Can quantum correlations be completely quantum?

Deficit of information zero-way was proposed in HorodeckiHHOSSS2005 as one of 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 above measure of quantumness of correlations and the measure of total correlations - mutual information. It 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 2x2 separable states. We also present numerical result suggesting that this interesting situation might also happen for 2x2 entangled states.