harrylin said:
Yes, sorry for the delay! I had to refresh my memory first.
De Raedt attempted to give a counter example to Bell's derivation method. His simple counter example is given on p.25, 26 of
http://arxiv.org/abs/0901.2546 :
In this second variation of the investigation, we let only two
doctors, one in Lille and one in Lyon perform the examina-
tions. The doctor in Lille examines randomly all patients of
types a and b and the one in Lyon all of type b and c each one
patient at a randomly chosen date. The doctors are convinced
that neither the date of examination nor the location (Lille or
Lyon) has any influence and therefore denote the patients only
by their place of birth. After a lengthy period of examination
they find
Γ(w) = Aa (w)Ab (w) + Aa (w)Ac (w) + Ab (w)Ac (w) = −3
They further notice that the single outcomes of Aa (w), Ab (w)
and Ac (w) are randomly equal to ±1. [..]
a single outcome manifests itself randomly in one city and [..]
the outcome in the other city is then always of opposite sign
That's not a counter-example.
What they claim to violate is Bell's original 1964 inequality. Bell's original inequality is something of an odd duckling in the zoology of Bell inequalities in that it relies on an extra (but entirely observable) assumption. Specifically, in their notation, and putting the locations back on (Lille = 1, Lyon = 2), the Bell inequality uses the assumption that A^{1}_{\mathbf{b}}(w) = A^{2}_{\mathbf{b}}(w). This is observable, since it implies that \langle A^{1}_{\mathbf{b}}(w) A^{2}_{\mathbf{b}}(w) \rangle = 1, and it just means that the correct way to state Bell's inequality should really be something like
\langle A^{1}_{\mathbf{a}}(w) A^{2}_{\mathbf{b}}(w) \rangle + \langle A^{1}_{\mathbf{a}}(w) A^{2}_{\mathbf{c}}(w) \rangle + \langle A^{1}_{\mathbf{b}}(w) A^{2}_{\mathbf{c}}(w) \rangle \geq -1 \quad \text{given that} \quad \langle A^{1}_{\mathbf{b}}(w) A^{2}_{\mathbf{b}}(w) \rangle = 1 \,.
Their counter-example isn't a counter-example because it has \langle A^{1}_{\mathbf{b}}(w) A^{2}_{\mathbf{b}}(w) \rangle = -1. Incidentally, if you try to read the inequality above in the same way as other Bell inequalities (i.e. without imposing a condition like A^{1}_{\mathbf{b}}(w) = A^{2}_{\mathbf{b}}(w)), then it's easy to see that its local bound is actually -3 (the same as the algebraic bound) instead of -1.
So they've demonstrated a "violation" of the 1964 Bell inequality in a way that breaks a necessary and verifiable condition for it to hold as a test of locality in the first place. Their approach simply wouldn't work for any other Bell inequality, such as CHSH, that doesn't rely on a condition like this.
Later, in section VII.B, they give another "counter-example" that similarly only works because they define their model such that E(\mathbf{b}, \mathbf{b}) = 4/\pi - 1 \neq 1. Of course, as they themselves point out, their model is incapable of violating the CHSH inequality.
On a side note, a concluding remark toward the end of section VII.A
Because no \lambda exists that would lead to a violation except a \lambda that depends on the index pairs (a, b), (a, c) and (b, c) the simplistic conclusion is that either elements of reality do not exist or they are non-local. The mistake here is that Bell and followers insist from the start that the same element of reality occurs for the three different experiments with three different setting pairs. This assumption implies the existence of the combinatorial-topological cyclicity that in turn implies the validity of a non-trivial inequality but has no physical basis. Why should the elements of reality not all be different? Why should they, for example not include the time of measurement? There is furthermore no reason why there should be no parameter of the equipment involved. Thus the equipment could involve time and setting dependent parameters such as \lambda_{\mathbf{a}}(t), \lambda_{\mathbf{b}}(t), \lambda_{\mathbf{c}}(t) and the functions A might depend on these parameters as well
reveals some basic confusions about what Bell's theorem actually implies and the assumptions underpinning it. Basically, if S is some Bell correlator that you could measure, with a local bound T_{\text{local}}, then the authors seem to be reading Bell's theorem as implying a
logical or
algebraic constraint on S:
S \leq T_{\text{local}} \,.
For certain simple correlation inequalities this holds, as the authors say, if you assume that the hidden variable \lambda is the same each time you do the test. But obviously we don't want to assume that, and what Bell's theorem actually proves is more like a bound on the
expectation value of S, i.e. something more like
\langle S \rangle \leq T_{\text{local}} \,.
This means that, according to locality, it is entirely possible to do a Bell test and measure a value for S that violates the local bound. It's just that the chance of this happening rapidly becomes very small if you do the test on (say) a very large number of entangled particles. So if you do a Bell test on a very large number of particles and get something
significantly above the local bound, the idea is that you can rule out locality with very high confidence.
Finally, the authors suggest that the hidden variable \lambda should be allowed to depend explicitly on time and possibly on the detector settings. Of course, letting \lambda depend explicitly on time doesn't really affect Bell's theorem (it holds regardless of the probability distribution \rho(\lambda) explicitly appearing in proofs of Bell inequalities, so allowing the probability distribution to change in time won't accomplish much). For a properly performed Bell test, letting \lambda depend on the detector settings is normally argued away on the basis of the so-called "free will" or "no conspiracies" assumption (it could also occur if you allow retrocausality, which is not included in the definition of Bell-locality).
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a interpretation of bell, aforegoing , bells interpretation of einstein..