  We again consider (as in a companion paper) an entangled two-particle state
that is produced from two independent down-conversion sources by the process of
"entanglement-swapping", so that the particles have never met. We show that
there is a natural extension of the Einstein-Pololsky-Rosen discussion of
"elements of reality" to include inefficient detectors. We consider inefficient
deterministic, local, realistic models of quantum theory that are "robust",
which we consider to be the minimum requirement for them to be taken seriously.
By robust, we mean they satisfy the following three criteria: (a) they
reproduce the quantum results for perfect correlations, if all particles are
detected; (b) they produce some counts for every setting of the angles (so they
don't describe some experiments that can easily be performed as "impossible");
(c) all their hidden variables are relevant (they must each produce a
detectable result in some experiment). For such models we prove a
Greenberger-Horne-Zeilinger (GHZ) type theorem for arbitrary detection
efficiencies, showing that any such theory is inconsistent with the quantum
mechanical perfect correlations. This theorem holds for individual events with
no inequalities. As a result, the theorem is also independent of any random
sampling hypothesis.
