## A new simulation of the EPR-Bohm correlations

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

### Re: A new simulation of the EPR-Bohm correlations

Joy Christian wrote:
Jochen wrote:See, I'm used to thinking about this in operational terms. You have two boxes, each of which takes either of two inputs, and produces one of two outputs. This is the level at which Bell inequalities are derived---whatever mechanism gives rise to the outcomes is irrelevant. If (and only if) those boxes can generate a violation of some Bell inequality without knowing about each other's inputs, you'll have shown that there is a local realistic model that can reproduce quantum mechanical predictions.

I find this argument very strange, if not simply wrong.
Neither quantum mechanics nor local realism has anything to do with such boxes. Quantum mechanics or its local-realistic alternative has to do with correlations observed in Nature.

This is not a contradiction. The experiments could be, in principle, done inside the boxes. All what is necessary is an earlier preparation step where one puts one of the two particles of a pair in each box.

Then, the box should contain some input possibility - which defines the direction of measurement - and a simple one bit output.

Joy Christian wrote:It takes only half a page of elementary calculation to show that these correlations can be explained within a manifestly local-realistic model:
http://arxiv.org/abs/1103.1879.

Laughable, not more.

Joy Christian wrote:If, however, the refutation of Bell's theorem presented in the above paper (or any other refutation of Bell's theorem for that matter) is not backed up by a computer simulation of boxes and outputs you describe, then that says nothing whatsoever about Nature. It only proves limitations of computers, programs, and programmers.

It says a lot, because a local realistic theory can be modelized on a computer, at least in principles (of course, one can invent local realistic theories which require computations beyond the ability of existing computers, no problem), and Bell's theorem is not about Nature but about the possibility to describe certain (quite simple) observations using an Einstein-causal realistic theory.

So, if your formulas cannot be implemented this way, this supports only what is obvious anyway - they are nonsense.
Schmelzer

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### Re: A new simulation of the EPR-Bohm correlations

Joy Christian wrote:PS: I will not be responding to any idiotic comments by the flatlanders.
Joy Christian
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### Re: A new simulation of the EPR-Bohm correlations

Jochen wrote:See, I'm used to thinking about this in operational terms. You have two boxes, each of which takes either of two inputs, and produces one of two outputs. This is the level at which Bell inequalities are derived---whatever mechanism gives rise to the outcomes is irrelevant. If (and only if) those boxes can generate a violation of some Bell inequality without knowing about each other's inputs, you'll have shown that there is a local realistic model that can reproduce quantum mechanical predictions.

I find this argument very strange, if not simply wrong.

Neither quantum mechanics nor local realism has anything to do with such boxes. Quantum mechanics or its local-realistic alternative has to do with correlations observed in Nature. It takes only half a page of elementary calculation to show that these correlations can be explained within a manifestly local-realistic model:

http://arxiv.org/abs/1103.1879 (see also this paper: http://arxiv.org/abs/1501.03393 and this simulation: http://challengingbell.blogspot.co.uk/2 ... f-joy.html).

If, however, the refutation of Bell's theorem presented in the above paper (or any other refutation of Bell's theorem for that matter) is not backed up by a computer simulation of boxes and outputs you describe, then that says nothing whatsoever about Nature. It only proves limitations of computers, programs, and programmers.
Joy Christian
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### Re: A new simulation of the EPR-Bohm correlations

All what has to be said about this variant of "refutation", where A, B are not +-1 valued functions, has been said in the comments of the last link.
Schmelzer

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### Re: A new simulation of the EPR-Bohm correlations

Schmelzer wrote:

All what has to be said about this variant of "refutation", where A, B are not +-1 valued functions, has been said in the comments of the last link.

Once again, complete and utter hogwash from the clueless flatlander.

As even a blind person can see from reading the above papers, the measurement variables A and B are nothing but +/-1 valued functions!!!

