In this work, we propose a soft covering problem for fully quantum channels using relative entropy as a criterion for operator closeness. We establish covering lemmas by deriving one-shot bounds on the achievable rates in terms of smooth min-entropies. In the asymptotic regime, we show that the infimum of the rate, defined as the logarithm of the minimum rank of the encoded input state, is given by the minimal coherent information between the reference and output systems that yields the target output state. Furthermore, we present a one-shot quantum decoupling theorem that also employs a relative-entropy criterion. Due to the Pinsker inequality, our one-shot results based on the relative-entropy criterion are tighter than the corresponding results based on the trace norm considered in the literature. In addition, we establish achievable error exponents and second-order rates for quantum soft covering under both trace-distance and relative-entropy criteria.

翻译：本文针对全量子信道提出了一种软覆盖问题，采用相对熵作为算子接近度的衡量准则。通过建立基于光滑最小熵的可达速率单次界，我们推导出覆盖引理。在渐近区域中，我们证明了速率的下确界（定义为编码输入态最小秩的对数）等于产生目标输出态时参考系统与输出系统之间的最小相干信息。此外，我们提出了同样采用相对熵准则的单次量子解耦定理。由于平斯克不等式的作用，基于相对熵准则的单次结果比文献中基于迹范数的相应结果更为紧致。另外，我们分别在迹距离和相对熵准则下建立了量子软覆盖的可达误差指数与二阶速率。