The capacity of a channel can usually be characterized as a maximization of certain entropic quantities. From a practical point of view it is of primary interest to not only compute the capacity value, but also to find the corresponding optimizer, i.e., the capacity-achieving input distribution. This paper addresses the general question of whether or not it is possible to find algorithms that can compute the optimal input distribution depending on the channel. For this purpose, the concept of Turing machines is used which provides the fundamental performance limits of digital computers and therewith fully specifies which tasks are algorithmically feasible in principle. It is shown for discrete memoryless channels that it is impossible to algorithmically compute the capacity-achieving input distribution, where the channel is given as an input to the algorithm (or Turing machine). Finally, it is further shown that it is even impossible to algorithmically approximate these input distributions.

翻译：通常，信道的容量可以表示为某些熵量的最大化。从实际角度来看，不仅计算容量值，而且找到相应的优化器（即容量可达输入分布）具有首要意义。本文探讨了一个普遍问题：是否能找到依赖于信道的算法来计算最优输入分布。为此，我们使用图灵机的概念，它提供了数字计算机的基本性能极限，从而完全规定了哪些任务在原则上可以通过算法实现。研究表明，对于离散无记忆信道，当信道作为算法（或图灵机）的输入给出时，在算法上无法计算容量可达输入分布。最后，进一步表明，即使对这些输入分布进行算法逼近也是不可能的。