Definition of "realistic" in "local-realistic"

[Joy Christian]

I made a terrible mistake in my post above.

I should have added the following prerequisites for anyone who tries to show that some "non-reality", non-locality", or lack of determinism is creeping in my derivation above:

(1) they should have some elementary training in basic calculus, in particular in what is known as "limits" in basic calculus;

(2) they should have some basic training in a non-calculus level physics, especially that concerning the laws of angular momenta;

(3) they should know something about the basic physics underlying the EPR-Bohm type experiments (nothing too sophisticated, just some basic idea);

and finally,

(4) they should actually read my paper linked above before making a fool of themselves.

With these prerequisites, Lockyer and his comments above are disqualified. I will not be responding to any further comments from him.

Needless to say, I would be quite happy to hear from anyone truly knowledgeable in the subject under discussion, and with a genuine objection.

[guest1202]

***
Below is the central derivation of the EPR-Bohm correlation from my latest paper, which predicts exactly what quantum mechanics predicts for such correlations.

Anyone still concerned about my use of "realistic", please tell us where in my derivation there is "non-reality", non-locality", or lack of determinism creeping in:

This is unreadable without definitions of the symbols. I tried without success to download the provided link to Dr. Christian's "latest paper" at the Academia website. This time nothing happened; on a previous trial the site said that in order to download it, I had to join something-or-other and to agree to a complicated legal document which would take hours to read in detail.

The paper does not seem to be in the arXiv, but I did find arXiv1103.1879v2 , provocatively titled "Disproof of Bell's Theorem", which contains a similar calculation, also with all symbols undefined. A footnote directs the reader to two other of Christian's papers for definitions. I had downloaded the most recent months ago. Whenever I download and read (often just skim) a paper, I write a brief summary for my records. The summary of that paper reads: "Almost unreadable because of undefined notation."

I imagine that the notation is defined somewhere, and even without definitions I can guess what some of the symbols might represent. However, until I find a concise, definitive source for the definitions, I wouldn't attempt to follow the calculations.

But it probably doesn't matter, because it seems unlikely that Christian's model satisfies the definition of "realistic" that I set forth in the post starting this thread. That definition was whimsically dubbed "1202realistic" to distinguish it from other inequivalent definitions such as Christian's definition "Crealistic" (which is still unknown to me). I believe that "1202realistic" is what nearly all authors mean by the "realistic" in "local-realistic model" for quantum mechanics.

To save readers the trouble of going back to that post, here in brief is the definition of a "1202realistic" model. We consider a situation in which observer Alice has a choice of two measurements, and , each yielding +1 or -1. Another oberver Bob similarly has a choice of measurements and . Quantum mechanics gives us the probability that if Alice uses measurement and Bob uses , Alice obtains result ,
and Bob obtains. Here each of and equals either +1 or -1. Quantum mechanics also gives three other similarly labeled probabilities and .

Let us write for the outcome which Alice obtains when she selects measurement and obtains result +1, with similar notation for the other possible measurement selections. Classically, besides the above probabilities given by quantum mechanics, we could also speak, for example, of the probability
that , and .
Then the four probabilities given by quantum mechanics would be marginals of .
For example, we would have



The definition of "1202realistic" is that there exists such a whose marginals are the four probability distributions given by quantum mechanics. This is the
definition of the "realistic" part of "local-realistic" which is used in almost 100% of the papers on the subject known to me, though it's often presented differently (see below). Its intuitive meaning is that Alice and Bob are measuring
things which are "real" in the sense that they have an existence independent of any measurement. An example is
whether or not one of the "quantum cakes" of Kwiat/Hardy (discussed in another post) has risen earlier than usual
halfway through the baking. Classically, we feel comfortable talking about that possibility whether or not we have
opened the oven to check.

I am quite sure that Dr. Christian's model cannot be 1202realistic. The reason is that there is a mathematical theorem with a very simple proof that for to exist with given marginals , etc., the marginals have to
satisfy the CHSH inequalities, and the marginals of quantum mechanics don't necessarily satisfy them. The proof of this theorem has been checked by thousands of mathematicians and physicists. I have never seen it questioned, including in the papers of Dr. Christian.

That does not imply that his calculations are wrong; I have no opinion on that. My impression without having available
precise definitions of the symbols, is that they are quite complicated and look enough like the traditional "hidden variable" model (which demonstrably cannot reproduce quantum mechanics) that someone unfamiliar with the subject
might think that they somehow define a "local-realistic" model.

