An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the maximal-entanglement-assisted zero-error one-shot classical capacity is uncomputable.

翻译：经典信道与量子信道容量理解的一个重要区别体现在以下问题上：是否存在一种算法能够计算（甚至高效计算）信道容量？尽管有大量证据表明量子信道容量可能是不可计算的，但对此类论断的形式化证明仍难以获得。我们首次系统研究了计算量子信道容量的计算复杂度问题。我们证明：对于一般量子信道，计算其量子容量属于QMA难问题；而最大纠缠辅助的零错误单次经典容量则是不可计算的。