Evidently, the flatlander is entirely clueless about my 3-sphere model. He has either never bothered to read a single one of my papers, or if he has tried, then he is clearly incapable of understanding what has been discussed extensively in my papers and books, which can all be found here: http://libertesphilosophica.info/blog/.

As I have said time and again, bogus claims about my work and bogus attacks on me are not going to validate his pet "ether theory" with the voodoo of "non-locality."
Joy Christian
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### Re: A new simulation of the EPR-Bohm correlations

Joy Christian wrote:
Schmelzer wrote:All what has to be said about this variant of "refutation", where A, B are not +-1 valued functions, has been said in the comments of the last link.

Once again, complete and utter hogwash from the clueless flatlander.
As even a blind person can see from reading the above papers, the measurement variables A and B are nothing but +/-1 valued functions!!!
Evidently, the flatlander is entirely clueless about my 3-sphere model.

I have to acknowledge that I'm indeed completely clueless about models of whatever where $A\in\{ \pm 1\}$ and $B\in \{\pm 1\}$ and $AB \in \{\pm 1\}$ but is a completely different point as in http://libertesphilosophica.info/blog/wp-content/uploads/2013/02/3sphere7.png.
Schmelzer

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### Re: A new simulation of the EPR-Bohm correlations

Schmelzer wrote:I have to acknowledge that I'm indeed completely clueless about models of whatever where $A\in \pm 1$ and $B\in \pm 1$ and $AB \in \pm 1$ but is a completely different point as in http://libertesphilosophica.info/blog/wp-content/uploads/2013/02/3sphere7.png.

You are completely clueless, period. Stop making bogus claims about my model.
Joy Christian
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### Re: A new simulation of the EPR-Bohm correlations

Jochen wrote:
FrediFizzx wrote:
Jochen wrote:Could you at least provide a link to the argument?

viewtopic.php?f=6&t=75&p=4528#p4528

The argument hinges on a couple of elementary misconceptions; however, Schmelzer seems to have dispensed with most of them quite handily, so I don't really know what I could add to the discussion there.

If you really think so then please post how quantum mechanics violates Bell's inequality. It can't. It is impossible for anything to violate it.
FrediFizzx
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### Re: A new simulation of the EPR-Bohm correlations

Joy Christian wrote:Stop making bogus claims about my model.

Stop making bogus claims that it has something to do with Bell's theorem. Then I will, of course, stop to make claims about it.
Schmelzer

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### Re: A new simulation of the EPR-Bohm correlations

FrediFizzx wrote:If you really think so then please post how quantum mechanics violates Bell's inequality. It can't. It is impossible for anything to violate it.

Use whatever introduction into QM to find out that it predicts for P(a,b) = -ab. Then compute yourself if this formula fulfills for small angles the Bell inequality
$1+ P(b,c) \ge |P(a,b)-P(a,c)|$.

The three values P(a,b), P(a,c), P(b,c) are, of course, obtained from different measurements. A mathematical triviality are the BI only in the case that all three results are given and the same experiment is used to measure all three P(a,b), P(a,c), P(b,c). If in each experiment only two of the three values are measured, a violation is in principle possible, and the BI are no longer a mathematical triviality.

Bell has derived predetermination of the results from Einstein causality and the EPR argument. This has allowed him to prove the theorem even if the expectation values P(a,b), P(a,c), P(b,c) are measured in different measurements.
Schmelzer

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### Re: A new simulation of the EPR-Bohm correlations

Schmelzer wrote:Bell has derived predetermination of the results from Einstein causality and the EPR argument. This has allowed him to prove the theorem even if the expectation values P(a,b), P(a,c), P(b,c) are measured in different measurements.

Bell's so-called "theorem" is based on elementary mathematical and conceptual mistakes: http://libertesphilosophica.info/blog/d ... orem-book/.

