Tight upper bound for the maximal quantum value of the Svetlichny operators

It is a challenging task to detect genuine multipartite nonlocality (GMNL). In this paper, the problem is considered via computing the maximal quantum value of Svetlichny operators for three-qubit systems and a tight upper bound is obtained. The constraints on the quantum states for the tightness of the bound are also presented. The approach enables us to give the necessary and sufficient conditions of violating the Svetlichny inequality (SI) for several quantum states, including the white and color noised Greenberger-Horne-Zeilinger (GHZ) states. The relation between the genuine multipartite entanglement concurrence and the maximal quantum value of the Svetlichny operators for mixed GHZ class states is also discussed. As the SI is useful for the investigation of GMNL, our results give an effective and operational method to detect the GMNL for three-qubit mixed states.

Figure 1

Consider the mixture of the GHZ state and white noise: ${\rho }_{\text{GHZ}}=p|\text{GHZ}\rangle \langle \text{GHZ}|+\frac{1-p}{8}I$, where $|\text{GHZ}\rangle =\frac{1}{\sqrt{2}}\left(\right|000\rangle +|111\rangle )$. By [22] we have that ${\rho }_{\text{GHZ}}$ will be genuinely multipartite entangled if and only if $0.428571<p\le 1$, while by our theorem one obtains that ${\rho }_{\text{GHZ}}$ will violate the SI if and only if $0.707107<p\le 1$.

Figure 2

Consider the mixture of the GHZ state and color noise: ${\sigma }_A\left(\rho \right)=p|\text{GHZ}\rangle \langle \text{GHZ}|+\frac{1-p}{4}{I}_2\otimes \tilde{I}$. By [23], one has that ${\sigma }_A\left(\rho \right)$ admits bilocal hidden models for $0\le p\le 0.416667$, and is genuinely multipartite entangled for $\frac{1}{3}<p\le 1$, while by our theorem one obtains that ${\sigma }_A\left(\rho \right)$ will be GMNL for $0.707107<p\le 1$.