APPENDIX ON THE TRADITIONAL "HIDDEN VARIABLE" MODEL

The above formulation of "1202realistic" in terms of marginals may be unfamiliar to many, because the majority of
papers use a superficially different, but logically equivalent, formulation. I will briefly set forth that formulation below
and explain why it is equivalent to the formulation given above. The explanation will use notation introduced by
Dr. Christian in a previous post, which is probably the most common notation in the literature.

Suppose that a source emits instructions to Alice and Bob to tell their instruments what to measure. The instructions
are traditionally called a "hidden variable", often denoted by the symbol . Often, papers consider a
more complicated situation in which the experimenters can measure spin in arbitrary directions, but just two different
directions are enough to derive CHSH, so I will assume that Alice can measure in directions and
and Bob in directions and . If Alice measures in direction , then the instructions determine the result +1 or -1 of the measurement. The result is denoted for Alice and for Bob.

The instructions in the "hidden variable" are determined by sampling from some given probability space. For example,
there might be a 1/5 probability that the hidden variable will instruct Alice to obtain +1 when she measures
in direction and also for Bob to obtain -1 when he measures in direction . Then
the quantum-mechanical distribution called above will be given by:



In the literature, instead of calculating joint probabilities like so-called "correlations"
are typically calculated and often expressed in integral notation. A "correlation" denoted is defined
as the probability that Alice and Bob get the same result minus the probability that they get opposite
results (when Alice measures along direction and Bob along ). When Alice and Bob are
spacelike separated (so it is assumed that their measurement results are independent), the correlation is expressed as

.

To finish up, if this assumed model could give the quantum-mechanical marginals etc., then
it would automatically give also the full probability function for which the , etc. are marginals Namely,

.

When only correlations are considered, as most papers do, this may not be so obvious.


But in general, there is no whose marginals are the quantum mechanical p_{ A B}, etc.
That is, quantum mechanics is not 1202realistic. Perhaps it could be considered "realistic" in some other sense, but not in the 1202realistic sense of the overwhelming majority of the literature.

[Joy Christian]

***
I was not aware of any difficulty in downloading the paper from Academia.Edu. Let me make it available here for a direct download from my blog:

http://libertesphilosophica.info/blog/l ... impExp.pdf (if anyone is still having difficulty in downloading the paper, then please do let me know).

In any case, I find the above comments by "guest1202" quite misplaced and misguided. My paper is quite clearly written, and with every definition made explicit.

More importantly, my model for the EPR-Bohm correlation presented in the above paper is manifestly local, traditionally realistic, and classically deterministic.

I have already made it quite clear that I am using the standard definition of local realism, as espoused by Einstein and later explicated by Bell. As I have already noted, if anyone has any doubt about this then they are welcome to consult Eq. (3.4) of the Clauser-Shimony report (Rep. Prog. Phys., Vol. 41, 1978), which is the standard reference on the subject (see the image below). The claim by "guest1202" that my model is somehow not realistic, or differently realistic, is ludicrous.

[FrediFizzx]

I think this is related to the problem that QM can't produce the negative cosine curve using +/- 1 outcomes. However if we are to believe Bell, then local-realistic models can't do it either. Only non-local hidden variable models (NLHV) can produce -a.b in R^3. However, Joy's classical local-realistic model can produce the negative cosine curve in S^3. So using that as a guide, QM should be able to produce it in S^3 which the experiments indicate that it does. Therefore QM is realistic as Michel was saying.

[FrediFizzx]

...To save readers the trouble of going back to that post, here in brief is the definition of a "1202realistic" model. We consider a situation in which observer Alice has a choice of two measurements, and , each yielding +1 or -1. Another oberver Bob similarly has a choice of measurements and . Quantum mechanics gives us the probability that if Alice uses measurement and Bob uses , Alice obtains result , and Bob obtains. Here each of and equals either +1 or -1. Quantum mechanics also gives three other similarly labeled probabilities and .

How does QM give those probabilities? Those are given by EPR via the Bell-CHSH argument.

