minkwe wrote:Nobody interested? In any case, let me share a little experiment I did and what I found. The question I wanted to answer was:

Is there any correlation between the pairs of random variables in {a, b, x, y} where {a,b} are the settings on both arms of the delft experiment and {x, y} are the outcomes. For the experiment, we essentially have a list of N settings pairs {a,b} and a list of N outcome pairs {x, y}, ie, a total of 4 lists of N elements. For this test, I used the raw data from the first Delft experiment, including their post-processing code which resulted N=4746 raw data items corresponding to "bell_trial_filter".

To measure correlation, I used a Mutual Information test. Since the lists contain discrete elements, this was easy to do. I calculated the mutual information between pairs of the lists above (a,b), (x, y), (a,y), (b,x), (a,x), (b, y). As can be imagined, the mutual information would be very small for lists that were purportedly randomly obtained. This was indeed observed. But then the next question which arose was, how do I determined what the expected mutual information should be? For this, I calculated the probability distribution of the elements for each list in the pair being compared, and then randomly generated 10,000 pairs of similar lists with the same probability distributions, calculated the mutual information on each of the 10,000 pairs, and then calculated the percentile of the observed mutual information relative to the 10,000. As you can imagine, this is a pretty good indication of how far from expected the observed mutual information was in the experimental data.

Thus, a percentile of 100% indicates that all of the mutual information values calculated from the 10,000 randomly generated pairs, are lower than the observed mutual information, an indication that the observation is clearly correlated more than expected. Similarly, a mutual information of 0% indicates that all the randomly generated mutual information values are higher than the observed one.

...

Key takeway: The first order mutual information in x is artificially low, while the second order is higher than expected. I short, there is something strange about this experiment. Hopefully you guys have better explanation of what is going on.

Splendid work. I do have an innocent explanation for Michel’s at first glance disturbing correlations.

His statistical analysis assumes iid data. In particular, identical distributions over time. Now, suppose that as time goes by, physical properties of all the systems involved in the experiment tend to drift and occasionally even suddenly jump. That can generate all kinds of spurious correlations. And with huge data-sets, they’ll be statistically highly significant.

That is why experimenters nowadays don’t use the conventional statistical analysis based on multinomial counts or Poisson distributions or naïeve resampling (bootstrap, cross validation, ...). Instead they have further developed and refined the martingale analysis which I pioneered in 2001. One gets less strong p-values but one also gets insurance against spurious correlations, against opportunistic early stopping, and more besides.

Anyway, Michel should publish, at least on arXiv. I can recommend him on arXiv, if he needs that. I can also help getting hold of data from researchers who like only to give their data to persons whom they consider to be reliable researchers. Fortunately, this kind of researcher is dying out. In the Netherlands there are national rules about availability of data coming from research which, let's face it, is usually paid for by the tax-payer, in one way or another. Those rules should obviously morally apply to anyone who participates in science.

Some further comments: Michel's variables a, b, x, y are all binary. There is nothing wrong with encoding them +/- 1 and looking at ordinary correlations. Of course we should realise that the settings a, b are just *labels* and there is not necessarily any particular relation between Alice's setting 1 and Bob's setting 1, Alice's 2 and Bob's 2. On the other hand, in the Bell-CHSH or Eberhard set-up we are comparing one of the *pairs* of settings to the other three. So one of Alice's settings and one of Bob's settings does have a special status.

In the Delft experiment, serious mistake of the experimenters, the settings were generated by quantum photonics! This is foolish. This means that (a) the physics of the setting-generation and of the source and detection are highly related; and (b) their randomness is not guaranteed, it is only verifed by extensive tests, but who knows what might go wrong the one time you are actually doing your experiment? With the martingale tests (alternatives to CHSH and to J) you rely heavily on the randomness of the settings. Drifts and jumps and memory effects in the other parts of the experiment are harmless. You can however "assume" that the deviation from randomness of the settings is smaller than some amount, at the cost of slightly decreasing the p-value.

I think one should use state-of-art pseudo random number generators and preferably a cascade of several different ones. Moreover, choose the random seeds for those generators by tossing some coins, etc. You never can avoid the conspiracy loophole but you can make its invocation pretty ridiculous, implausible, bad physics.

The Delft experiment is much too small. There are too many things which can go wrong, and if you test for all of them, you are bound to find some which are at least as strong as the actual physical signal which you want to determine.

An interesting new analysis of several of the loophole-free experiments is PHYSICAL REVIEW A 99, 022112 (2019), Very strong evidence in favour of quantum mechanics and against local hidden variables from a Bayesian analysis, Yanwu Gu, Weijun Li, Michael Evans, and Berthold-Georg Englert.

https://arxiv.org/abs/1808.06863However they assume that the iid (independence and identical distributions) assumptions are OK so that one can reduce the data to the 16 totals N(x, y | a, b)