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 by two of the present authors. 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态，提出量子软覆盖问题，其核心在于寻找逼近该信道输出所需输入态的最小秩。随后，通过利用量子香农理论中的退耦合技术，我们以光滑最小熵的形式证明了一次性量子覆盖引理。该覆盖结果被证明等价于两位作者先前提出的后验（反向）信道失真准则下的率失真编码定理。这两个一次性结果直接推导出关于信道相干信息的独立同分布渐近推论。量子覆盖引理的效力通过另外两个应用得到展示：首先，我们构建量子信道可分辨性问题，并给出一次性及渐近上界与下界；其次，我们提出量子信道无限制识别容量与同时识别容量的新上界，特别是首次将同时识别容量与无限制识别容量区分开来，从而证明了最后一位作者的一个长期猜想。