  There is a significant interest in testing quantum entanglement and Bell
inequality violation in high-energy experiments. Since the analyses in
high-energy experiments are performed with events statistically averaged over
phase space, the states used to determine observables depend on the choice of
coordinates through an event-dependent basis and are thus not genuine quantum
states, but rather "fictitious states." We prove that if Bell inequality
violation is observed with a fictitious state, then it implies the same for a
quantum sub-state. We further show analytically that the basis which
diagonalizes the spin-spin correlations is optimal for constructing fictitious
states, and for maximizing the violation of Bell's inequality.
