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：针对一般量子信道及其一个输出态，本文提出量子软覆盖问题：寻找逼近给定信道输出所需的输入态的最小秩。通过利用量子香农理论中的解耦技术，我们以平滑最小熵的形式证明了单次量子覆盖引理。该覆盖结果被证明等价于后验（反向）信道失真准则下的率失真编码定理[Atif, Sohail, Pradhan, arXiv:2302.00625]。这两个单次结果直接推导出关于独立同分布渐近性的推论，其形式为信道的相干信息。我们的量子覆盖引理通过两个额外应用展示其效力：首先，我们构建了量子信道可解性问题，并给出了单次以及渐近的上下界；其次，我们提供了量子信道无限制和同时识别容量的新上界，尤其首次将同时识别容量与无限制识别容量分离，证明了最后一位作者长期存在的猜想。