I am not an experimentalist but I doubt it could be acceptable because the experiment would not represent the theoretical derivation.

Maybe I did not understand your point well but the proof I currently have is not analytic but based on simulation of a system following a given statistical behaviour. Bell like inequalities do not impose anything about behaviour of the observed system. In the original Bell inequality, every event has an outcome and my model cannot be applied there. However in CHSH inequality, due to detector efficiency this rule is relaxed. So taking advantage of this, a detector is free to signal an outcome or stay silent while obeying the locality rule. In this logic I defined a rule in detectors for in which condition it generates an outcome or stays silent.

A possible experiment reflecting this is about randomly (but having a statistical distribution logic) blinking leds in two colors (blue and red) mounted on two rotating wheels in synchronized manner. The correlation is to observe the same color (only blue or red but not a violet) on two wheels in spots defined by angles a, b, a', b' at the same time. For example by cameras programmed to take pictures every second.

You said: "Bell like inequalities do not impose anything about behaviour of the observed system."

Then the Bell inequality would be unfalsifiable and it does not make sense to perform a CHSH experiment. If that's the case, then violations, or not violations, of the inequality by experiments would not mean anything.

I know of people who believe that is in fact the case. For instance: Article Is the Moon There If Nobody Looks: Bell Inequalities and Phy...

Thank you for the article. By "Bell like inequalities do not impose anything about behaviour of the observed system.", I meant the observed entities can behave in any possible way, one may think, they can even be the decision of mathematicians but they cannot behave depending on the condition of the remote station.

I understand that the inequality is violated according to quantum mechanics because a measurement made on one entangled entity affects the outcome of its paired entity, thus violating locality. I am not arguing on that, but I am saying that one should take account of non signaling (zero) cases, otherwise CHSH can be violated within locality by using hidden variables. I saw that Kupczynski's article mentioned the "0" outcomes beside +-1, but not followed the equations yet.

Yes, then you are talking about experimental loopholes and I think you are right. Local hidden variables can violate the inequality because of these loopholes.

So my question is could these rules satisfy the CHSH measurement?

1.A source sends some data to two stations without knowing their configurations. This data is supposed to be emitted with a constant average density.

2. Stations cannot communicate with each other and cannot send data to source.

3. Stations have no memory.

4. Stations send signals which consist of +1 and -1 to a center which calculate correlations it receives from two stations. Stations also send their setup info as (x) or (x') where x is the station identifier a or b. Stations are free to how to interpret the data they receive from the source. However they should send their results without delays.

5.Stations can switch their configurations (x, x') upon a schedule or randomly by their own.

6. The center expects valid signals are only ones which coming in pairs (in a brief time window) and discard any unpaired signal.

If condition 6 ensures your measurements come from the same entangled particles then local hidden variables are supposed to satisfy the inequality.

The experimental details are rather complicated and I do not know but from a theoretical point of view, it is very simple. The question is to what extent the ideal conditions assumed in the theoretical derivation can be realized in an actual experiment.

"If condition 6 ensures your measurements come from the same entangled particles then local hidden variables are supposed to satisfy the inequality."

But I made a numeric analysis in excel sheet where the inequality is violated by using hidden variables. The hidden variable is a geometric angle which determines the probability of detectors to produce +1 or - 1 as one would expect but also determines the probability of the detectors can detect the signal. I will review my work and will give the details.

The signals issued by the source are bichromatic, say red and blue. They are always polarized orthogonal to each other. On detectors, this bichromatic signal is passed from a polarizer (filter) where a color signal can pass from filter by %100 probability if it is polarized in same angle of the filter and 0% if orthogonal and intermediate values varies by cos² of the angle difference. As the other color is always polarized by 90° of difference, its detection probability varies by sin². The logic in the detector is that one detected color produces +1 and the other -1. Detector sums these numbers and only sends to the correlator (center) if it is not zero. This means when it detect a red or a blue signal but not the violet.

So the correlator receives sometimes paired signals from detectors and counts and correlates, sometimes only one signal from a detector and discards it, and sometimes no signal at all, so nothing to do.

A simulation of this process typically produces S=2.5, in violation of the equality.

The problem I see is that your detectors do not have 100% efficiency. In the case those null results are deterministic, i.e., the same combinations of settings and lambda give always cero, then if you count those events, the inequality should hold.

However, you should consult an experimentalist or some who knows about the experimental issues of Bell tests. You must find someone who really knows. There are many who pretend to know about these things. I am sorry but I can't help you because I am not an expert.

The efficiency is about 75% here because 25% of the time, both red and blue are detected and these events are discarded. On the figure blue-red we can see the distribution of blue and red signals while lamba varies from 0 to pi when the filter angle (theta) is set to zero degree. Each hair corresponds to a signal passed from the filter. A signal can pass from the filter by the probability by cos²(theta-lambda) for blue signal and sin² for red. In many events around lambda=pi/4 red and blue signals come together and are discarded.

The second figure (4correlations) summarizes four session results. Since each station only send 75% (in average) of signals to the correlator, the correlated signal count is the square of this efficiency. These lost cases mostly coincide with anti correlations in sessions AB, AB', A'B, and to the correlation in session A'B'. So the correlation is about 0.63 for first three sessions and - 0.63 for the last and this gives S = 6.3 * 4 = 2.5.

So this hidden local variable selectively varies the detector "efficiency" and produce an extraordinary CHSH sum.