  We investigated the super quantum discord based on weak measurements. The
super quantum discord is an extension of the standard quantum discord defined
by projective measurements and also describes the quantumness of correlations.
We provide some equivalent conditions for zero super quantum discord by using
quantum discord, classical correlation and mutual information. In particular,
we find that the super quantum discord is zero only for product states, which
have zero mutual information. This result suggests that non-zero correlations
can always be detected using the quantum correlation with weak measurements. As
an example, we present the assisted state-discrimination method.