Let us write for the outcome which Alice obtains when she selects measurement and obtains result +1, with similar notation for the other possible measurement selections. Classically, besides the above probabilities given by quantum mechanics, we could also speak, for example, of the probability
that , and .
Then the four probabilities given by quantum mechanics would be marginals of .
For example, we would have



The definition of "1202realistic" is that there exists such a whose marginals are the four probability distributions given by quantum mechanics. This is the definition of the "realistic" part of "local-realistic" which is used in almost 100% of the papers on the subject known to me, though it's often presented differently (see below). Its intuitive meaning is that Alice and Bob are measuring things which are "real" in the sense that they have an existence independent of any measurement. An example is whether or not one of the "quantum cakes" of Kwiat/Hardy (discussed in another post) has risen earlier than usual halfway through the baking. Classically, we feel comfortable talking about that possibility whether or not we have opened the oven to check.

I am quite sure that Dr. Christian's model cannot be 1202realistic. The reason is that there is a mathematical theorem with a very simple proof that for to exist with given marginals , etc., the marginals have to satisfy the CHSH inequalities, and the marginals of quantum mechanics don't necessarily satisfy them. The proof of this theorem has been checked by thousands of mathematicians and physicists. I have never seen it questioned, including in the papers of Dr. Christian...

Do you have an online reference for this theorem? However, it may not be necessary as you seem to be confusing the marginals as coming from QM then you say right here that the QM marginals "...don't necessarily satisfy them". Quite frankly, it is all a matter of whether the terms in the Bell-CHSH inequality are dependent or independent. If you make it so that the QM marginals are dependent, then they will satisfy Bell-CHSH. See the argument in this paper.

http://cds.cern.ch/record/433857/files/0004037.pdf
"Violating" Clauser-Horne Inequalities Within Classical Mechanics
***

[guest1202]

FrediFizzx wrote:

Do you have an online reference for this theorem [that given probability distributions
are marginals of some if and only if they satisfy the CHSH inequalities]? However, it may not be necessary as you seem to be confusing the marginals as coming from QM then you say right here that the QM marginals "...don't necessarily satisfy them". Quite frankly, it is all a matter of whether the terms in the Bell-CHSH inequality are dependent or independent. If you make it so that the QM marginals are dependent, then they will satisfy Bell-CHSH. See the argument in this paper.

These results are over 30 years old, published before the Internet was commonly used and before the arXiv existed,
so they may not be easy to find online. The references are:

A. Fine, "Joint distributions, quantum correlations, and commuting observables" Journal of Mathematical Physics 23, 1306 (1982); doi: 10.1063/1.525514. This one is available online at http://dx.doi.org/10.1063/1.525514. The result that you ask about is his Theorem 4.

J. F. Clauser and M. A. Horne, "Experimental consequences of objective local theories, Phys. Rev. D. 10 (1974), 526-535

The result that you ask about is proved in the Fine paper, but his equivalent condition is what he calls the "Bell-CH" inequalities instead of the CHSH inequalities. The equivalence of Bell-Ch with CHSH is proved in an appendix in the Clauser/Horne paper, around their equation (B2).

I am puzzled by your reference to terms in the Bell-CHSH inequality as being "dependent" or "independent". One can speak of two events in a probability space as "independent", or two random variables as "independent", but the terms in a CHSH inequality like



are numbers like (say) . I don't know what it means to say that two numbers are independent.

Could it be that you are using "independent" in some nontechnical sense in referring to the way that the numbers were obtained? If so, the various correlations are (and must necessarily be) observed in separate (meaning "independent"?) experiments. If that is your meaning of "independent" in this context, then
he quantum-mechanical (QM) marginals would necessarily be "independent". Then "dependent" QM marginals would be impossible and there would be no point to ask if they satisfy CHSH.

I'm not trying to be pedantic about the language. I mention it because I have often been puzzled by some regular posters' use of "independence". Often, I simply can't figure out what they are trying to say. If they want their posts to be read and understood, they ought to realize that outside of technical probability theory, the term "independent" is susceptible to many meanings, and which one they are using should be made explicit.

I don't know what you think I may be confusing with what. The theorem of Fine quoted above has nothing to do with quantum mechanics. Its hypothesis is that certain "marginals" are given, and in particular,those marginals could come from classical physics or from quantum mechanics. It conclusion is that these "marginals" come from some (and so really are "marginals"!) if and only if they satisfy the Bell-CH (equivalently, the CHSH) inequalities.

[Joy Christian]

***
In his or her comments "guest1202" tries to give a false impression to the reader that he or she has actually read my papers, let alone understood them. It is quite easy to see that this is a false impression by simply reading my papers, for example this one, where all of the quantum mechanical probabilities for the EPR-Bohm type experiment are explicitly reproduced in a purely local-realistic manner, with further support from several event-by-event computer simulations.

