  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.
