了文献中的标准有界梯度/Hessian 条件，因此现有的凸优化方法及其收敛保证不能直接适用。在本文中，我们提出了一类新的凸优化方法，称为具有 Polyak 步长的镜像下降。我们证明了它们在弱条件下的收敛性，表明它们可证明收敛以最小化量子 Renyi 散度。数值实验结果表明，具有 Polyak 步长的熵镜下降在最小化量子 Renyi 散度方面收敛速度很快。