Conditions for monogamy of quantum correlations: Greenberger-Horne-Zeilinger versus $W$ states

Quantum correlations are expected to respect all the conditions required for them to be good measures of quantumness in the bipartite scenario. In a multipartite setting, sharing entanglement between several parties is restricted by the monogamy of entanglement. We take over the concept of monogamy to an information-theoretic quantum correlation measure, and find that it violates monogamy in general. Using the notion of interaction information, we identify necessary and sufficient conditions for the measure to obey monogamy, for arbitrary pure and mixed quantum states. We show that while three-qubit generalized Greenberger-Horne-Zeilinger states follow monogamy, generalized $W$ states do not.

Figure 1

Discord monogamy for generalized $W$. We plot ${\delta }_M$ (in bits) against $(\theta ,\phi )$ (dimensionless). Here, ${\delta }_M\left({\rho }_{ABC}\right)=D\left({\rho }_{AB}\right)+D\left({\rho }_{AC}\right)-D\left({\rho }_{A:BC}\right)$.

Figure 2

Discord monogamy for mixed states. Left: ${\delta }_M$ (in bits) against the mixing parameter $p$ (dimensionless), for different generalized $W$ states. Right: ${\delta }_M$ is plotted against $p$ and $\alpha$ (dimensionless) for all generalized GHZ states.