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Theoretical considerations of Bell-inequality experiments usually assumeidentically prepared and independent pairs of particles. Here we consider pairsthat exhibit both intra- and inter-pair entanglement. The pairs are taken froma large many-body system where all the pairs are generally entangled with eachother. Using an explicit example based on single mode entanglement and anancillary Bose-Einstein condensate, we show that the Bell-inequality violationin such systems can display statistical properties that are remarkablydifferent from those obtained using identically prepared, independent pairs. Inparticular, one can have probabilistic violation of Bell's inequalities inwhich a finite fraction of all the runs result in violation, even though therecould be no violation when averaging over all the runs. Whether or not aparticular run of results will end up being local realistically explainable is"decided" by a sequence of quantum (random) outcomes.

Bell's experiment with intra- and inter-pair entanglement: Single-particle mode entanglement as a case study