minkwe wrote:δ > 0 (M1); LG (-)
δ ≥ 4 − 3/γ (M2); LG (17)
| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 6/γ − 4 (M3); LG (20)
My labels M1, M2, M3 (Michel) and LG formula numbers
Michel Fodje thinks that LG (Larsson and Gill (2004);
http://arxiv.org/abs/quant-ph/0312035, contains untrue statements.
He determines this by looking at three equations in the paper and then trying out the test value gamma = 0.5.
Let's take a look at the context.
Suppose that we are given a LHV model with coincidence-interval post-selection, no other loopholes, the sample size is infinite - we talk just about probabilities, not about relative frequencies. Population means, not sample averages. Probability theory, not statistics. The statistical bridge between theory and experiment is another matter.
delta is defined in LG formula (10) so Michel's first statement that it is non-negative is a triviality: it is the minimum of four different conditional probabilities. In fact we know it lies between 0 and 1.
LG's Theorem 1 (the only theorem in their paper) states that under clearly defined conditions,
| E(AC′|ΛAC′) + E(AD′|ΛAD′)| + |E(BC′|ΛBC′) − E(BD′|ΛBD′)|≤ 4 - 2 delta (11)
He has not indicated anything wrong with the theorem. Obviously we also know, trivially, that the left hand side is nonnegative (it's the sum of two absolute values of various quantities. As delta varies between 0 and 1 (as obviously, it could, by varying the LHV model), the bound on the right hand side varies (in the opposite direction) between 2 and 4. A rather pretty extension of standard CHSH which has delta = 1 and bound = 2, that's one end of the spectrum. At the other end we have delta = 0 and bound = 4. The trivial bound which holds because the four "E(..|..) quantities lies between -1 and +1.
As a corollary to Theorem 1 which includes (11), LG proceed to derive (M2), which is LG's (20). On the way they of course have to define the new quantity gamma (there was no gamma in Theorem 1, but there is one in (M2) = (20)
gamma is defined in LG's (16). It is obviously a number between 0 and 1 since it is the minimum of four probabilities.
The proof that delta >= 4 - 3 / gamma is pretty trivial, Fodje gave no objections to it. This is LG's (17) and Fodjes (M2).
From this we trivially obtain Fodje's (M3) which is LG's (20).
From the context it is clear that LG claim that (20) is true if the conditions of Theorem 1 are true. No more and no less, at this point.
The fact that LG's (17) = Fodjes (M2) is trivially true if gamma <= 0.75 is not a problem. The corollary to Theorem 1 is trivially true if gamma <= 0.75.
So far, LG have simply said nothing whatsoever of any interest in the case that gamma <= 0.75. They have said something novel in the case that gamma > 0.75 because for any gamma between 0.75 and 1.00 their theorem tells us something which was not known before.
Next, LG say "let us see whether this bound is necessary and sufficient". They should have said, let us see if this bound is sharp.
They do not state any more formal theorems but merely give an example. The example shows that if you pick a number l between 0 and 1, you can arrange that gamma = (3 + l) / 4 and S = 4 * (3 - l) / (3 + l) which is the same as saying that S = 6/gamma - 4. Thus: gamma van be chosen freely between 3/4 and 1, and for each choice there is a LHV model with that value of gamma and with S = 6/gamma - 4.
This takes care of all of the values of gamma between 3/4 and 1. There is no need to consider smaller gamma, since for gamma = 3/4 the bound S <= 4 can be attained. So trivially we can attain the bound S <= 4 with gamma < 3/4.
Thus I conclude that there is no error at all in LG (2005) though to be sure, we had to cut off quite a few corners in order (a) to keep under the four page limit and (b) not burden the physicist reader with "pedantic" mathematical precision and formality.
Still I congratulate Michel on his careful reading of our paper which certainly brought up some points which would need clarification for non-experts in the field. Really, ever result likes this needs to be pulbished in two versions: one for the physicists, one for the mathematicians. However, I would say that a competent and knowledgable mathematician can make the translation from physics language to mathematics language without any difficulty, hence it is not even interesting, as pure mathematics; while the physicist doesn't understand the proof anyway - they just need to get the general idea, understand the main results, and hope that the referees did a good job checking math details. I can assure you, they did - they were very nit-pickish, but they too understood the conflicting constraints on a letter written by mathematicians with a somewhat more mathematical content than usual in europhysics letters... I think we tried PRL first but got completely supid referee reports and there was no point in wasting time arguing with them ... the editor seemed totally uninterested, anyway. Obviously this is more of a European than a US subject. Indeed, the Europeans continue to be the leaders in the field. The US physicists did not catch on till it was a bit late.