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 entanglement-assisted zero-error capacity under some restrictions is uncomputable; indicative of the fact that quantum channel capacities may generally be undecidable.

翻译：在理解经典信道与量子信道容量的一个关键区别中，以下问题至关重要：是否存在能计算（甚至高效计算）信道容量的算法？尽管有压倒性证据表明量子信道容量可能是不可计算的，但此类论断的形式化证明始终难以捉摸。我们开创性地研究了计算量子信道容量的难度问题。结果表明，对于一般量子信道，计算其量子容量是QMA-困难的，且在特定约束下，纠缠辅助零错误容量是不可计算的——这暗示量子信道容量在一般情况下可能是不可判定的。