It has been disproven many times over, with an explicit local-realistic model for ALL quantum correlations (not just EPR-Bohm), contradicting the so-called "theorem":

viewtopic.php?f=6&t=115&start=30#p3977,

http://arxiv.org/abs/1405.2355.

Moreover, explicit, numerical, event-by-event simulations of this theoretical model for the special case of EPR-Bohm correlation have also been presented:

http://rpubs.com/jjc/84238,

http://arxiv.org/abs/1501.03393,

http://libertesphilosophica.info/blog/,

It is highly irresponsible, unethical, and unscientific to dismiss this vast body of evidence by making disingenuous and bogus claims about it without making any serious effort of studying it, as some irresponsible Bell-devotees have been doing. In fact, in my opinion such irresponsible actions are manifestations of scientific misconduct:

Joy Christian
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### Re: A new simulation of the EPR-Bohm correlations

Joy Christian wrote:
Bell's so-called "theorem" is based on elementary mathematical and conceptual mistakes: http://libertesphilosophica.info/blog/d ... orem-book/.

It has been disproven many times over, with an explicit local-realistic model for ALL quantum correlations (not just EPR-Bohm), contradicting the so-called "theorem":

viewtopic.php?f=6&t=115&start=30#p3977,

http://arxiv.org/abs/1405.2355.

Moreover, explicit, numerical, event-by-event simulations of this theoretical model for the special case of EPR-Bohm correlation have also been presented:

http://rpubs.com/jjc/84238,

http://arxiv.org/abs/1501.03393,

http://libertesphilosophica.info/blog/,

It is highly irresponsible, unethical, and unscientific to dismiss this vast body of evidence by making disingenuous and bogus claims about it without making any serious effort of studying it, as some irresponsible Bell-devotees have been doing. In fact, in my opinion such irresponsible actions are manifestations of scientific misconduct:

The vast body of incontrovertible evidence I refer to is the one I have presented over the past eight years: http://arxiv.org/find/all/1/au:+Christi ... /0/all/0/1.

There has not been a single valid argument presented against my work to date. Not a single. Only bogus and straw-man arguments have been presented, and they have all been debunked as such by me, as can be verified from the list of my papers linked above, as well as from many responses I have posted on this very forum.

Therefore I repeat:

It is highly irresponsible, unethical, and unscientific to dismiss this vast body of evidence by making disingenuous and bogus claims about it without making any serious effort of studying it, as some irresponsible Bell-devotees have been doing. In fact, in my opinion such irresponsible actions are manifestations of scientific misconduct:

Last edited by Joy Christian on Sat Jun 27, 2015 8:15 am, edited 1 time in total.
Joy Christian
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### Re: A new simulation of the EPR-Bohm correlations

Joy Christian wrote:There has not been a single valid argument presented against my work to date. Not a single. Only bogus and straw-man arguments have been presented, and they have all been debunked as such by me, as can be seen from the list of my papers linked above, as well as from many responses I have posted on this very forum.

Since you automatically label any arguments againts your "model" as either bogus or straw-man, arguing with you is pointless. We just note that in the seven or eight years since you first published your "theory" on arXiv, no one has followed up. Your papers are not cited in any meaningful way. Eight years after Einstein published his 1905 papers, he was universally acknowledged as one of the most brilliant physiscists of his time. Say no more.
Heinera

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### Re: A new simulation of the EPR-Bohm correlations

Heinera wrote:
Joy Christian wrote:There has not been a single valid argument presented against my work to date. Not a single. Only bogus and straw-man arguments have been presented, and they have all been debunked as such by me, as can be seen from the list of my papers linked above, as well as from many responses I have posted on this very forum.

Since you automatically label any arguments againts your "model" as either bogus or straw-man, arguing with you is pointless. We just note that in the seven or eight years since you first published your "theory" on arXiv, no one has followed up. Your papers are not cited in any meaningful way. Eight years after Einstein published his 1905 papers, he was universally acknowledged as one of the most brilliant physiscists of his time. Say no more.

