  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.
