Due to the infinite population space, a statistical experiment can not exhaust the entire population space, only finite samples of the population space are obtained. Thus when the finite sample is used to represent the population space in (\ref{eqn:replace}), one necessary assumption is that \(\hat{p}_{++}\) derived from a finite set of sample events (i.e. a subset of entire population space) does properly represent \(p_{++}\) of the entire population space, or alternatively, the sampling is fair.
As fair sampling assumption arises when a subset is used to represent the entire population space, it is independent of the detection efficiency in an experiment. Even if the detector efficiency is 100%, the collected experimental data set is still a subset of the population space, and fair sampling is still required when representing the population with a data set from a 100% detector. Thus fair sampling loophole can not be closed, unless one is not doing a statistical experiment.
The fair sampling loophole originates from the statistical nature of Bell’s inequality, which intends to test the local realism/entanglement hypothesis, whereas the underlying physics concept of entanglement (i.e. the subject to be tested) is specified with certainty. There has been extensive interest in studying Bell’s inequality, as many experiments collected batches of data and computed the correlations after the test runs, but they do not attempt to directly test the concept of entanglement/local real model, i.e. to observe in real time whether change on one side has impact on the other side.
But the real fundamental question is about the physical essence of entanglement. If one could directly observe the spooky action at a distance, it would be a conclusive support of entanglement, which would leave test of Bell’s inequality (by computing correlations after the happening moment of entanglement) superfluous. If on the other hand, an experiment has many aspects manifesting local realism (the opposite to entanglement), simple violation of Bell’s inequality may not be effective.
As existing work show, transition of wave function from odd parity to even parity suggests that spin on one side can be independently changed without impact on the other side; operations of rotating spin followed by measurement disentangle the wave function to be measured; and as a basic requirement of representing the entire population with finite sample in any statistical experiment, fair sampling loophole can not be closed; thus the reported experiments are more of a proof of local real model than evidence of entanglement.
\acknowledgments