And how many years after the ground-breaking publication of his book was Alfred Wegener universally acknowledged as one of the most brilliant scientists of his time?

Here is what I mean by straw-man fallacy: https://www.youtube.com/watch?v=v5vzCmURh7o. How many of my critics even mention S^3 in their criticism of my work?

I have yet to come across a single one of my critics who has any understanding of what S^3, or a parallelized 3-sphere is: http://arxiv.org/abs/1405.2355.
Joy Christian
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### Re: A new simulation of the EPR-Bohm correlations

Joy Christian wrote:It is highly irresponsible, unethical, and unscientific to dismiss this vast body of evidence by making disingenuous and bogus claims about it without making any serious effort of studying it, as some irresponsible Bell-devotees have been doing. In fact, in my opinion such irresponsible actions are manifestations of scientific misconduct:

Given that my answer to this accusation has been deleted by the administration, it seems time to leave this forum, where I do not even have the right to defend myself against accusations of scientific misconduct.

Joy Christian, wish you a lot of fun in future with your interesting social experiments with your "true believers".

I do not wait until I will be banned, so, if there appears something new, instead of the endless repetition of errors identified long ago, I have at least the possibility to return and answer - at least if somebody else informs me about this.

Good bye
Schmelzer

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### Re: A new simulation of the EPR-Bohm correlations

Schmelzer wrote:Given that my answer to this accusation has been deleted by the administration, it seems time to leave this forum, where I do not even have the right to defend myself against accusations of scientific misconduct.

For the record, Schmelzer's response included some name calling and ad hominem attack on me, so I reported his post to the administration.
Joy Christian
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### Re: A new simulation of the EPR-Bohm correlations

Joy Christian wrote:
Jochen wrote:See, I'm used to thinking about this in operational terms. You have two boxes, each of which takes either of two inputs, and produces one of two outputs. This is the level at which Bell inequalities are derived---whatever mechanism gives rise to the outcomes is irrelevant. If (and only if) those boxes can generate a violation of some Bell inequality without knowing about each other's inputs, you'll have shown that there is a local realistic model that can reproduce quantum mechanical predictions.

I find this argument very strange, if not simply wrong. Neither quantum mechanics nor local realism has anything to do with such boxes. Quantum mechanics or its local-realistic alternative has to do with correlations observed in Nature.

Well, it's a standard way of thinking about this in the quantum info community. Its virtue is precisely that it is wholely independent of the underlying implementation. The reasoning is that, if you are committed to the existence of predetermined values for all observables (i.e. realism), then, if they are not disturbed by the act of measurement, you can find a joint probability distribution $P(A,B,C,D)$; and as shown above, such a joint distribution, no matter how the outcomes are generated, is a sufficient condition for Bell inequalities holding.

If you now say you have a local realistic strategy that nevertheless violates Bell inequalities, then there is some recipe to calculate, from just the hidden variables and the local input of one party, the outcome of every experiment they could perform; that is, there is a function only of the hidden variables and the input of one party such that that parties outcomes are reproduced by that function. So all you have to do is build a box that implements this strategy---either by simulation (which is possible if there is such a recipe for calculation), or by some actual physical system. If you can do this such that neither box has knowledge of the input to the other, then (and only then) will you have shown that there is a local realistic model for quantum correlations.

It takes only half a page of elementary calculation to show that these correlations can be explained within a manifestly local-realistic model: http://arxiv.org/abs/1103.1879.

And you know very well that not everybody accepts these calculations as correct. But if you could show such a two-boxes implementation of your recipe, there would not be any doubt: because the impossibility of such boxes is just what Bell's theorem asserts.

FrediFizzx wrote:
Jochen wrote:The argument hinges on a couple of elementary misconceptions; however, Schmelzer seems to have dispensed with most of them quite handily, so I don't really know what I could add to the discussion there.

