Determining whether a noisy quantum channel can be used to reliably transmit quantum information is a challenging problem in quantum information theory. This is because it requires computation of the channel’s coherent information for an unbounded number of copies of the channel. In this paper, we devise an elementary perturbative technique to solve this problem in a wide variety of circumstances. Our analysis reveals that a channel’s ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces. We exploit this link to develop easy tests which can be used to detect positivity of quantum channel capacities simply by comparing the channels’ input, output, and environment dimensions. Several noteworthy examples, such as the depolarizing and transpose-depolarizing channels (including the Werner-Holevo channel), dephasing channels, generalized Pauli channels, multi-level amplitude damping channels, and (conjugate) diagonal unitary covariant channels, serve to aptly exhibit the utility of our method. Notably, in all these examples, the coherent information of a single copy of the channel turns out to be positive.

确定是否可以使用嘈杂的量子信道可靠地传输量子信息是量子信息论中的一个具有挑战性的问题。这是因为它需要为无限数量的通道副本计算通道的相干信息。在本文中，我们设计了一种基本的微扰技术来解决各种情况下的这个问题。我们的分析表明，通道传输信息的能力与其输入、输出和环境空间的相对大小密切相关。我们利用此链接开发简单的测试，可用于通过比较通道的输入、输出和环境维度来检测量子通道容量的积极性。几个值得注意的例子，例如去极化和转置去极化通道（包括 Werner-Holevo 通道）、去相位通道、广义泡利通道、多级振幅阻尼通道和（共轭）对角单一协变通道，用于恰当地展示我们方法的效用. 值得注意的是，在所有这些示例中，通道的单个副本的连贯信息被证明是积极的。