We propose quantum soft covering problems for fully quantum channels and classical-quantum (CQ) channels using relative entropy as a criterion of operator closeness. We prove covering lemmas by deriving one-shot bounds on the rates in terms of smooth min-entropies and smooth max-divergences, respectively. In the asymptotic regime, we show that for quantum channels, the rate infimum defined as the logarithm of the minimum rank of the input state is the coherent information between the reference and output state; for CQ channels, the rate infimum defined as the logarithm of the minimum number of input codewords is the Helovo information between the input and output state. Furthermore, we present a one-shot quantum decoupling theorem with relative entropy criterion. Our results based on the relative-entropy criterion are tighter than the corresponding results based on the trace norm considered in the literature due to the Pinsker inequality.

翻译：我们提出了针对全量子信道和经典-量子（CQ）信道的量子软覆盖问题，并以相对熵作为算子接近度的准则。通过光滑最小熵和光滑最大散度，我们分别推导出单次覆盖率的界限，从而证明了覆盖引理。在渐近情况下，我们证明：对于量子信道，输入态的最小秩的对数定义的率下确界为参考态与输出态之间的相干信息；对于CQ信道，输入码字最小数量的对数定义的率下确界为输入态与输出态之间的霍列沃信息。此外，我们提出了基于相对熵准则的单次量子解耦定理。由于Pinsker不等式，基于相对熵准则的结果比文献中考虑迹范数的相应结果更为紧致。