Non-zero total correlation means non-zero quantum correlation

• We use the weak measurement to characterize quantum correlation.
• We provide a necessary and sufficient condition for vanishing super discord.
• We find there are some quantum correlation even in classical state.
• The application in state discrimination is discussed.

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.