Boole vs. Bell - the latest paper of De Raedt et al

[DrChinese]

...Again, this is exactly what local realism is all about. It is not a hidden assumption and it is not true in general if local realism condition is violated. Testing this assumption is exactly what Bell tests are about. ...

At the same time it is essential that $\{ S_{1,\alpha}, S_{2,\alpha}, S_{3,\alpha} \}$ and $\rho(s_{1},s_{2},s_{3}|abc)$ are purely mathematical artefacts with no physical meaning attached to them. The triples, which might exist in theory, can never be measured (nor do they need to be). The result of Bell's derivation contains only expectations of pairwise correlations P(a,b), P(a,c) and P(b,c), and these can be easily measured in separate experiments. ...

, Doc, you forced me to go through this paper again. It's a mess. It hurts my brain.

More later
DK

Sorry 'bout that!

I like your analysis, although you refer to things a little differently than I.

Everyone tends to define realism a bit differently. The thing I require is that a) it be presented as some reasonable and specific (mathematical) requirement for a local realistic candidate model; b) it leads to some prediction which is inconsistent with QM. If you never offer a decent a), then you can't call yourself a local realist (in my book).

So the triples (which exist in principle but as you say do not need to be discoverable in practice) do not exist which are consistent with QM per above. I mean, to me, it is really just that simple. The thing that shocks me (that few seem to get) is that you only need to see that QM correctly predicts the cos^2 correlation function to get the experimental support. The whole Bell inequality thing is overblown, as you don't need to compare pairs of doubles or anything like that. That is purely for show.

And I cannot but imagine that the EPR authors would have agreed had known about Bell. Actually, Rosen knew about it quite well and later wrote: "Bell showed that assuming locality leads to a disagreement with quantum mechanics." He lived to 1996 and was an advisor to the highly respected Asher Peres.

 

[Delta Kilo]

What do you want to achieve with an example that, as we all know, doesn't work?

Well, the iron bar works in a sense that it obeys local realism and the original Bell's inequality. At the same time it has apparent problems with De Raedt's. The point was to highlight the differences in assumptions (the existence of hidden symmetry assumption in EBBI).

Section 2 is supposed to be Boollean logic, similar to 1+1=2. If you say that according to experimental data 1+1=/=2 (or more precisely, eq.16 is not obeyed) then logically there is a something wrong with your data. The only alternative is that De Raedt's derivation is erroneous so that his eq.16 is not similar to 1+1=2 (then where is the error?).

The precondition of De Raedt's inequality is the existence of triples. No triples means the rest is not applicable. There is nothing wrong with the data and there is nothing wrong with the math.

However, in most discussions of Bell's Theorem this is not regarded as essential for the inequality, because an appropriate rotation of reference angle on one side turns a perfect anti-correlation into a perfect correlation. Also in the little debate in this thread between Billschneider and DrC they agreed on an example with a perfect correlation. You could emulate that for your iron bar example with a +/- inverter on one side.

Yes, you could, but what would it do to the physical meaning of the triple of results? It is already a strain to attach a meaning to a triple of results (or a joint probability of such a triple) in symmetrical case, the anti-symmetric case is even further from reality. E.g. a triple {x,y,z} roughly translates into the following: "you measure (a,b) you get {x,-y} but if instead you had measured (b,c) you would get {y,-z}", all that while you know that you cannot measure both at the same time. This is a purely artificial construct with no physical meaning attached to it at all.

And that's the whole point. The pre-condition for De Raedt's EBBI is the existence of triples (or a joint probability of such triples). I maintain that these are mathematical artifacts. They don't make any physical sense, they cannot be physically measured or observed. Because of that it is impossible to tell whether the assumptions hold or not by just looking at the experimental setup, their possible existence must be deduced by other means. In particular, Bell's assumption of local realism provides such a missing link.

From the truth table of this example and following their second variation, I see a perfect anti-correlation in Lille and Lyon for each patient.

Erhm, are we looking at the same table here? I was referring to the table I on page 26. Lille is 1 and Lyon is 2. On the same day we have $A^{1}_{a}=A^{2}_{a}$, $A^{1}_{b}=-A^{2}_{b}$, $A^{1}_{c}=A^{2}_{c}$. Where do you see perfect anti-correlation?

What De Raedt et al argue, I think, is that Bell's inequalities are a restriction of EBBI (eq.16) and that EBBI can never be broken. You could convince me that that is wrong by giving real detailed data that break EBBI and not Bell's inequality.

EBBI can never be broken as long as the pre-condition of existence of triples (or their joint probability) holds. Since these triples are contrived and unphysical, the validity of the pre-condition (and therefore the applicability of EBBI) cannot be easily deduced from experiment physical setup. Likewise, if you see EBBI violations in a two-outcome system it just means that some artificially-constructed function is not positive everywhere. So you don't have a well-defined joint probability of thee values which never exist together at the same time anyway, so what is the significance attached to it? Who cares?

In other words: EBBI are only useful as long as you can attach physical meaning to the precondition. Eg. if you know that your two-outcome experiment is in fact producing 3 symmetrical outcomes but only 2 are available for measurement then yes, EBBI are applicable and useful. However, most of the time the reverse is not true: you cannot really derive any useful physical meaning from the (non)-violation of EBBI.

On the other hand, Bell's assumptions are explicit and physical. Their violation by QM actually tells you something important.

Regards
DK

 

[harrylin]

[... quickly getting to the elephant in the room:]
The authors have missed an elephant in the room here. The existence (in theory) of $\{ A(\vec{a},\lambda), A(\vec{b},\lambda), A(\vec{c},\lambda) \}$ follows directly from Bell's assumption of local realism, that is from the definition of $A(\vec{a},\lambda)$. Basically independence of $A(a,\lambda)$ from settings on the other end means we can measure $b$ but could have measured $c$ and the result for $a$ would have been the same. As a result we can write joint probability distribution $\rho(s_{1},s_{2},s_{3}|abc)=\int \rho(\lambda) \frac{1+s_{1} A(a,\lambda)}{2}\frac{1+s_{2} A(b,\lambda)}{2}\frac{1+s_{3} A(c,\lambda)}{2} d\lambda$.
Again, this is exactly what local realism is all about.
It is not a hidden assumption and it is not true in general if local realism condition is violated. Testing this assumption is exactly what Bell tests are about.
[rearrange:]
Since the authors [STRIKE]blindly refuse to see[/STRIKE] do not see [STRIKE]the elephant in the room[/STRIKE] the connection between local realism assumption and the separability of outcomes/existence of triples/joint probability, the austors wrongly conclude that Bell is just a consequence of EBBI and since EBBI are sometimes violated, then Bell's inequality violation is also not such a big deal.

Actually, you here point to the elephant in the room that most followers of Bell overlook but of which De Raedt is well aware; it just happens that he decided not to emphasize it in his last paper that is discussed here. To my embarrassment, I saw that elephant only this week, and only after it was pointed out to me. My only excuse for overlooking such a big mistake is that my probability calculation skills are (were) very rusty and based on a poor formulation of the product rule.
So, if this hasn't been discussed yet, we should start a topic on it!

