Bell's Theorem was developed on the basis of considerations
involving a linear combination of spin correlation functions, each
of which has a distinct pair of arguments. The simultaneous
presence of these different pairs of arguments in the same
equation is investigated, and the implicit counterfactual
assumption in Bell's theorem is discussed.
We show how an explicit contextuality can arise from a model
displaying unfair sampling, and we discuss it in the
framework of David Mermin's cleverly simple version of Bell's
theore, which pinpoints in a very straightforward way how
interpreting entanglement from a realistic point of view can be
problematic. We present an extended version of Mermin's device
that can actually be given a straightforward realistic
interpretation.
We stress that the low efficiency of detectors in all experiments
with photons makes the use of the fair sampling assumption
unavoidable. Since this very assumption is false in all existing
local realistic models based on inefficient detection, we thus
question its validity. We show that it is no more reasonable to
assume fair sampling than it is impossible to test, and we
actually propose an experimental test which would provides clear
cut results in case of unfair sampling
We then analyze optical EPR experimental data performed by Weihs et al in Innsbruck 1997-1998. We show that for some linear
combinations of the raw coincidence rates, the experimental
results display some anomalous behavior that a more general source state (like non-maximally entangled state) cannot
straightforwardly account for. We use the fair sampling
assumption, and assume explicitly that the detection efficiencies
for the pairs of entangled photons can be written as a product of
the two corresponding detection efficiencies for the single
photons. We show that this explicit use of fair sampling cannot be
maintained to be a reasonable assumption as it leads to an
apparent violation of the no-signalling principle.