Distribution of Bell-inequality violation versus multiparty-quantum-correlation measures

Violation of a Bell inequality guarantees the existence of quantum correlations in a shared quantum state. A pure bipartite quantum state, having nonvanishing quantum correlation, always violates a Bell inequality. Such correspondence is absent for multipartite pure quantum states in the case of multipartite correlation function Bell inequalities with two settings at each site. We establish a connection between the monogamy of Bell-inequality violation and multiparty quantum correlations for shared multisite quantum states. We believe that the relation is generic, as it is true for a number of different multisite measures that are defined from radically different perspectives. Precisely, we quantify the multisite-quantum-correlation content in the states by generalized geometric measure, a genuine multisite entanglement measure, as well as three monogamy-based multiparty-quantum-correlation measures, viz., 3-tangle, quantum-discord score, and quantum-work-deficit score. We find that generalized Greenberger-Horne-Zeilinger states and another single-parameter family of states, which we refer to as the special Greenberger-Horne-Zeilinger states, have the status of extremal states in such relations.

Figure 1

Plot of Bell-inequality-violation monogamy score (${\delta }_{B^V}$) vs GGM ($\mathcal{E}$) for three-qubit pure states. ${\delta }_{B^V}$ is plotted along the ordinate while $\mathcal{E}$ is along the abscissa. We separately generate ${10}^5$ states from two distinct classes, the GHZ class (green dots) and the W class (magenta dots), and find that for both classes of states the special GHZ state (red solid line) gives the lower boundary. The blue dotted curve corresponds to the generalized GHZ state, which gives the boundary for those states whose maximum eigenvalue comes from the nodal part. All axes are dimensionless.

Figure 2

${\delta }_{B^V}$ (ordinate) against $\mathcal{E}$ (abscissa) of arbitrary four-qubit pure states. We separately generate $5\times {10}^5$ four-qubit states for all nine classes. We find that ${|\text{SGHZ}\rangle }_4$ (red solid line) gives the lower boundary. The dotted blue line represents the ${|\text{GGHZ}\rangle }_4$ state. All axes are dimensionless.

Figure 3

${\delta }_{B^V}$ ($y$ axis) vs ${\delta }_{{\mathcal{C}}^2}$ ($x$ axis) for three-qubit nondistributive pure states (green dots) with $B_{12}^V=B_{13}^V=0$. All the points lie above the ${|\text{GGHZ}\rangle }_3$ state (blue dotted line). Both the axes are dimensionless.

Figure 4

Plot of ${\delta }_{B^V}$ (ordinate) with ${\delta }_{{\mathcal{C}}^2}$ (abscissa) for randomly generated three-qubit GHZ class states. All the points lie above the ${|\text{SGHZ}\rangle }_3$ state (red solid line). Both axes are dimensionless.

Figure 5

Bell-violation monogamy score (${\delta }_{B^V}$) vs quantum-discord monogamy scores ${\delta }_{\mathcal{D}}$ for three-qubit pure states. ${\delta }_{B^V}$ is in the ordinate while ${\delta }_{\mathcal{D}}$ is along the abscissa. Two distinct classes of three-qubit pure states, the GHZ class (green dots) and the W class (magenta dots) of states, are separately generated. (a) We make the measurements on the nodal part and observe that all the points are bounded from below by ${|\text{SGHZ}\rangle }_3$ (red solid curve), while the blue dotted line corresponds to the ${|\text{GGHZ}\rangle }_3$ state that lies above it. On the other hand, (b) for ${\delta }_{\mathcal{D}}$, i.e., when the measurements in discord are performed on the non-nodal observers, the ${|\text{GGHZ}\rangle }_3$ state gives the boundary (dotted blue line). Units in abscissa are in bits while ${\delta }_{B^V}$ is dimensionless.