At the same time it is essential that $\{ S_{1,\alpha}, S_{2,\alpha}, S_{3,\alpha} \}$ and $\rho(s_{1},s_{2},s_{3}|abc)$ are purely mathematical artefacts with no physical meaning attached to them. The triples, which might exist in theory, can never be measured (nor do they need to be). The result of Bell's derivation contains only expectations of pairwise correlations P(a,b), P(a,c) and P(b,c), and these can be easily measured in separate experiments.

What this all means is in general EBBI are not appicable to all experiments with 2 outcomes exactly because the triples are not always guaranteed to exist together. It is Bell's local realism assumption (plus hidden symmetry assumption) which provides sufficient condition for such triples to exist.
However the authors failed to realize this connection.

They certainly realize that connection and, if I understand it correctly, they hold that the same is true for Bell's assumptions. Definitely we should start a thread on Bell's assumptions.

 

[harrylin]

Well, the iron bar works in a sense that it obeys local realism and the original Bell's inequality. At the same time it has apparent problems with De Raedt's. The point was to highlight the differences in assumptions (the existence of hidden symmetry assumption in EBBI).

OK. Probably a little confusion of terms had slipped in: your iron bar model fails to work for Bell's requirement that it should not obey his inequality.

The precondition of De Raedt's inequality is the existence of triples. No triples means the rest is not applicable. There is nothing wrong with the data and there is nothing wrong with the math.

OK. The triples are possible experiences at three angles (or twice three angles); that corresponds perfectly with Bell's approach which is based on probability calculus. However, for some reason the authors decided not to discuss that point in this paper. I'm afraid that that choice wasn't helpful...
Correction: They do mention just that, in "Relation to Bell’s work": "class of probabilistic models that form the core of Bell’s work". See also next.

[about the theoretically similar problem of perfect symmetry]
Yes, you could, but what would it do to the physical meaning of the triple of results? It is already a strain to attach a meaning to a triple of results (or a joint probability of such a triple) in symmetrical case, the anti-symmetric case is even further from reality. E.g. a triple {x,y,z} roughly translates into the following: "you measure (a,b) you get {x,-y} but if instead you had measured (b,c) you would get {y,-z}", all that while you know that you cannot measure both at the same time. This is a purely artificial construct with no physical meaning attached to it at all.

And that's the whole point. The pre-condition for De Raedt's EBBI is the existence of triples (or a joint probability of such triples). I maintain that these are mathematical artifacts. They don't make any physical sense, they cannot be physically measured or observed. [..]

Isn't that also the point of De Raedt, concerning measurement data? Note however that possible measurements at three angles does make physical sense for a realistic model, and for which Bell starts with a probability analysis.

Erhm, are we looking at the same table here? I was referring to the table I on page 26. Lille is 1 and Lyon is 2. On the same day we have $A^{1}_{a}=A^{2}_{a}$, $A^{1}_{b}=-A^{2}_{b}$, $A^{1}_{c}=A^{2}_{c}$. Where do you see perfect anti-correlation?

In the second variant, Lille (1) measures a and b, and Lyon (2) measures b and c.
And I am looking at p.26, table I. The only measurement results that can be compared are for patients from Brasil (b):

Even (Lille, Lyon) = (+1, -1)
Odd (Lille, Lyon) = (-1, +1)

EBBI can never be broken as long as the pre-condition of existence of triples (or their joint probability) holds. Since these triples are contrived and unphysical, the validity of the pre-condition (and therefore the applicability of EBBI) cannot be easily deduced from experiment physical setup. Likewise, if you see EBBI violations in a two-outcome system it just means that some artificially-constructed function is not positive everywhere. So you don't have a well-defined joint probability of thee values which never exist together at the same time anyway, so what is the significance attached to it? Who cares?
[..] you cannot really derive any useful physical meaning from the (non)-violation of EBBI.

Perhaps the authors want you to see that and they argue that this is the same for Bell's inequalities, for they write:
"these EBBI express arithmetic relations between numbers that can never be violated by a mathematically correct treatment of the problem [...] In the original EPRB thought experiment, one can measure pairs of data only, making it de-facto impossible to use Boole’s inequalities properly."

On the other hand, Bell's assumptions are explicit and physical. Their violation by QM actually tells you something important.
[...]

I now think that De Raedt et al agree in principle with that; however they hope that it will tell you something that they already knew, which is that Bell's treatment of the problem cannot have been correct.

 

[Delta Kilo]

OK. The triples are possible experiences at three angles (or twice three angles); that corresponds perfectly with Bell's approach which is based on probability calculus.

Well, no. There is no such thing as possible experience at three angles in Bell's work. The term is meaningless because the apparatus only has 2 outputs and 2 angle settings.
Besides, the original Bell test is anti-symmetrical, and there is simply no way to build a consistent triple for that.

Correction: They do mention just that, in "Relation to Bell’s work": "class of probabilistic models that form the core of Bell’s work".

There is very little (if any) relation to Bell's work in this section. For a start, when De Raedt writes down eqs (49) claiming them to be Bell's, he [STRIKE]sneakily[/STRIKE] quietly drops the minuses (cf. Bell's eq (14) ), thereby converting anti-symmetric case into symmetric one. He never mentiones symmetry requirement in his work, so either he is oblivious to it's importance, or he is intentionally avoiding it because it raises awkward questions.

Isn't that also the point of De Raedt, concerning measurement data?

Sorry but I struggle to see his point. His [STRIKE]stream of consciousness[/STRIKE] logic seems to go like that:
a) EBBIs are always correct by design.
b) Bell's inequality is equivalent to EBBI.
c) Despite (a), EBBIs are shown to fail, including in artificial examples such as doctors/patients
d) Because of (a), failure (c) can ony mean one thing: the data is wrong because it does not come in triples
e) Because of (b), it is assumed that (d) is applicable to Bell's inequality as well.
f) Because of all of the above, experimental violation of Bell's inequaluty is simply brushed off as not applicable.

However there are quite a few problems with this:

a) EBBI are correct only as long as pre-conditions are satisfied, including existence of joint probability of triples and hidden symmetry assumption. The existence of said joint probability is difficult to establish in practice since it is a mathematical artefact which has no immediate physical interpretation.
b) Not quite. Bell's assumptions are a lot more clear-cut and physical than EBBI ones. However Bell's assumption of local realism can be used to construct EBBI's joint probability function.
c) The author repeatedly shoots himself in the foot because he does not seem to understand his own hidden assumptions (eg. symmetry). Some of the examples are simply plain wrong.
d) In order to explain (c) a trick of words is introduced: the requirement for joint probability function $f^{(3)}(s_{1},s_{2},s_{3})$ to merely exist theoretically (that is be non-negative and sum up to 1) is replaced with much stronger requirement for the actual triples of data $\Upsilon^{(3)}$ to exist experimentally. That is, all sorts of things are explained away with a sweeping phrase "the data must come in triples".
e) Bell's explicit assumptions are suffitient. There is no need to actually provide the (non-existent) triples.
f) Because of (e) Bell's violations actually prove "spooky actions at a distance".

Note however that possible measurements at three angles does make physical sense for a realistic model, and for which Bell starts with a probability analysis.

