Weak-measurement induced quantum discord or super quantum discord (SQD) is a generalization of the normal quantum discord and is defined as the difference between quantum mutual information and classical correlation obtained by weak measurements in a given quantum system. This correlation is an information-theoretic measure and is, in general, different from entanglement-separability measures such as entanglement. Super quantum discord may be nonzero even for certain separable states. So far, SQD has been calculated explicitly only for a limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for SQD for X states, a seven real-parameter family of two-qubit states and investigate its monogamy properties. The monogamy behaviour of SQD depends on the measurement strength. The formalism can be easily extended to N-qubit X states.
We investigate the general monogamy and polygamy relations satisfied by quantum correlation measures. We show that there exist two real numbers $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ such that for any quantum correlation measure
Q,Qx is monogamous if $x\ensuremath{\ge}\ensuremath{\alpha}$ and polygamous if $0\ensuremath{\le}x\ensuremath{\le}\ensuremath{\beta}$ for a given multipartite state $\ensuremath{\rho}$. For $\ensuremath{\beta}<x<\ensuremath{\alpha}$, we show that the monogamy relation can be superactivated by finite
m copies ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}m}$ of $\ensuremath{\rho}$ for nonadditive correlation measures. As a detailed example, we use the negativity as the quantum correlation measure to illustrate such superactivation of monogamy properties. A tighter monogamy relation is presented at last.