Quantum channel capacity is a fundamental quantity in order to understand how good can quantum information be transmitted or corrected when subjected to noise. However, it is generally not known how to compute such quantities, since the quantum channel coherent information is not additive for all channels, implying that it must be maximized over an unbounded number of channel uses. This leads to the phenomenon known as superadditivity, which refers to the fact that the regularized coherent information of $n$ channel uses exceeds one-shot coherent information. In this article, we study how the gain in quantum capacity of qudit depolarizing channels relates to the dimension of the systems considered. We make use of an argument based on the no-cloning bound in order to proof that the possible superadditive effects decrease as a function of the dimension for such family of channels. In addition, we prove that the capacity of the qudit depolarizing channel coincides with the coherent information when $d\rightarrow\infty$. We also discuss the private classical capacity and obain similar results. We conclude that when high dimensional qudits experiencing depolarizing noise are considered, the coherent information of the channel is not only an achievable rate but essentially the maximum possible rate for any quantum block code.

翻译：量子信道容量是衡量量子信息在噪声环境中传输或纠错能力的基本量。然而，由于量子信道的相干信息并非对所有信道均具有可加性，导致其需在无限制的信道使用次数上最大化，这使得该容量的计算通常难以实现。这一现象被称为超可加性，即$n$次信道使用的正则化相干信息超过单次使用的相干信息。本文研究了qudit退极化信道量子容量增益与系统维度的关系。我们基于不可克隆界限的论证，证明了对于此类信道族，可能的超可加性效应随维度增加而递减。此外，我们证明了当$d\rightarrow\infty$时，qudit退极化信道的容量与相干信息相等。我们还讨论了私有经典容量并得到类似结论。最终得出结论：在高维qudit经历退极化噪声时，该信道的相干信息不仅是一个可达速率，而且本质上也是任何量子分组码的最大可能速率。