In the second variant, Lille (1) measures a and b, and Lyon (2) measures b and c.
And I am looking at p.26, table I. The only measurement results that can be compared are for patients from Brasil (b):

Even (Lille, Lyon) = (+1, -1)
Odd (Lille, Lyon) = (-1, +1)

This is incompatible with EBBI's hidden symmetry assumption. If we assume the existence of a triple (a,b,c), which b should we pick, the one from Lille or the opposite one from Lyon?

Perhaps the authors want you to see that and they argue that this is the same for Bell's inequalities, for they write:
"these EBBI express arithmetic relations between numbers that can never be violated by a mathematically correct treatment of the problem [...] In the original EPRB thought experiment, one can measure pairs of data only, making it de-facto impossible to use Boole’s inequalities properly."

Bell != Boole. Bell has extra assumption which allows one to work with pairs not triples.

I now think that De Raedt et al agree in principle with that; however they hope that it will tell you something that they already knew, which is that Bell's treatment of the problem cannot have been correct.

No, Bell's treatment is OK, it's their treatment of Bell's treatment which is not correct .

 

[Delta Kilo]

What this all means is in general EBBI are not appicable to all experiments with 2 outcomes exactly because the triples are not always guaranteed to exist together. It is Bell's local realism assumption (plus hidden symmetry assumption) which provides sufficient condition for such triples to exist.
However the authors failed to realize this connection.

They certainly realize that connection and, if I understand it correctly, they hold that the same is true for Bell's assumptions.

Well, I dunno. If they do realize it, how can they say:

From our work above it is then an immediate corollary that Bell’s inequalities cannot be violated; not even by influences at a distance.

PS: Their "Extended EPR experiment" (FIG 4) is just so wrong, it's not funny anymore. Do they seriously suggest measuring the spin of the same particle in directions $\vec{b}$ and $\vec{c}$ simultaneously?

DK

 

[harrylin]

[I wrote, not you:] "Note however that possible measurements at three angles does make physical sense for a realistic model, and for which Bell starts with a probability analysis."

Well, no. There is no such thing as possible experience at three angles in Bell's work. The term is meaningless because the apparatus only has 2 outputs and 2 angle settings.

There are on each side three possible angle settings, and only one of them can be chosen on each side for each measurement. Bell called two of those three angles a subsample, and all three of the angles he called a whole sample.

Besides, the original Bell test is anti-symmetrical, and there is simply no way to build a consistent triple for that.

There is very little (if any) relation to Bell's work in this section. For a start, when De Raedt writes down eqs (49) claiming them to be Bell's, he [STRIKE]sneakily[/STRIKE] quietly drops the minuses (cf. Bell's eq (14) ), thereby converting anti-symmetric case into symmetric one. He never mentiones symmetry requirement in his work, so either he
is oblivious to it's importance, or he is intentionally avoiding it because it raises awkward questions.

Bell called that difference "a trivial one" and I think that he was right about that: if I'm not mistaken, you only need to rotate the detector references relative to each other by an certain angle to make the test perfectly symmetrical. Do you think that a simple change of coordinates can affect the possibilities?

Sorry but I struggle to see his point. His [STRIKE]stream of consciousness[/STRIKE] logic seems to go like that: [ ... some good points]

I now think that both of us did not understand this paper's arguments! I'm afraid that we skipped too fast over the introduction:
"Should the EBBI be violated, the logical implication is that one or more of the necessary conditions to prove these inequalities are not satisfied. As these conditions do not refer to concepts such as locality or macroscopic realism, no revision of these concepts is necessitated by Bell’s work."
More about that in my next post.

[about the observation data (+1, -1) and (-1, +1) in (Lille, Lyon) for patients from Brasil:]
This is incompatible with EBBI's hidden symmetry assumption. If we assume the existence of a triple (a,b,c), which b should we pick, the one from Lille or the opposite one from Lyon?

I don't know what you mean with "existence of a triple (a,b,c)". a b and c are three groups of patients, and in the two cities only one patient is selected each day. I think that De Raedt et al only wanted to illustrate how the grouping of triples in pairs can lead to unexpected and magical looking results if there is an unknown common cause that is not accounted for.

PS: Their "Extended EPR experiment" (FIG 4) is just so wrong, it's not funny anymore. Do they seriously suggest measuring the spin of the same particle in directions $\vec{b}$ and $\vec{c}$ simultaneously?
DK

Not at the same time, but sequentially. I dunno, perhaps DrC has an opiniion?

 

[harrylin]

Well, I dunno. If they do realize [the connection between EBBI and Bell], how can they say:

From our work above it is then an immediate corollary that Bell’s inequalities cannot be violated; not even by influences at a distance.

In fact it was also not clear to me! But it now appears to me that just as Bell threw a theorem at Einstein et al, De Raedt et al now throw a theorem of their own at Bell.

The RHM theorem (please correct me if my paraphrasing of p.11-12 is wrong):

If one makes the same assumptions* as Bell, then it immediately follows from RHM's derivations that Bell’s inequalities cannot be violated; not even by influences at a distance.

*the implicit assumptions of Bell's inequalities according to RHM:
- the existence of identical elements of reality for each of the three angle pairs.
- these elements are dichotomic variables that follow the algebra of integers.

In other words, according to RHM:

- influences at a distance cannot explain a violation of Bell's inequalities
- only an inappropriate grouping in pairs can lead to a violation of such inequalities.

Cheers,
Harald

 

[DrChinese]

...
- influences at a distance cannot explain a violation of Bell's inequalities
...

You have wondered so far from the proper assessment of the situation with Bell, it is hard to know where to start. So I won't. But it doesn't take a genius to figure out that action at a distance can ALWAYS be invoked as an explanation for entanglement. Of course, that does not make it so.

Just remind yourself that in all this, there are NO local realistic theories on the table for anyone to even discuss at this point in time. They have all been soundly refuted by experiment (and I don't just mean Bell tests). So that should tell you how ridiculous this entire line of reasoning is.

And that is being kind, I must be feeling good today.

 

[harrylin]

[..] it doesn't take a genius to figure out that action at a distance can ALWAYS be invoked as an explanation for entanglement. Of course, that does not make it so.

It's certainly not about the question if action at a distance can be invoked as an explanation for entanglement. As yous say, it doesn't take a genius to figure out that action at a distance can ALWAYS be invoked as an explanation for entanglement.

Their argument as I now understand it, is that Bell used his inequality to argue that "action at a distance" occurs; and now dRHM use their similar inequality as evidence that Bell's argument is erroneous, because according to them the violation or not of such an inequality has nothing to do with "action at a distance" or "no action at a distance".

As they put it:

"A violation of the EBBI cannot be attributed to influences at a distance. The only possible way that a violation could arise is if grouping is performed in pairs"

As none of us had fully understood their argument, so far also none of us have checked their derivation to see if their theorem is solid or not.

Cheers,
Harald

PS Delta Kilo and I would be grateful if you can answer his question about the "Extended EPR experiment" (FIG 4)

 

[harrylin]

Interesting further input from another thread:

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. [..]

