  Given a classical channel---a stochastic map from inputs to outputs---the
input can often be transformed to an intermediate variable that is
informationally smaller than the input. The new channel accurately simulates
the original but at a smaller transmission rate. Here, we examine this
procedure when the intermediate variable is a quantum state. We determine when
and how well quantum simulations of classical channels may improve upon the
minimal rates of classical simulation. This inverts Holevo's original question
of quantifying the capacity of quantum channels with classical resources. We
also show that this problem is equivalent to another, involving the local
generation of a distribution from common entanglement.
