Can quantum correlations be completely quantum?

Deficit of information zero-way was proposed in Horodecki (2005 Phys. Rev. A 71 062307) as one of the possible measures of quantumness of correlations. Numerical calculations suggested that there exist such states for which this quantity is almost equal to mutual information. In this paper, we present a family of states for which we have equality between the above measure of quantumness of correlations and the measure of total correlations-mutual information. This means that whole correlations in these states have, in some sense, quantum character and that quantum correlations do not necessarily imply classical correlations. We prove this intriguing feature for a subclass of 2 x 2 separable states. We also present numerical results suggesting that this interesting situation might also happen for 2 x 2 entangled states