Bell relied on the fact that deterministically $A^{1}_{\mathbf{b}}(w) = A^{2}_{\mathbf{b}}(w)$; and while it is generally held that the sign isn't important, De Raedt reproduced similar observables in his illustration. However:

[..] Bell derived some inequalities for the case where $\langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{b}} \rangle = -1$. You can alternatively derive some similar but not identical inequalities for the case where $\langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{b}} \rangle = +1$. The particular inequality that de Raedt et. al. considered is derived assuming $\langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{b}} \rangle = +1$, and there is simply no reason to expect it should be satisfied if $\langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{b}} \rangle = -1$.

Specifically, if you assume $\langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{b}} \rangle = -1$, you can derive the following four inequalities:
$$\begin{eqnarray} \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{b}} \rangle + \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{c}} \rangle - \langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{c}} \rangle &\geq& -1 \,, \\ \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{b}} \rangle - \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{c}} \rangle + \langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{c}} \rangle &\geq& -1 \,, \qquad (*) \\ \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{b}} \rangle + \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{c}} \rangle + \langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{c}} \rangle &\leq& +1 \,, \\ \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{b}} \rangle - \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{c}} \rangle - \langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{c}} \rangle &\leq& +1 \,. \qquad (*) \end{eqnarray}$$
The second and fourth of these inequalities, which I've marked (*), are the ones Bell derived in 1964. Specifically, they're equivalent to Eq. (15) of Bell's 1964 paper [1]. The other two can easily be derived in an analogous manner (or, alternatively, just by flipping the sign of $A^{2}_{\mathbf{c}}$).

If you instead set $\langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{b}} \rangle = +1$, you get four slightly different inequalities, which you can basically all derive by flipping the sign on $A^{1}_{\mathbf{b}}$:
$$\begin{eqnarray} \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{b}} \rangle + \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{c}} \rangle + \langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{c}} \rangle &\geq& -1 \,, \qquad (\#) \\ \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{b}} \rangle - \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{c}} \rangle - \langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{c}} \rangle &\geq& -1 \,, \\ \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{b}} \rangle + \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{c}} \rangle - \langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{c}} \rangle &\leq& +1 \,, \\ \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{b}} \rangle - \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{c}} \rangle + \langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{c}} \rangle &\leq& +1 \,. \end{eqnarray}$$
The first of these, (#), is the one that de Raedt et. al. tested.
[..][1] J. S. Bell, Physics 1 3 195--200 (1964).

I had overlooked that the inequality that De Raedt gave as example is one of Boole - and not exactly one of Bell. Thanks for pointing that out!

So, he merely wanted to illustrate how that kind of inequalites (Boole/Bell) can be broken with local realism, if applied in the peculiar manner of Bell. And it appears to me that Boole did not assume a certain outcome result; according to the presentation, the Boole inequality of eq.113 in De Raedt's paper must be valid for all proper pair combinations, no matter what the products are. But instead of lingering on that point, for this discussion it will be interesting to test Bell's inequality (his equation no.15) on De Raedt's illustration.

Now, it looks to me that your representation here above of Bell's original inequality is still not quite right: an absolute sign is lacking. According to my copy, Bell's eq.15 for locations 1 and 2 is (rearranged):

$$\begin{eqnarray} |\langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{b}} \rangle - \langle A^{1}_{\mathbf{a}} A^{2}_{\mathbf{c}} \rangle| - \langle A^{1}_{\mathbf{b}} A^{2}_{\mathbf{c}} \rangle &\leq& +1 \,. \qquad \end{eqnarray}$$

And the same for location pairs (1,3) and (2,3).

Here are the fictive measurement results once more, for locations 1-3 on even and odd days:

... Even ...|.. Odd
L ...1...2...3.|..1...2...3
A_a +1 +1 +1.| -1. -1. -1
A_b +1. -1 +1.| -1 +1. -1
A_c. -1. -1. -1.|+1 +1 +1

Computing from the results for location pair (1,2), I obtain as outcomes: +1, -1.
That location pair does not break Bell's inequality, the average is 0.

For location pair (2,3), I obtain as outcomes: +1, +1. Average +1.
Also no breaking of Bell's inequality.

For location pair (1,3), I obtain as outcomes: +3, +3. Average +3.
If I'm not mistaken, this pair very strongly breaks Bell's inequality!

Thus it's easy to modifiy De Raedt's illustration for Bell's original inequality: just take Lille=1, Lyon=3.

Now, it's a bit of a weak point that this effect is not homegeneous; but while unrealistic for Lille and Lyon, we can imagine a random fluctuation of such funny properties between all locations. Let's see what that gives for the average result of all locations:

(0 + 1 + 3) / 3 = 4/3

Thus, Bell's inequality applied on that refined illustration, gives according to me (I may have made an error of course):

4/3 <= 1
Obviously that inequality is broken.

In conclusion, it still looks to me that De Raedt's modified illustration with patients does show how inequalites like those of Bell can be broken with local realism.

 

[DrChinese]

In conclusion, it still looks to me that De Raedt's modified illustration with patients does show how inequalites like those of Bell can be broken with local realism.

This is a bit of sleight of hand. We are talking about quantum particles, not patients and doctors. The DrChinese challenge, as applied to this scenario is, becomes:

a) Give me any local realistic sample you care to invent.
b) I get to pick what to measure in any particular trial. I will do this "randomly" as long as the sample is not overly cherry picked. Obviously since I get to see the data before I pick, I can always cheat but I agree not to unless your success depends on me selecting a particular set of measurement to make.
c) It must satisfy the perfect correlations condition so as to imply the existence of hidden variables. IE When I pick the same attribute to observe at both spots, I get the same answer.
d) And as Delta Kilo has pointed out, there must be at least 3 choices of things for me to measure (per b).

De Raedt's modified illustration may look one way to you, but that view won't be shared by most.

 

[harrylin]

This is a bit of sleight of hand. We are talking about quantum particles, not patients and doctors. The DrChinese challenge, as applied to this scenario is, becomes:

a) Give me any local realistic sample you care to invent.
b) I get to pick what to measure in any particular trial. I will do this "randomly" as long as the sample is not overly cherry picked. Obviously since I get to see the data before I pick, I can always cheat but I agree not to unless your success depends on me selecting a particular set of measurement to make.
c) It must satisfy the perfect correlations condition so as to imply the existence of hidden variables. IE When I pick the same attribute to observe at both spots, I get the same answer.
d) And as Delta Kilo has pointed out, there must be at least 3 choices of things for me to measure (per b).

De Raedt's modified illustration may look one way to you, but that view won't be shared by most.

That illustration comes close doing that - but obviously it was not intended to address the "DrChinese challenge".

 

[harrylin]

[concerning the quote "From our work above it is then an immediate corollary that Bell’s inequalities cannot be violated; not even by influences at a distance.":]

[..] Bell's inequalities are most certainly violated by influences at a distance. [...]

What they meant was not clear to me until now; it may be that they explained it, but it wasn't clear to me. However, they also refer to papers by Accardi.and just now morrobay put our attention to a pre-print by him, in which that point is explained clearer IMHO.

