Inadequacy of the CHSH Bell inequality for an application to its associated photon correlation experiments
Résumé
We analyse the proof of a CHSH Bell-type inequality and show that this inequality cannot be applied to the experiments it has been designed for. In the example we study, two polarization
values are attributed to each of the two photons of a correlated photon pair, such that
four values are assigned simultaneously to a single pair. But in the experiments these four values cannot be measured simultaneously on a single photon pair, such that the assumptions
underlying the inequality do not properly translate the
experimental situation into mathematics. The CHSH inequality is therefore
based on wrong modelling.
To derive an inequality that correctly reflects the experimental situation
and does contain the experimentally measured probabilities $p(a_{j} \wedge b_{k})$,
a combination of
four distinct photon pairs (and, where applicable, two single photons to measure
$p(a_{2})$ and $p(b_{2})$ in independent runs)
must be considered rather than just a single pair.
Using the same methodology as was used for the CHSH inequality one can derive
a new inequality, which is now based on correct modelling and properly translates the physics into mathematics. But is has different, less stringent boundaries with the effect that it is no longer violated by the experiments.
Larsson has tried to rebut this within the context of the wrong model that has been used
to derive the CHSH inequality. Within this wrong context he presented the different boundaries obtained within the correct model as an experimental loophole
caused by statistical fluctuations when the counting statistics are not good enough.
Individual violations of the inequality would be possible but the averages, i.e.
the measured probabilities
respect the inequality. This is wrong because what is at stake in the wrong model is a systematic theoretical error caused by the wrong modelling. It has nothing to do with statistical fluctuations and can therefore not be healed by improving the counting statistics. But
based on his idea about the averages
Larsson is able to prove the inequality for the measured probabilities.
However, this is a proof locked up inside the premises of the wrong model,
while a comparison between the wrong and the correct model
shows that their probability distributions are normalized differently, such
that the wrong model is smuggling in a normalization error.
The same modelling error occurs
in the derivation of other types of Bell inequalities such that none of them can be applied
to the experimental data they were designed for. The violation of these Bell inequalities can therefore not be considered
as a proof for the existence of entanglement.
Domaines
Physique Quantique [quant-ph]
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