If you really think so then please post how quantum mechanics violates Bell's inequality. It can't. It is impossible for anything to violate it.

Well, Bell inequalities are statements about a certain kind of theories, where the observed probabilities always form a convex polytope. Their violation merely says that quantum mechanics is not such a theory.

Let me illustrate that. Take two propositions, $A$ and $B$, which are about $\{+1,-1\}$-valued observables $\mathbf{A}$ and $\mathbf{B}$, and assert '$\mathbf{A}$ is $+1$' and '$\mathbf{B}$ is $+1$' respectively. This could, for instance, simply be independent coin throws. Then form the conjunction $A \wedge B$, which asserts '$\mathbf{A}$ is $+1$ and $\mathbf{B}$ is $+1$'. We can construct the truth table for our propositions (I couldn't get the array-environment to work):

Code: Select all
 A | B | A^B-----------0 | 0 |  01 | 0 |  00 | 1 |  01 | 1 |  1

Now, each row contains a possible assignment of values, and all rows togethern contain all possible assignments. We can now construct probability distributions over these assignments: say, $pr_1 + (1-p)r_2$, where $r_i$ is the row $i$, means that with probability p, all observables are not $+1$, while with probability (1-p), $A$ is $+1$. Plainly, the most general probability distribution on our system is $P=\sum_i\lambda_ir_i$, where $0\leq\lambda_i\leq 1$ and $\sum_i\lambda_i=1$. This equation defines a convex polytope (actually a simplex) in three dimensions, the so-called correlation polytope. All admissible correlations of the system lie in this polytope, and thus, one will only ever observe probability distributions such that they can be written in the above form. Note here that these are statistical predictions: the only way to have access to them is to carry out a sufficiently large number of measurements on identically prepared systems.

Or at least so one once thought, until quantum mechanics came around. Because note that in the above derivation, we have made the implicit assumption that all of $\mathbf{A}$ and $\mathbf{B}$ have definite values; only this made it possible to derive the correlation polytope. If we drop this assumption, and instead calculate the probabilities using the quantum mechanical prescription $p(A)=\mathrm{tr}(\Pi_A^+\rho)$, where $\Pi_A^+$ is the projector onto the $+1$-eigenspace of $\mathbf{A}$, then we get a different---and actually larger---convex body, of which the classical correlation polytope is merely a subset.

What does this have to do with Bell inequalities? Well, polytopes have two ways of characterizing them: via their vertices, as I have done, and via their faces. These faces are given by inequalities: anything that's larger than a certain value is outside the polytope, e.g. These inequalities are exactly Bell's inequalities. Hence, their derivation hinges on the assumption of definite values; and thus, in theoris where that assumption is violated, as it is in quantum mechanics, there is no reason for Bell inequalities to hold---and indeed, they are violated.

Now you're probably wondering that I haven't said anything about locality. That's because that really only comes in once one tries to give the inequalities---which are just equivalent to the mathematical statement that there exists a joint probability distribution for all variables, which one gets from the assumption that there are definite values and that measurements are non-disturbing---any operational or empirical content. In reality, there are many ways for the assumption that there is a joint PD to fail: if my measurement of $A$ influenced the likelihood of $B$ coming up $+1$, for instance. So, you must introduce an auxiliary assumption in order to make testable predictions, which is also the reason that hidden-variable theories per se can never be excluded. There are three basic ways to make this assumption:

• Macroscopic realism: essentially, you assume that a system is always in one of the states available to it (corresponding to having always one fixed set of responses to all possible measurements), and that measurements carried out at different points in time do not influence one another. This gives you the Leggett-Garg inequalities.
• Noncontextuality: observables that mutually commute do not influence one another; hence, the value you obtain for $A$ is independent of whether you measure it together with $B$ or $C$, which constitute the so-called measurement context. This gives you the Kochen-Specker theorem.
• Locality: measurements performed in spacelike separation don't influence one another, thus Bob's outcomes are independent of Alice's choices. You can view this as a special case of noncontextuality, since operators at spacelike separation always commute. This gives you Bell's theorem.