Accardi finds that with Bell's assumptions about probability distributions the Bell inequality must even hold for non-local processes. - p.8 of http://arxiv.org/pdf/quant-ph/0007005v2.pdf
I suppose that that is also what De Raedt means.

PS: I also found an interesting monograph by Gill contra Accardi: http://www.jstor.org/stable/4356235

 

[stevendaryl]

[concerning the quote "From our work above it is then an immediate corollary that Bell’s inequalities cannot be violated; not even by influences at a distance.":]What they meant was not clear to me until now; it may be that they explained it, but it wasn't clear to me. However, they also refer to papers by Accardi.and just now morrobay put our attention to a pre-print by him, in which that point is explained clearer IMHO.

Accardi finds that with Bell's assumptions about probability distributions the Bell inequality must even hold for non-local processes. - p.8 of http://arxiv.org/pdf/quant-ph/0007005v2.pdf
I suppose that that is also what De Raedt means.

I'm getting several different threads about Bell's inequalities mixed up. I thought I had given an example to the contrary.

Bell's inequality is proved under the assumption that joint probabilities can be written in this form:

$P(A \wedge B | \alpha \wedge \beta) = \sum_\lambda P(\lambda) P(A | \lambda \wedge \alpha) P(B | \lambda \wedge \beta)$

where $A, B$ are the results at the two detectors, $\alpha, \beta$ are the settings of the two detectors, and $\lambda$ is the hidden variable. Note that the conditional probability for $A$ depends only on $\lambda$ and $\alpha$, but not $\beta$. The conditional probability for $B$ does not depend on $\alpha$.

If you allow nonlocal interactions, then a more general expression is possible, that is still a "realistic hidden-variables" theory:

$P(A \wedge B | \alpha \wedge \beta) = \sum_\lambda P(\lambda) P(A | \lambda \wedge \alpha \wedge \beta) P(B | \lambda \wedge \alpha \wedge \beta)$

You can certainly violate Bell's inequalities with a realistic model of this form. To give a simple example:

Let $P(\lambda) = 1$ for $0 \leq \lambda \leq 1$
Let $P(A | \lambda \wedge \alpha \wedge \beta) = 1$ for $\lambda \leq \frac{1}{2}$ and 0 otherwise.
Let $P(B | \lambda \wedge \alpha \wedge \beta) = 1$ for $\frac{1}{2} sin^2(\frac{1}{2}(\beta - \alpha)) \leq \lambda \leq \frac{1}{2} (1 + sin^2(\frac{1}{2}(\beta - \alpha)))$ and 0 otherwise.

This model reproduces exactly the predictions of QM for the spin-1/2 twin-pair EPR experiment, and violates Bell's inequality.

 

[harrylin]

[..] I thought I had given an example to the contrary.

Bell's inequality is proved under the assumption that joint probabilities can be written in this form:

$P(A \wedge B | \alpha \wedge \beta) = \sum_\lambda P(\lambda) P(A | \lambda \wedge \alpha) P(B | \lambda \wedge \beta)$

where $A, B$ are the results at the two detectors, $\alpha, \beta$ are the settings of the two detectors, and $\lambda$ is the hidden variable. Note that the conditional probability for $A$ depends only on $\lambda$ and $\alpha$, but not $\beta$. The conditional probability for $B$ does not depend on $\alpha$.

If you allow nonlocal interactions, then a more general expression is possible, that is still a "realistic hidden-variables" theory:

$P(A \wedge B | \alpha \wedge \beta) = \sum_\lambda P(\lambda) P(A | \lambda \wedge \alpha \wedge \beta) P(B | \lambda \wedge \alpha \wedge \beta)$

You can certainly violate Bell's inequalities with a realistic model of this form. [..]

At first sight, I see no disagreement between these statements of yours (incl. your example) and theirs.

What De Raedt seems to argue (and probably what he and others have shown), is that in order to break such an inequality one must have a joint probability that differs from the one that Bell assumed for deriving that inequality. That is not only true if the model is local (an option that Bell found hard to imagine) but even if the model is non-local. Do you disagree with that?

 

[DrChinese]

Accardi finds that with Bell's assumptions about probability distributions the Bell inequality must even hold for non-local processes. - p.8 of http://arxiv.org/pdf/quant-ph/0007005v2.pdf
I suppose that that is also what De Raedt means.

That doesn't even make sense (that they must apply to non-local theories as they do local ones). And that is acknowledged in EPR.

The point of a non-local realistic theory is that there is a relationship between distant objects and a mechanism whereby there is mutual instantaneous influence. No counterfactual measurement of the pair is possible due to the importance of the mutual influence. There is no product state by definition.

 

[harrylin]

That doesn't even make sense (that they must apply to non-local theories as they do local ones). And that is acknowledged in EPR.

The point of a non-local realistic theory is that there is a relationship between distant objects and a mechanism whereby there is mutual instantaneous influence. No counterfactual measurement of the pair is possible due to the importance of the mutual influence. There is no product state by definition.

I take it that you are saying that what De Raedt et all apparently mean (according to me) makes no sense according to you.

It looks to me that EPR's opinions are not taken as authority by De Raedt or Accardi (or Bell). Are you saying that it is impossible to create a non-local model (thus with influence at a distance) that adheres to Bell's assumptions for local models so that it can't break Bell's inequality? I guess not, for then you'd likely have stated that what they claim is wrong. Maybe you simply don't see the point that they made?

 

[stevendaryl]

At first sight, I see no disagreement between these statements of yours (incl. your example) and theirs.

What De Raedt seems to argue (and probably what he and others have shown), is that in order to break such an inequality one must have a joint probability that differs from the one that Bell assumed for deriving that inequality. That is not only true if the model is local (an option that Bell found hard to imagine) but even if the model is non-local. Do you disagree with that?

I'm not sure exactly what you're saying. As I said, the point of locality is that it allows you to assume that the conditional probability for Alice's result depends only on Alice's settings (and the shared hidden variable) while the conditional probability for Bob's result depends only on Bob's settings (and the shared hidden variable). If Alice's result depends on Bob's setting, or Bob's result depends on Alice's setting, then you can't derive Bell's inequality. So it seems to me that locality is pretty important.

There is an assumption made by Bell, which he expounds on his "theory of local be-ables", that in a realistic setting, the probability for an event should only depend on facts about the causal past of that event (where "causal past" means "past lightcone", if there are no faster-than-light influences). In other words, if I have complete information about the state of the universe in the causal past of an experiment, then I have the most information possible about the possible results of the experiment. Facts about regions of the universe that are not in the causal past of the experiment are only relevant in that they reveal facts about the causal past.

For instance, if I put a $1 bill and a $10 bill into two identical white envelopes, and give one to Alice and another to Bob, and they separate and open their envelopes, the knowledge that Bob found a $1 bill in his envelope tells me something about what Alice will find in her envelope. But the complete description of the causal past of Alice's envelope includes a specification of what bill was put into it. So if you had complete information about the causal past of Alice's envelope, knowledge about Bob's envelope would tell you nothing new.

In other words, Bell's assumption is basically that all state information about the universe is localized. There are no "nonlocal" facts that can't be factored into a collection of local facts.

