Entanglement and Complementarity

We show that states that have more correlations among complementary observables must be entangled. The reverse is false: general entangled states do not have more correlations on complementary observables than separable ones. We either prove or conjecture that this is true for different measures of correlation: the mutual information, the sum of conditional probabilities and the Pearson correlation coefficient. We also show that states with nonzero discord typically have less correlation than classically correlated states.