What are some experimental verifications of Born's rule in quantum mechanics? Born's rule in quantum mechanics states that when measuring a system using a measuring device that can detect (=project onto) an orthogonal basis of states, the probability of obtaining a certain outcome equals to the square of the modulus of the projection of the (normalized) system's state onto the corresponding basis state. 
What are some experimental verifications of Born's rule in quantum mechanics? 
 A: Born's rule (square of the modulus) has been verified experimentally numerous times. However, only the overall averages have been verified. For example if the prediction is 60% probability, then over large number of trials, the average outcome will approach the predicted value of 60%. This has been verified by measuring particle spin at angle A relative to the angle of its previously known spin angle. The prediction is square of cos(A/2). These predictions have also been verified with entangled pairs (Bell's state) where the same spin prediction is square of sin(A/2).
What has not been verified is whether the outcomes are due to independent probability, or they are guided by some balancing mechanism.
This level of analysis has not been done, at least not been published by the experimentalists yet.
I did such analysis on data from one recent entanglement experiment, and I am currently doing similar analysis on data from another recent experiment. The completed analysis indicates presence of some kind of balancing mechanism. The preliminary analysis of second experiment has also given same indications. I should be completing the second one in a few days.
The first one can be read at http://vixra.org/abs/1609.0237
Another point is that at least in one state (one of Bell's states), a perfect anti correlation is predicted. That means anti correlation with probability of 1. There is no such thing as probability of 1. If probability is one, that has to be enforced by a law, not by probability. So, if one scenario has to be enforced by a law (e.g. conservation of angular momentum), then other scenarios have to be enforced by law as well. Those other scenarios are enforced by the laws at an average level and so may appear to be probabilistic even if they are not really so. It depends upon how much the law lets deviate the outcomes from the averages. And higher level analysis shows the deviations are within probabilistic limits. Only a detailed analysis can differentiate between a true probabilistic nature and a balancing mechanism enforced by the conservation laws.
Because you have questioned verification of a QM related rule, your question is likely to be marked duplicate, or voted for close as unclear. In my experience, high rank members on this site do not like questioning anything related to QM. Not because they want to discourage you, but because the mathematics is too convincing and they genuinely believe in the formalism beyond any verification. I think the strength of mathematics should be a reason to welcome verification instead of deflecting it.
