Trade-off relation among genuine three-qubit nonlocalities in four-qubit systems

We study the trade-off relations satisfied by the genuine tripartite nonlocality in multipartite quantum systems. From the reduced three-qubit density matrices of the four-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states and W states (4-qubit entangled state), we find that there exists a trade-off relation among the mean values of the Svetlichny operators associated with these reduced states. Namely, the genuine three-qubit nonlocalities are not independent. For four-qubit generalized GHZ states and W states, the summation of all their three-qubit maximal (squared) mean values of the Svetlichny operator has an upper bound. This bound is better than the one derived from the upper bounds of individual three-qubit mean values of the Svetlichny operator. Detailed examples are presented to illustrate the trade-off relation among the three-qubit nonlocalities.

Figure 1

For $\theta \in [0,\frac{\pi }{4}]$, the upper bound of the sum of violations of the Svetlichny inequality for four reduced three-qubit states is $16|cos2\theta |$. It is less or equal to $16\max \{{cos}^4\theta ,{sin}^4\theta \}=16{cos}^4\theta$ derived from (8). The blue line is the bound from Theorem 1. The yellow dashed one comes from (8). When $\theta =0$, two bounds are equal.

Figure 2

The blue line is for the upper bound of (15), and the yellow dashed line is for the upper bound of (17) for $\theta \in (\pi /2,3\pi /2)$.

Figure 3

In the range of $\gamma \in [0,1]$, the bound $F$ is smaller than $G$.

Figure 4

Top left figure: violation of Svetlichny operators for states $W_{abc}$ and $W_{abd}$. Top right figure: violation of Svetlichny operators for states $W_{acd}$ and $W_{bcd}$. Bottom figure: trade-off relation among the nonlocallity of W-class states. For $\gamma \in [0,1]$, the quantities ${\langle S_{W_{abc}}\rangle }^2(={\langle S_{W_{abd}}\rangle }^2)$ and ${\langle S_{W_{acd}}\rangle }^2(={\langle S_{W_{bcd}}\rangle }^2)$ vary in a way such that their summation kept to be bounded by $F$.