The long term behaviour of a quantum channel under iterations (i.e. under repeated applications of itself) yields a plethora of interesting properties. These include ergodicity, mixing, eventual scrambling, becoming strictly positive, and the vanishing of its one-shot zero error capacities. We derive relations between these seemingly different properties and find novel bounds on indices which quantify the minimum number of iterations needed for the onset of some of these properties. We obtain a lower bound on the one-shot zero-error classical capacity of $n$ iterations of an ergodic channel (for any positive integer $n$) in terms of the cardinality of its peripheral spectrum. We also find upper bounds on the minimum number of iterations needed for the one-shot capacities of any channel to stabilize. We consider two classes of quantum channels, satisfying certain symmetries, for which upper bounds on the above indices are optimal, since they reduce to the corresponding indices for a stochastic matrix (for which the bounds are known to be optimal). As an auxiliary result, we obtain a trade-off relation between the one-shot zero error classical and quantum capacities of a quantum channel.

翻译：在迭代（即重复应用自身）作用下，量子信道的长期行为展现出大量有趣性质，包括遍历性、混合性、最终混洗、严格正性以及单次零错误容量的消失。我们推导出这些看似不同性质之间的关系，并得到了量化某些性质出现所需最少迭代次数的指标的新界限。对于遍历信道的$n$次迭代（$n$为任意正整数），我们利用其外围谱的基数给出了单次零错误经典容量的下界，同时找到了任何信道单次容量稳定所需最少迭代次数的上界。我们考虑了两类满足特定对称性的量子信道，对于它们而言，上述指标的上界是最优的，因为这些界限可简化为随机矩阵的相应指标（已知该情形下的界限最优）。作为辅助结果，我们获得了量子信道单次零错误经典容量与量子容量之间的权衡关系。