However, for the class of states considered, it is shown that no violations of local realism can arise within the unconditional ensemble constructed by selecting the results obtained by two of the observers (say, by observers 1 and 2) irrespective of the result obtained in the corresponding measurement by the third observer. As will presently be seen, this unrestricted selection of the data provided by the results of observers 1 and 2 involves the incoherent mixture of two pure states $|\Psi^{+}_{12}\rangle$ and $|\Psi^{-}_{12}\rangle$ for particles 1 and 2, with weighting factors given by the probability of getting the result $z_3$ and $-z_3$, respectively, in a spin measurement on particle 3 along the axis $\vec{e}_3 (\vartheta_{3},\varphi_{3})$. Such a mixture, however, turns out to be completely equivalent to a mixture of two product states for particles 1 and 2, and then no violation of a Bell inequality will occur for the above-defined unconditional ensemble.

Unconditional two-particle correlations for the class of triorthogonal states
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In order to determine under which conditions the correlation between the measurement results for particles 1 and 2 can lead to a violation of Bell’s inequality it is convenient to introduce another set of basis states $\{|z_3 \rangle^{\textstyle 