 

[harrylin]

I'm not sure exactly what you're saying. As I said, the point of locality is that it allows you to assume that the conditional probability for Alice's result depends only on Alice's settings (and the shared hidden variable) while the conditional probability for Bob's result depends only on Bob's settings (and the shared hidden variable). If Alice's result depends on Bob's setting, or Bob's result depends on Alice's setting, then you can't derive Bell's inequality. So it seems to me that locality is pretty important.

Certainly, for Bell "locality" (as well as what he understood with "realism") was important to motify the separation of variables. Now, as the title of his paper suggests, De Raedt considers there such inequalities in the broader mathematical framework of Boole. That framework does not depend on such concepts as locality; what really matters are the conditional probabilities themselves.

There is an assumption made by Bell, which he expounds on his "theory of local be-ables", that in a realistic setting, the probability for an event should only depend on facts about the causal past of that event (where "causal past" means "past lightcone", if there are no faster-than-light influences). In other words, if I have complete information about the state of the universe in the causal past of an experiment, then I have the most information possible about the possible results of the experiment. Facts about regions of the universe that are not in the causal past of the experiment are only relevant in that they reveal facts about the causal past.

For instance, if I put a $1 bill and a $10 bill into two identical white envelopes, and give one to Alice and another to Bob, and they separate and open their envelopes, the knowledge that Bob found a $1 bill in his envelope tells me something about what Alice will find in her envelope. But the complete description of the causal past of Alice's envelope includes a specification of what bill was put into it. So if you had complete information about the causal past of Alice's envelope, knowledge about Bob's envelope would tell you nothing new.

In other words, Bell's assumption is basically that all state information about the universe is localized. There are no "nonlocal" facts that can't be factored into a collection of local facts.

That looks good to me - and I guess also to De Raedt. What he apparently tried to do is to make people think "outside of the box"; in this case, the "box" is the particular EPR setting and subsequent reasoning of Bell. Sometimes that helps to get a fresh look at puzzles like these.

 

[DrChinese]

I take it that you are saying that what De Raedt et all apparently mean (according to me) makes no sense according to you.

It looks to me that EPR's opinions are not taken as authority by De Raedt or Accardi (or Bell). Are you saying that it is impossible to create a non-local model (thus with influence at a distance) that adheres to Bell's assumptions for local models so that it can't break Bell's inequality? I guess not, for then you'd likely have stated that what they claim is wrong. Maybe you simply don't see the point that they made?

Not so much taking issue with you as the idea that non-local theories must adhere to the same requirements as local ones. Non-local theories can break a Bell Inequality because it violates the assumption of measurement/device independence. Ie a measurement choice here does affect an outcome there.

Obviously de Raedt et al are making a point that I don't think stands, sure it is possible I don't really understand it. I am not a poster boy for non-local theories anyway.

Accardi makes a lot of points I think are either wrong or irrelevant too. Example being his chameleon analogy, which like the Doctor/Patients analogy does not come close to addressing Bell. (Although perhaps Boole...) Sadly, many writers fail to play devil's advocate against their own position and end up far down the creek without a paddle.

 

[audioloop]

Two years ago an intriguing paper of De Raedt's team concerning Bell's Theorem appeared in Europhysics Letters (http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.0767v2.pdf).

Now (officially next month), an elaboration on those ideas has been published:

Hans De Raedt et al: "Extended Boole-Bell inequalities applicable to quantum theory"
J. Comp. Theor. Nanosci. 8, 6(June 2011), 1011
http://www.ingentaconnect.com/content/asp/jctn/2011/00000008/00000006/art00013

Full text also in http://arxiv.org/abs/0901.2546

De Raedt et al do not pretend to be the first to discuss these issues, and they refer to quite a number of earlier papers by other authors that bring up similar points.

Below I present a little summary of their very elaborated explanations.

It all looks very plausible to me since I tend regard Bell's Theorem as a magician's trick - we tend to interpret a miracle as a trick, even if nobody can explain how the trick is done. Now, this paper appears to explain "how it's done" and I like to hear if there are valid objections.

Before we discuss their criticism about Bell's "element of reality", it may be good to discuss Boole's example of patients and illnesses, which De Raedt et all reproduce in this paper. They show that by failing to account for unknown causes for the observations, similar inequalities can be drawn up as those of Bell, without a valid reason to infer a spooky action at a distance - although it appears that way.

Does anyone challenge the correctness of that claim?

Regards,
Harald

--------------------------------------------------------
Abstract:
We address the basic meaning of apparent contradictions of quantum theory and probability frameworks as expressed by Bell's inequalities. We show that these contradictions have their origin in the incomplete considerations of the premises of the derivation of the inequalities. A careful consideration of past work, including that of Boole and Vorob'ev, has lead us to the formulation of extended Boole-Bell inequalities that are binding for both classical and quantum models. The Einstein-Podolsky-Rosen-Bohm gedanken experiment and a macroscopic quantum coherence experiment proposed by Leggett and Garg are both shown to obey the extended Boole-Bell inequalities. These examples as well as additional discussions also provide reasons for apparent violations of these inequalities.

The above summary is IMHO a rather "soft" reflection of its contents: the way I read it, basically this paper asserts to show that Bell's theorem is wrong! It does this in an elaborate way, here are some fragments of the text (the below is copied from the ArXiv version):

"the Achilles heel of Bell's interpretations: [..] all of Bell's derivations assume from the start that ordering the data into triples as well as into pairs must be appropriate and commensurate with the physics. [..] From our work above it is then an immediate corollary that Bell's inequalities cannot be violated; not even by influences at a distance."

The paper next discusses such things as "Filtering-type measurements on the spin of one spin-1/2 particle", "Application to quantum flux tunneling", "Application to Einstein-Podolsky-Rosen-Bohm (EPRB) experiments" (in particular Stern-Gerlach).

To top it off, illustrations of apparent Bell violations are given, even of a similar inequality with "a simple, realistic every-day experiment involving doctors who perform allergy tests on patients". [..] "Together these examples represent an infinitude of possibilities to explain apparent violations of Boole-Bell inequalities in an Einstein local way." Special attention is given to "EPR-Bohm experiments and measurement time synchronization".

"It is often claimed that a violation of such inequalities implies that either realism or Einstein locality should be abandoned. As we saw in our counterexample which is both Einstein local and realistic in the common sense of the word, it is the one to one correspondence of the variables to the logical elements of Boole that matters when
we determine a possible experience, but not necessarily the choice between realism and Einstein locality."
[..]
"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."

The -IMHO- most important conclusion of the paper is that "A violation of the Extended Boole-Bell inequalities cannot be attributed to influences at a distance"; they argue that a violation only can arise from a grouping in pairs.

interesting paper:

The irrelevance of Bell inequalities in Physics.
"Comments on the Extended Boole-Bell Inequalities Applicable to Quantum Theory"
http://hal.archives-ouvertes.fr/docs/00/82/41/24/PDF/RHM-hans.pdf

"the violation of the celebrated Bell inequalities in quantum mechanics is due only to a rather elementary, even if somewhat subtle error made in the way the statistical data are handled"



.

