Relativistic three-partite non-locality

Bell-like inequalities have been used in order to distinguish non-local quantum pure states by various authors. The behavior of such inequalities under Lorentz transformation has been a source of debate and controversies in the past. In this paper, we consider the two most commonly studied three-particle pure states, that of W and GHZ states which exhibit distinctly different type of entanglement. We discuss the various types of three-particle inequalities used in previous studies and point to their corresponding shortcomings and strengths. Our main result is that if one uses Svetlichny's inequality as the main measure of non-locality and uses the same angles in the rest frame ($S$) as well as the moving frame ($S^{\prime}$), then maximally violated inequality in $S$ will decrease in the moving frame, and will eventually lead to lack of non-locality ( i.e. satisfaction of inequality) in the $v \rightarrow c$ limit. This is shown for both GHZ and W states and in two different configurations which are commonly studied (\textbf{Case $I$} and \textbf{Case $II$}). Our results is in line with a more familiar case of two particle case. We also show that the satisfaction of Svetlichny's inequality in the $v\rightarrow c$ limit is independent of initial particles' velocity. We use Czachor's relativistic spin operators instead of Pauli spin operators in our calculations. Our results draws a clear picture of three-particle non-locality making its general properties consistent with previous studies on two-particle systems regardless of the W state or the GHZ state is involved.