The problem of converting noisy quantum correlations between two parties intonoiseless classical ones using a limited amount of one-way classicalcommunication is addressed. A single-letter formula for the optimal trade-offbetween the extracted common randomness and classical communication rate isobtained for the special case of classical-quantum correlations. The resultingcurve is intimately related to the quantum compression with classical sideinformation trade-off curve $Q^*(R)$ of Hayden, Jozsa and Winter. For a generalinitial state we obtain a similar result, with a single-letter formula, when weimpose a tensor product restriction on the measurements performed by thesender; without this restriction the trade-off is given by the regularizationof this function. Of particular interest is a quantity we call ``distillablecommon randomness'' of a state: the maximum overhead of the common randomnessover the one-way classical communication if the latter is unbounded. It is anoperational measure of (total) correlation in a quantum state. Forclassical-quantum correlations it is given by the Holevo mutual information ofits associated ensemble, for pure states it is the entropy of entanglement. Ingeneral, it is given by an optimization problem over measurements andregularization; for the case of separable states we show that this can besingle-letterized.
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