So, it's in the following sense that quantum mechanics violates Bell inequalities, while classical mechanics doesn't: Bell inequalities are really statements about theories in which all variables have a joint probability distribution; for such theories, they hold necessarily. However, this is not something you can require in general, as any influence of one variable on another spoils it. Hence, you must make an additional assumption in order to produce testable predictions. The weakest such assumption is given by locality, since we have good reason (from special relativity) to believe that information can't be transmitted faster than light. So under the joint assumptions of realism and locality, you can justify the assumption of the existence of a joint PD.

But this need not hold: a given theory could well violate either assumption, in which case it isn't constrained by Bell inequalities. So, the experimental violation of BIs then tells us that quantum mechanics must not obey one of the assumptions. We can either give up locality, if we want to be able to claim that there is a definite value for every measurement to discover; then, we would have a theory essentially like quantum mechanics in that the space of admissible probability distributions is a convex polytope, there just isn't a single such polytope for each experiment. Or, we could give up the assumption of definite values; then, we get a theory for which the set of admissible probability distributions isn't a polytope, but a more general convex body (this can be fully characterized using convex optimization methods). Which one to choose is up to individual preferences.
Jochen

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### Re: A new simulation of the EPR-Bohm correlations

Jochen wrote:The reasoning is that, if you are committed to the existence of predetermined values for all observables (i.e. realism), then, if they are not disturbed by the act of measurement, you can find a joint probability distribution $P(A,B,C,D)$; and as shown above, such a joint distribution, no matter how the outcomes are generated, is a sufficient condition for Bell inequalities holding.

Hello Jochen and welcome to SPF. I have a few questions about what you posted above.

1) I agree that the existence of the joint probability distribution P(A,B,C,D) is the only assumption required to obtain the inequalities. Do you agree that neither locality nor hidden variables are required?

2) In an EPRB experiment, a series of 2 spin-half particles are measured at angles (a,b) to produce the joint probability distribution P(A,B) for these particles. Since the act of measurement destroys those particles. A different set of particle pairs are measured at angles (c,d) to produce the joint probability distribution P(C,D) for those particles. How is it possible to measure the joint probability distribution P(A,B,C,D) for a series of particles, if the outcomes (A,B,C,D) are never jointly measured for any of the particle pairs?

3) Finally, QM makes a prediction for the expectation value <AB> of the joint measurement at angles (a,b). Why should any model which tries to reproduce this prediction care about the joint probability distribution P(A,B,C,D), when all that is needed to calculate <AB> is P(A,B), and no experiment will ever be able to measure P(A,B,C,D)?

4) One more. Why do you care whether anything is "disturbed by the act of measurement", when we know already for a fact that the particles cease to exist after measurement?

Thanks.
minkwe

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### Re: A new simulation of the EPR-Bohm correlations

Jochen wrote:If you now say you have a local realistic strategy that nevertheless violates Bell inequalities, then there is some recipe to calculate, from just the hidden variables and the local input of one party, the outcome of every experiment they could perform; that is, there is a function only of the hidden variables and the input of one party such that that parties outcomes are reproduced by that function. So all you have to do is build a box that implements this strategy---either by simulation (which is possible if there is such a recipe for calculation), or by some actual physical system. If you can do this such that neither box has knowledge of the input to the other, then (and only then) will you have shown that there is a local realistic model for quantum correlations.

I believe I have already shown this, both by simulation,

http://rpubs.com/jjc/84238,

which may be reformulated in terms of two separate computers as you have described (although I have not shown this),

and by actual physical system,

http://arxiv.org/abs/1211.0784 and http://arxiv.org/abs/1501.03393,

which describes a classical, macroscopic EPR-B-type experiment, which I predict will "violate" the Bell inequality (or more precisely produce the strong correlation).

