Michel:
gill1109 wrote:Bell's derivation does not assume that particles can be measured more than once. Bell shows that the assumption of a local hidden variables theory implies certain limits on correlations which can be observed in Nature, if Nature could be described by such a theory. There is no need to "rescue" his 1964 paper, but there certainly was possibility to remove ambiguities and sharpen the results. Bell's 1980 (?) "Bertlmann" paper already improves and sharpens Bell (1964) in numerous respects.
My own recent work is a further *strengthening* of Bell's. Bell's 1980 results, which improve on those from 1964, are a *corollary* of mine. I derive finite N probability bounds, but Bell only has infinite N limits.
A local hidden variables *theory* implies the mathematical existence, simultaneously, within the theory, of outcomes of different potential measurements. The measurements don't need to be *done*. They aren't *done* within the derivation of the famous inequalities.
When deriving the CHSH inequality we are not talking about measurements at all, and certainly not about multiple measurements on the same particles. We are talking about mathematical relations between functions A(a, lambda), B(b, lambda) and a probability distribution rho(lambda). The functions A and B only take the values -1 and +1. We determine that the functions E(a, b) = int A(a, lambda) B(b, lambda) rho(lambda) d lambda are not completely arbitrary but have to satisfy certain relations, in particular we find
E(a, b) + E(a, b') + E(a', b) - E(a', b') <= 2
In quantum theory, correlations are computed in a different way using various Hillbert space objects (operators, states, ...). One can also work within quantum theory and assuming a product system, pairs of POVM measurements on each system with outcomes -1 and +1 only, and an arbitrary joint quantum state rho, prove the Tsirelson inequality
E(a, b) + E(a, b') + E(a', b) - E(a', b') <= 2 sqrt 2.
If we only assume no action at a distance (at the surface level), one can only prove the PR inequality (Popescu-Rohrlich)
E(a, b) + E(a, b') + E(a', b) - E(a', b') <= 4
All of this theoretical work has got nothing whatsoever to do with actual experiments. The calculations do not presuppose some weird experiment during which all kinds of impossible things are done. They show that if you believe that a certain kind of theory underlies the correlation which we see in Nature, then those correlations will satisfy certain properties. They can't be just anything.
When we do experiments we get to learn something about E(a, b), up to statistical error, by many times doing a particular measurement on pairs of particles. We learn about E(a, b') by measuring other particles in a different way.
It's very hard for me to understand why this is so hard to understand ... but perhaps it would help to have some understanding of statistics and real experiments.
For instance if we are interested in whether people are getting more intelligent, we could take a sample of fathers and a sample of sons and look at the difference between the average IQ's of the fathers and of the sons. Alternatively we could take a sample of father-son pairs and look at the average of the differences. For given sample sizes, the second route is more accurate than the first route, but the first route is not invalid. Of course we do have to take care that our sample of fathers and our sample of sons are random samples from the same population as the sample of father-son pairs, otherwise things go wrong.
Exactly as in Bell-EPR experiments. If we measure a lot of time one pair of settings, and then another lot of time another pair of settings, maybe things have changed in the lab between the first batch and the second batch and the two samples of pairs of particles are not samples from the same population.
Or if we reject particles which arrive too early or too late at our detectors, maybe the particles which are left are not a random sample from all particles.
This is why good EPR-B experiments are done with delayed-choice settings and event-ready detectors. It's like the preference for a randomized double blind clinical trial above an observational study when testing new medications. We can't test both treatments on the same patient. We can only treat each single patient in one way. Yet we do clinical trials and tell the world that the new treatment is better than the old one.
Seems that Watson, Adenaur and now Fodje have no belief whatsoever in evidence based medicine. Because we can't both give a breast-cancer patient a mastectomy and a breast-preserving operation, we can never tell which treatment is better.