 

[stevendaryl]

interesting paper:

The irrelevance of Bell inequalities in Physics.
"Comments on the Extended Boole-Bell Inequalities Applicable to Quantum Theory"
http://hal.archives-ouvertes.fr/docs/00/82/41/24/PDF/RHM-hans.pdf

"the violation of the celebrated Bell inequalities in quantum mechanics is due only to a rather elementary, even if somewhat subtle error made in the way the statistical data are handled"



.

So there are papers saying that Bell is wrong, because it's easy for a local hidden variables theory to violate the inequalities. Then there are other papers saying that Bell is wrong because nothing can violate the inequalities, not even quantum mechanics.

 

[harrylin]

interesting paper:

The irrelevance of Bell inequalities in Physics.
"Comments on the Extended Boole-Bell Inequalities Applicable to Quantum Theory"
http://hal.archives-ouvertes.fr/docs/00/82/41/24/PDF/RHM-hans.pdf

"the violation of the celebrated Bell inequalities in quantum mechanics is due only to a rather elementary, even if somewhat subtle error made in the way the statistical data are handled"
.

OK I see that it's a commentary on De Raedt's paper, and it addresses the same rather obscure conclusion that I discussed in post #154. Thanks!

So there are papers saying that Bell is wrong, because it's easy for a local hidden variables theory to violate the inequalities. Then there are other papers saying that Bell is wrong because nothing can violate the inequalities, not even quantum mechanics.

Hmm yes at first sight it looks to me that that commentary exaggerates quite a bit! However, the point that De Raedt made is also here: such inequalities are purely mathematical, so that in order to break them one has to break one of the mathematical conditions on which they are based.
Of course, Bell never pretended otherwise; it's just a thing not to forget.

 

[stevendaryl]

OK I see that it's a commentary on De Raedt's paper, and it addresses the same rather obscure conclusion that I discussed in post #154. Thanks!

Hmm yes at first sight it looks to me that that commentary exaggerates quite a bit! However, the point that De Raedt made is also here: such inequalities are purely mathematical, so that in order to break them one has to break one of the mathematical conditions on which they are based.
Of course, Bell never pretended otherwise; it's just a thing not to forget.

The condition that must be broken to violate the inequality is locality. If the probability distribution for Alice's result depends on Bob's detector settings, then there is no reason for the inequality to hold.

 

[harrylin]

The condition that must be broken to violate the inequality is locality. [..]

If so, then De Raedt's illustration that I just discussed in post #151 is "non-local"?!

 

[DrChinese]

The irrelevance of Bell inequalities in Physics.
"Comments on the Extended Boole-Bell Inequalities Applicable to Quantum Theory"
http://hal.archives-ouvertes.fr/docs/00/82/41/24/PDF/RHM-hans.pdf

"the violation of the celebrated Bell inequalities in quantum mechanics is due only to a rather elementary, even if somewhat subtle error made in the way the statistical data are handled"

Hmmm, takes 20 pages and 64 formulae to correct an "elementary" error.

I will definitely add this to my pantheon of "why Bell is wrong/misguided/etc" links. Each one with a completely different critique, and all equally well accepted*.


*Not.

 

[stevendaryl]

If so, then De Raedt's illustration that I just discussed in post #151 is "non-local"?!

I have to admit that I didn't read that in any kind of detail, so I can't comment. Past efforts on my part to understand papers that claim to refute Bell have all ended in frustration, because the authors almost always end up proving something that is beside the point. But for the sake of the discussion, I guess I can try once again with De Raedt's example.

Is there a definitive statement of what the example is and what it shows? Or can you just summarize it here? The post that you pointed to seems to start in the middle.

 

[stevendaryl]

EPR Challenge

This picture illustrates the challenge for a local hidden-variables explanation for the spin-1/2 twin-pair EPR experiment: Is it possible to simulate the quantum mechanical prediction using nonquantum means?

What would be sufficient to disprove Bell's claims would be to write three computer programs of the following type:

1. Generator(i): computes the ith value for λ, where λ is a floating point number (I'm assuming that any other reasonable type of value can be "encoded" into a real.
2. Detector_A(α, λ): takes a pair α, λ, where α is a detector orientation (chosen by Alice), and λ is the output of Generator.
3. Detector_B(β, λ): takes a pair β, λ, where βis a detector orientation (chosen by Bob), and λ is the output of Generator.

For a large number of rounds i (enough that there are good statistics for various pairs of α and β), have Alice and Bob randomly choose α_i and β_i, respectively, and have the Generator randomly choose λ_i. Record the outputs A_i and B_i. Then compute statistics:

P(A, B | α, β) = N'/N
P(A, $\neg$B | α, β) = N''/N
P($\neg$A, B | α, β) = N'''/N
P($\neg$A, $\neg$B | α, β) = N''''/N

where N = the number of rounds i such that α_i = α,
β_i = β, and where

• N' = the number of those rounds such that A_i = +1, B_i = +1,
• N'' = the number of those rounds such that A_i = +1, B_i = -1,
• N''' = the number of those rounds such that A_i = -1, B_i = +1,
• N'''' = the number of those rounds such that A_i = -1, B_i = -1,

The claim is that no matter what programs are used, you will not get

P(A, B | α, β) = 1/2 sin²(θ/2)
P(A, $\neg$B | α, β) = 1/2 cos²(θ/2)
P($\neg$A, B | α, β) = 1/2 cos²(θ/2)
P($\neg$A, $\neg$B | α, β) = 1/2 sin²(θ/2)

(If you want, you can add more inputs to the detectors to represent local randomness.)

Mucking about with Bell's inequalities is a waste of time, it seems to me. The bottom line is really the nonexistence of three programs that would reproduce the predictions of QM. And there are no such programs.

Now, you could try messing with the requirements. For instance, you can say that the detectors sometimes output a "null" value, rather than +1 or -1. Or you can say that (as someone, maybe De Raedt, suggested), you can say that occasionally, Alice's detector or Bob's detector gets the wrong λ; maybe Alice gets λ_i while Bob gets λ_i+1. I don't have an opinion about whether such generalizations could allow a better simulation of QM.

 

[harrylin]

I have to admit that I didn't read that in any kind of detail, so I can't comment. Past efforts on my part to understand papers that claim to refute Bell have all ended in frustration, because the authors almost always end up proving something that is beside the point. But for the sake of the discussion, I guess I can try once again with De Raedt's example.

Is there a definitive statement of what the example is and what it shows? Or can you just summarize it here? The post that you pointed to seems to start in the middle.

Sure. First of all, for the context: I gave summary of the paper under discussion here in my first post of this thread:
https://www.physicsforums.com/showthread.php?t=499002

And after you asked me in the other thread, I summarized that simple example for you as follows
(https://www.physicsforums.com/showthread.php?t=697939&page=3):

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

Perhaps the weakest point of that example is that the freely chosen detector position of Bell tests with anti-correlation is not fully matched by it. And it is still unclear to me if that is impossible to implement in an example, or only difficult to do. Consequently, the question is for me still open if Bell's assumptions about local realism were valid or not.

However, as wie brought up that the inequality in DeRaedt's paper does not exactly match equation 15 of Bell 1964, I re-analyzed that simple illustration with that inequality in post #151 here.
- https://www.physicsforums.com/showthread.php?p=4465579

[..]