Jochen wrote:
• Noncontextuality: observables that mutually commute do not influence one another; hence, the value you obtain for $A$ is independent of whether you measure it together with $B$ or $C$, which constitute the so-called measurement context. This gives you the Kochen-Specker theorem.
• Locality: measurements performed in spacelike separation don't influence one another, thus Bob's outcomes are independent of Alice's choices. You can view this as a special case of noncontextuality, since operators at spacelike separation always commute. This gives you Bell's theorem.

In my view it is meaningless to consider concepts like "locality" and "contextuality" without first considering the geometrical and topological properties of the physical space in which events such as Alice's observations and Bob's observations are occurring: http://arxiv.org/abs/1405.2355.

My starting point is very different from yours. Since all events we ever observe are occurring in the physical space, my first question is: What are the properties of this physical space? On physical grounds, from our deep knowledge of general relativity (or Einstein's theory of gravity), we know that the physical space is more naturally described by S^3, not by R^3 as usually assumed. I have discussed this change from R^3 to S^3 quite extensively in many papers, so let me not repeat the details here. But once this transition is made, a manifestly local-realistic derivation of the strong correlation follows at once, as can be readily seen from what I have linked above.

In short, you and I seem to be living in two different worlds: You in R^3 and I in S^3. What you see as quantum correlation in R^3, I see as classical correlation in S^3.
Joy Christian
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### Re: A new simulation of the EPR-Bohm correlations

Jochen wrote:[*] Macroscopic realism: essentially, you assume that a system is always in one of the states available to it (corresponding to having always one fixed set of responses to all possible measurements), and that measurements carried out at different points in time do not influence one another. This gives you the Leggett-Garg inequalities.

Does that assumption imply that a second independent system similar to the first has the same single fixed set of responses to all possible measurement?

Noncontextuality: observables that mutually commute do not influence one another; hence, the value you obtain for $A$ is independent of whether you measure it together with $B$ or $C$, which constitute the so-called measurement context.This gives you the Kochen-Specker theorem.

If our system refers physically to the set of particles we measure our observables on, does the measurement of P(A,B) on the system commute with the measurement of P(A,C) on the same system? How can they commute if it is physically impossible to do both measurements on the same system? Or do you think the KS theorem applies to separate disjoint systems, for which all measurements necessarily commute?

[*] Locality: measurements performed in spacelike separation don't influence one another, thus Bob's outcomes are independent of Alice's choices. You can view this as a special case of noncontextuality, since operators at spacelike separation always commute. This gives you Bell's theorem.

Again when you see Bell's inequalities

$1 + \gte | -|$
Do the terms <BC>, <AB>, <AC> in that inequality refer simultaneously to a single system of particle pairs $\{i \to N\}$ or does the terms each refer to separate disjoint systems of particle pairs $\{i \to N\},\{j \to N\}, \{k \to N\}$ respectively? Do you see any differences between the fact that those observables commute for disjoint systems of particle pairs but do not commute for a single system of particle pairs?

So, it's in the following sense that quantum mechanics violates Bell inequalities

Do the terms <BC>, <AB>, <AC> predicted by QM all refer simultaneously to refer to a single system of particle pairs, or to 3 seperate disjoint and independent systems?

Finally, do you now appreciate the fact that the reason QM and experiments appears to violate Bell's inequalities is trivially because the terms <BC>, <AB>, <AC> in the inequalities refer simultaneously to a single system of particle pairs, which is impossible to measure in practice, but tjewhile the terms <BC>, <AB>, <AC> from QM and experiments refer to or are measured from separate independent disjoint systems of particle pairs.

If you believe the inequalities should apply to independent disjoint systems of particle pairs, please could you derive the inequalities starting from
$P(a,b) - P(a,C) = \frac{1}{N}\sum_i A_iB_i - \frac{1}{N}\sum_j A_jB_j$
minkwe

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