Quantum relative entropy programs are convex optimization problems which minimize a linear functional over an affine section of the epigraph of the quantum relative entropy function. Recently, the self-concordance of a natural barrier function was proved for this set. This has opened up the opportunity to use interior-point methods for nonsymmetric cone programs to solve these optimization problems. In this paper, we show how common structures arising from applications in quantum information theory can be exploited to improve the efficiency of solving quantum relative entropy programs using interior-point methods. First, we show that the natural barrier function for the epigraph of the quantum relative entropy composed with positive linear operators is optimally self-concordant, even when these linear operators map to singular matrices. Compared to modelling problems using the full quantum relative entropy cone, this allows us to remove redundant log determinant expressions from the barrier function and reduce the overall barrier parameter. Second, we show how certain slices of the quantum relative entropy cone exhibit useful properties which should be exploited whenever possible to perform certain key steps of interior-point methods more efficiently. We demonstrate how these methods can be applied to applications in quantum information theory, including quantifying quantum key rates, quantum rate-distortion functions, quantum channel capacities, and the ground state energy of Hamiltonians. Our numerical results show that these techniques improve computation times by up to several orders of magnitude, and allow previously intractable problems to be solved.

翻译：量子相对熵程序是一类凸优化问题，其目标是在量子相对熵函数图上集的仿射截面上最小化线性泛函。最近，该集合上一个自然障碍函数的自协调性得到了证明。这为使用非对称锥规划的内点法求解此类优化问题开辟了可能性。本文展示了如何利用量子信息论应用中常见的结构来提升使用内点法求解量子相对熵程序的效率。首先，我们证明了由正线性算子复合的量子相对熵图上集的自然障碍函数具有最优自协调性，即使这些线性算子映射到奇异矩阵时亦然。与使用完整量子相对熵锥建模问题相比，这使我们能够从障碍函数中移除冗余的对数行列式项，并降低整体障碍参数。其次，我们展示了量子相对熵锥的某些截面如何展现出有益特性，应尽可能利用这些特性来更高效地执行内点法的关键步骤。我们演示了这些方法如何应用于量子信息论中的具体问题，包括量子密钥率量化、量子率失真函数、量子信道容量以及哈密顿量的基态能量计算。数值结果表明，这些技术可将计算时间提升数个数量级，并使先前难以处理的问题得以求解。