  The problem of converting noisy quantum correlations between two parties into
noiseless classical ones using a limited amount of one-way classical
communication is addressed. A single-letter formula for the optimal trade-off
between the extracted common randomness and classical communication rate is
obtained for the special case of classical-quantum correlations. The resulting
curve is intimately related to the quantum compression with classical side
information trade-off curve $Q^*(R)$ of Hayden, Jozsa and Winter. For a general
initial state we obtain a similar result, with a single-letter formula, when we
impose a tensor product restriction on the measurements performed by the
sender; without this restriction the trade-off is given by the regularization
of this function. Of particular interest is a quantity we call ``distillable
common randomness'' of a state: the maximum overhead of the common randomness
over the one-way classical communication if the latter is unbounded. It is an
operational measure of (total) correlation in a quantum state. For
classical-quantum correlations it is given by the Holevo mutual information of
its associated ensemble, for pure states it is the entropy of entanglement. In
general, it is given by an optimization problem over measurements and
regularization; for the case of separable states we show that this can be
single-letterized.
