Bell inequalities for three particles

We present tight Bell inequalities expressed by probabilities for three four- and five-dimensional systems. A tight structure of Bell inequalities for three $d$-dimensional systems (qudits) is proposed. Some interesting Bell inequalities of three qubits reduced from those of three qudits are also studied.

Figure 1

Numerical results for the generalized GHZ states ${\mid {\psi }_2⟩}_{\mathit{GHZ}}=\cos \xi \mid 000⟩+\sin \xi \mid 111⟩$, which violate inequality (18) except $\xi =0$, $\pi ∕2$.

Figure 2

(Color online) Numerical results for the generalized $W$ states ${\mid \psi ⟩}_W=\sin \beta \cos \xi \mid 100⟩+\sin \beta \sin \xi \mid 010⟩+\cos \beta \mid 001⟩$ which violate inequality (18) for different $\xi$ and $\beta$. Here the cases $\beta =\pi ∕12$, $\pi ∕6$, $\pi ∕4$, $\pi ∕3$, $5\pi ∕12$, and $\pi ∕2$ are considered.

Figure 3

(Color online) Violation of two Bell inequalities for the generalized GHZ state of three qubits with different value of $\xi$, where curve $A$ is for the inequality given in Ref. 12 and curve $B$ is for our inequality (18).