We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a one-shot quantum covering lemma in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory. This covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion [Atif, Sohail, Pradhan, arXiv:2302.00625]. Both one-shot results directly yield corollaries about the i.i.d. asymptotics, in terms of the coherent information of the channel. The power of our quantum covering lemma is demonstrated by two additional applications: first, we formulate a quantum channel resolvability problem, and provide one-shot as well as asymptotic upper and lower bounds. Secondly, we provide new upper bounds on the unrestricted and simultaneous identification capacities of quantum channels, in particular separating for the first time the simultaneous identification capacity from the unrestricted one, proving a long-standing conjecture of the last author.

翻译：我们针对给定的一般量子信道及其一个输出状态，提出了量子软覆盖问题——即寻找能逼近该信道输出所需输入状态的最小秩。随后，通过利用量子香农理论中的解耦技术，我们以平滑最小熵形式证明了一个一次性量子覆盖引理。该覆盖结果被证明等价于后验（逆向）信道失真准则下的率失真编码定理[Atif, Sohail, Pradhan, arXiv:2302.00625]。上述两个一次性结果直接导出了关于独立同分布渐近性的推论，其形式以信道的相干信息表示。我们的量子覆盖引理通过两项额外应用展示了其效力：首先，我们构建了量子信道可解析性问题，并给出了一次性及渐近上下界；其次，我们给出了量子信道无限制识别容量与同步识别容量的新上界，首次区分了同步识别容量与无限制识别容量，证实了最后一位作者长期以来的猜想。