We propose an iterative method for approximating the capacity of
classical-quantum channels with a discrete input alphabet and a finite
dimensional output, possibly under additional constraints on the input
distribution. Based on duality of convex programming, we derive explicit upper
and lower bounds for the capacity. To provide an $\varepsilon$-close estimate
to the capacity, the presented algorithm requires $O(\tfrac{(N \vee M) M^3
\log(N)^{1/2}}{\varepsilon})$, where $N$ denotes the input alphabet size and
$M$ the output dimension. We then generalize the method for the task of
approximating the capacity of classical-quantum channels with a bounded
continuous input alphabet and a finite dimensional output. For channels with a
finite dimensional quantum mechanical input and output, the idea of a universal
encoder allows us to approximate the Holevo capacity using the same method. In
particular, we show that the problem of approximating the Holevo capacity can
be reduced to a multidimensional integration problem. For families of quantum
channels fulfilling a certain assumption we show that the complexity to derive
an $\varepsilon$-close solution to the Holevo capacity is subexponential or
even polynomial in the problem size. We provide several examples to illustrate
the performance of the approximation scheme in practice.

本文提出基于凸规划对离散输入字母和有限维输出的经典量子通道容量的迭代估算方法，得到 Holevo 容量的近似值，并且可将其扩展到有界连续输入和有限维输出的通道。