I will investigate the capacities of noisy quantum channels through a combined analytical and numerical approach. First, I introduce novel flagged extension techniques that embed a channel into a higher-dimensional space, enabling single-letter upper bounds on quantum and private capacities. My results refine previous bounds and clarify noise thresholds beyond which quantum transmission vanishes. Second, I present a simulation framework that uses coherent information to estimate channel capacities in practice, focusing on two canonical examples: the amplitude damping channel (which we confirm is degradable and thus single-letter) and the depolarizing channel (whose capacity requires multi-letter superadditivity). By parameterizing input qubit states on the Bloch sphere, I numerically pinpoint the maximum coherent information for each channel and validate the flagged extension bounds. Notably, I capture the abrupt transition to zero capacity at high noise and observe superadditivity for moderate noise levels.

翻译：本研究通过结合解析与数值方法，探究噪声量子信道的容量特性。首先，我们提出新颖的标记扩展技术，将信道嵌入更高维空间，从而建立量子容量与私密容量的单字母上界。所得结果改进了现有界值，并明确了量子传输消失的噪声阈值。其次，我们构建基于相干信息的仿真框架，用于实际信道容量估计，重点分析两个典型信道：振幅阻尼信道（经证实具有可降级性，因而存在单字母解）与退极化信道（其容量需通过多字母超可加性求解）。通过在布洛赫球面上参数化输入量子比特态，我们数值确定了各信道的最大相干信息，并验证了标记扩展界。研究特别揭示了高噪声下容量突变为零的现象，并观测到中等噪声水平下的超可加性特征。