This picture illustrates the challenge for a local hidden-variables explanation for the spin-1/2 twin-pair EPR experiment: Is it possible to simulate the quantum mechanical prediction using nonquantum means?

What would be sufficient to disprove Bell's claims would be to write three computer programs of the following type:

[..]

Mucking about with Bell's inequalities is a waste of time, it seems to me. The bottom line is really the nonexistence of three programs that would reproduce the predictions of QM. And there are no such programs.

That it's very difficult to write such a set of computer programs is well known (although Accardi apparently claims to have done it). In fact, it was already known that it's very difficult to come up with a fitting "local realistic" model, and therefore Bell came up with his famous inequality which he claimed cannot be broken by such a model. Searching for such programs is a waste of time if Bell was right.
It would also be sufficient to disprove Bell's claims by giving an example that does what he claims to be impossible: breaking his inequality with an example that uses no "spooky action at a distance". The topic of De Raedt's paper under discussion in this thread happens to relate to that claim about inequalities; efforts to come up with an impossible(?) program are discussed in other papers.

Now, you could try messing with the requirements. For instance, you can say that the detectors sometimes output a "null" value, rather than +1 or -1. Or you can say that (as someone, maybe De Raedt, suggested), you can say that occasionally, Alice's detector or Bob's detector gets the wrong λ; maybe Alice gets λ_i while Bob gets λ_i+1. I don't have an opinion about whether such generalizations could allow a better simulation of QM.

Certainly any simulation about what could be realistic must account for anything that could significantly influence the results in reality. But I also don't know what may matter and what not.

 

[harrylin]

I found two journal articles by other authors that refer to De Raedt et al's Boole/Bell paper (I just found them now):

http://www.ingentaconnect.com/content/asp/jctn/2011/00000008/00000006/art00012

ABSTRACT
We discuss the connection of a violation of Bell's inequality and the non-Kolmogorovness of statistical data in the EPR-Bohm experiment. We emphasize that nonlocalty and "death of realism" are only sufficient, but not necessary conditions for non-Kolmogorovness. Other sufficient conditions for non-Kolmogorovness and, hence, a violation of Bell's inequality can be found. We find one important source of non-Kolmogorovness by analyzing the axiomatics of quantum mechanics. We pay attention to the postulate (due to von Neumann and Dirac) on simultaneous measurement of quantum observables given by commuting operators. This postulate is criticized as nonphysical. We propose a new interpretation of the Born-von Neumann-Dirac rule for the calculation of the joint probability distribution of such observables. A natural physical interpretation of the rule is provided by considering the conditional measurement scheme. We use this argument (i.e., the rejection of the postulate of simultaneous measurement) to provide a motivation for the non-Kolmogorovness of the probabilistic structure of the EPR-Bohm experiment. and
http://iopscience.iop.org/1402-4896/2012/T151/014007

ABSTRACT
In the given controversy, Einstein was right; the Copenhagen quantum mechanics has been based on physically unacceptable assumptions. And also later, Bell's inequalities have been mistakenly interpreted: holding true only in the classically deterministic model and not for the Schrödinger solutions when the initial state of the evolving system is represented by a (not fully known) set of different classical states; and the measured results in individual events are statistically distributed. The structure of Hilbert space formed by the solutions of the corresponding Schrödinger equation cannot be arbitrarily defined; it must be adapted to the corresponding physical system. Any Schrödinger state is then equivalent to a superposition of the solutions of the corresponding Hamilton equations, while all solutions of these equations form a greater set. However, the usual energy quantization approach represents phenomenological characteristics only, and the proper cause should be interpreted on other physical grounds. The actual source of quantum phenomena may hardly be explained without the participation of all interactions between the corresponding physical objects; their not yet fully known properties surely play an important role. The results obtained in experiments when the mutual (mainly elastic) collisions of the corresponding particles are studied might surely be very helpful.

PS I hit a broken link in the second paper, and found the new address:
http://www.cost.eu/domains_actions/mpns/Actions/MP1006

 

[stevendaryl]

Sure. First of all, for the context: I gave summary of the paper under discussion here in my first post of this thread:
https://www.physicsforums.com/showthread.php?t=499002

And after you asked me in the other thread, I summarized that simple example for you as follows
(https://www.physicsforums.com/showthread.php?t=697939&page=3):

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 :

I find De Raedt's writing almost incomprehensible. If he has a point, it'll take me a while to discover it.

 

[harrylin]

I find De Raedt's writing almost incomprehensible. If he has a point, it'll take me a while to discover it.

I find that true for most papers on this topic; however it looks to me that his simple illustration is easy to verify - it's literally as simple as 1+1 (only more elaborated).

 

[stevendaryl]

I find that true for most papers on this topic; however it looks to me that his simple illustration is easy to verify - it's literally as simple as 1+1 (only more elaborated).

Well, what it seems to me is that he is describing a deterministic function

$A(x,l,e) = +/- 1$

where

• $x$ = $a$, $b$, or $c$ (country of the patient's birth)
• $l$ = $1$, $2$, or $3$ (city where the patient is tested),
• $e$ = even or odd, depending on the day the test is given

This seems like a completely straight-forward "hidden variables" model to me. What De Raedt does with this model is to arrange for certain subsets of the triples $\langle x, l, e \rangle$ to produce the appearance of a non-local interaction. Okay. What this shows is that the criterion for what's a non-local interaction has to be formulated in a way that is insensitive to such subsetting. That's sort of an interesting point, but as I have said several times, what's of interest is not whether a particular inequality holds or not, it's whether the predicted QM results can be explained in terms of a local model. I don't see that De Raedt is shedding any light on that.

 

[harrylin]

Well, what it seems to me is that he is describing a deterministic function

$A(x,l,e) = +/- 1$

where

• $x$ = $a$, $b$, or $c$ (country of the patient's birth)
• $l$ = $1$, $2$, or $3$ (city where the patient is tested),
• $e$ = even or odd, depending on the day the test is given

This seems like a completely straight-forward "hidden variables" model to me. What De Raedt does with this model is to arrange for certain subsets of the triples $\langle x, l, e \rangle$ to produce the appearance of a non-local interaction. Okay. What this shows is that the criterion for what's a non-local interaction has to be formulated in a way that is insensitive to such subsetting. That's sort of an interesting point, but as I have said several times, what's of interest is not whether a particular inequality holds or not, it's whether the predicted QM results can be explained in terms of a local model. I don't see that De Raedt is shedding any light on that.

The topic happens to be inequalities, and in particular the one of Bell; however his other two examples shed some light on QM results. I did not (yet) study those simply because it takes some time to do and his particle model of light is not much to my liking.

Meanwhile I suddenly hit on an Arxiv paper that describes a classical (and straightforward) computer simulation of the Malus-law coincidence + breaking of Bell inquality in optical experiments(!); however I don't know if it has been officially published. It refers to a journal paper of 1996 that describes a demonstration of an EPRB-like experiment with LED's, but not breaking Bell's inquality. So it's not clear yet if I found material for a new topic on this forum...