The basic issue with quantum measurements is that once an observer measures two entangled spin states from the same source, that observer cannot then know that those spin states always existed for each particle and were simply hidden. This means that before observation, quantum spin states exist as a superposition of both spin states and not just one no matter how far apart the two particles are.
Classical spin states represent revealed knowledge in that once revealed by observation, the observer knows for certain that the spin states existed even before their measurement. In the absence of perturbations, measuring a classical spin state coming from a source necessarily means that spin state existed prior to the measurement.
Bell's theorem uses a particle pair correlation to test whether a spin state existed prior to its measurement. Quantum and classical correlations agree at certain fixed angles; 0, 90, 180, 270, and 360, but at other angles, the probability is a product of the entangled superposition states. This quantum product is what results in the quantum cos(angle) correlation instead of the classical linear correlation shown in the figure.
This is a model of correlated particle pairs that are polarized either spin up or spin down by degrees from 0 to 360. Therefore all of this spin information exists and is just hidden until revealed by the plots and there are no quantum superposition states in the model. Angles from 90 to 270 are +1 and 270 are -1. The model reports polarization correlation between those particle pairs and in the absence of any noise, what the result is perfect correlation. Even with 6% noise added, the correlation remains very good since 6% noise flips very few spins. When the model adds ever larger amounts of noise, though, more and more spins flip and randomize the polarization of selected spin pairs.
Therefore the noise function seems to have some kind of role, but the noise seems a little odd. The noise function maximizes at 90 and 270 and minimizes at 0, 180, and 360 and so that means spin pairs with increasing noise amplitude will randomize around 90 and 270 before 0, 180, and 360. Thus, the model noise function is wired to randomize spins around 90 and 270 before spins at 0, 180, and 360. It is not clear how this noise is physical since the noise at 90 and 270 should be the same as the noise at 0, 180, and 360.
When you add just enough noise to flip ~28% of the spins and truncate the measurement, you lose 28% of the spins preferentially around 90 and 270 and so rescale +/-1 and therefore match the cos(angle) of quantum correlation. Classical realism measures perfectly correlated spin states as +/-50% for all angles since each particle can only be either +1 or -1. However, quantum superposition entangles the particle pairs and makes the likelihood +/-71% (1/sqrt(2)).
If the model noise function were flat with angle instead of peaked, the model would simply reveal its hidden classical values even down to one bit of information. There are many ways for the classical noise of chaos to resemble quantum phase noise by selecting populations of particles. This model has selected a subpopulation of particle pairs that happens to then average to the quantum result of Bell's theorem under particular conditions. When the model applies a polarization to its random noise, the noise is weighted by the slope of that angle...i.e., but the cos(angle). Therefore the model is built with the cos(angle) dependence by weighting the noise.
You argue that once the polarization information reduces to one bit, this represents the classical noise limit of Shannon Capacity Theorem and that limit explains why quantum effects are really classical after all. I actually do not really like the Bell's theorem approach since it unnecessarily complexifies quantum superposition and therefore can be gamed by any number of ways of hiding values such as this model shows.
This model result seems to be due to a careful choice of a classical noise, not to a physical principle.Attachment #1: mceachernCorrelate.jpg