Relating monogamy of quantum correlations and multisite entanglement

There are two important paradigms for defining quantum correlations in quantum information theory, viz. the information-theoretic and the entanglement-separability ones. We find an analytical relation between two measures of quantum correlations, one in each paradigm, and show that only a certain cone-like region on the two-dimensional space spanned by these measures is accessible to pure three-qubit states. The information-theoretic multiparty quantum correlation measure is related to the monogamy considerations of a bipartite information-theoretic quantum correlation measure, while the entanglement-separability multiparty measure is the generalized geometric measure, a genuine multiparty entanglement measure. We also find an analytical relation between two multiparty entanglement measures, and again obtain a cone-like accessible region in this case. One of the multisite measures in this case is related to the monogamy of a bipartite entanglement measure, while the other is again the generalized geometric measure.

Figure 1

Multiparty entanglement vs quantum monogamy score. Here, quantum monogamy score is identified with the entanglement monogamy score, and multiparty entanglement is quantified by generalized geometric measure, which is a genuine multisite entanglement. The units of the axes are chosen as follows. The vertical axis is in units for which the GGM of the GHZ state (generalized GHZ state with $\alpha =\frac{1}{\sqrt{2}}$) is $\frac{1}{2}$, while the horizontal axis is such that the concurrence of the singlet state is unity. For a randomly chosen pure state of three qubits, its entanglement monogamy score is taken as the abscissa and its GGM as the ordinate, and the pair is then plotted on the horizontal and vertical axes of the figure. We randomly chose $2.5\times {10}^4$ points. The envelope of the points thus plotted forms a two-dimensional conical shape, with one axis coinciding with the vertical axis (GGM axis, i.e., entanglement monogamy score $=0$), and the other being formed out of generalized GHZ states. The tangent to the curved boundary at the origin of the two axes forms a nonzero angle with the horizontal axis (entanglement monogamy score axis, i.e., GGM $=0$).

Figure 2

Multiparty entanglement vs discord monogamy score. The notation is just as in Fig. 1, except that the horizontal axis now depicts the discord monogamy score and is measured in bits. We randomly generated $5\times {10}^4$ points. The envelope this time is formed of two curved lines, of which the right one is again (i.e., as in Fig. 1) formed out of generalized GHZ states, while the one on the left is the mirror image of the one on the right with respect to the GGM axis. The tangent to the curves at the origin of the two axes in this case coincides with the GGM $=0$ line.