Contradicting these constructive demonstrations by means of explicit counterexamples to Bell's theorem, "guest1202" falls back to the so-called proofs of Bell-CHSH type, which cannot possibly go through without resorting to unphysical juxtapositions of three or four mutually exclusive incompatible experiments. I have repeatedly made this point clear in my papers, for example in this paper. But "guest1202" has either not bother to read my papers, or does not have a background or intellectual capacity to understand them.

In any case, all one really has to understand is an elementary point that no Bell type theorem can go through without considering three or four mutually exclusive incompatible experiments, thus making such theorems quite irrelevant for any experimentally realizable physics. The last point has been discussed to death in this forum, especially by "minkwe" --- see for just one example these comments by "minkwe": viewtopic.php?f=21&t=34#p948.

***

[guest1202]

guest1202 wrote:
...To save readers the trouble of going back to that post, here in brief is the definition of a "1202realistic" model. We consider a situation in which observer Alice has a choice of two measurements, and , each yielding +1 or -1. Another oberver Bob similarly has a choice of measurements and . Quantum mechanics gives us the probability that if Alice uses measurement and Bob uses , Alice obtains result , and Bob obtains. Here each of and equals either +1 or -1. Quantum mechanics also gives three other similarly labeled probabilities and .

FreddiFizzx asks:

How does QM give those probabilities? Those are given by EPR via the Bell-CHSH argument.

First of all, recall that the Hilbert space for joint Alice/Bob experiments is a tensor product of the individual Hilbert spaces for Alice and Bob. Let be a state in this joint Hilbert space (such as the singlet state). Suppose that Alice is measuring the spin in direction (obtaining +1 or -1). In quantum mechanics, this measurement corresponds to a quantum "observable" , which is a projector operator on Alice's Hilbert space. Similarly, Bob measuring a spin in direction corresponds to a projector on his Hilbert space. A joint measurement corresponds to the projector on the joint tensor product. Denoting by Alice's measurement and by Bob's,
the probability previously denoted is given by:



and is given by

, etc.,
where denotes the identity operator.

You might enjoy calculating the explicit values when is the singlet state.

[FrediFizzx]

I'm not trying to be pedantic about the language. I mention it because I have often been puzzled by some regular posters' use of "independence". Often, I simply can't figure out what they are trying to say. If they want their posts to be read and understood, they ought to realize that outside of technical probability theory, the term "independent" is susceptible to many meanings, and which one they are using should be made explicit.

Well, then I will make it more explicit. From the CH paper we have the following expression right before their eq. (4),

-1 ≤ p1(λ, a)p2(λ, b) - p1(λ, a)p2(λ, b') + p1(λ, a')p2(λ, b) + p1(λ, a')p2(λ, b') - p1(λ, a') - p2(λ, b) ≤ 0

Now we will label each of the probability terms with A, A', B, and B' so we have,

-1 ≤ AB - AB' + A'B + A'B' - A' - B ≤ 0

So it is easy to see that the terms AB and AB' depend on the same A. And so forth with the other terms. IOW, the terms in the inequality are dependent on elements of each other and that is what makes the upper bound of 0 mathematically impossible to violate by anything. Now from the "Violating" Clauser-Horne... paper that I linked to above, you will see that the QM probabilities can be formulated in such a way as to have that same dependency between the terms thus satisfying Bell-CH and not violating it. I am sure an equivalent could be done for Joy's local-realistic model. So as I said before, the proof of the theorem doesn't matter. But thanks for the references anyways.

[guest1202]

In a very recent post (I don't know how to produce a link to it),
Dr. Christian presents a page of calculations with no symbols defined
and asks readers to specify

"where in my derivation there is 'non-reality', non-locality',
or lack of determinism creeping in" .

I've read the paper and have several comments on it.
In summary these are:

(1) The paper's conclusion reproduces the *correlations* of quantum mechanics for the singlet state (only), but probably not all predictions of quantum mechanics. Therefore even if one agrees that it satisfies some "traditional" definition of "realism" (which I don't), in its present form, it cannot be said to provide a "local-realistic" model for quantum mechanics.

(2) Some important mathematical details of the derivation are not clear
to me, and I am not sure that the derivation is correct.

This post will deal with the more important issue of (1).
A subsequent post will discuss (2).

BEGIN DISCUSSION OF (1)

For example, Christian's model does not seem to predict the probability (denoted in a previous post) that if Alice selects measurement (say measuring the spin in direction )
and Bob selects measurement (say measuring spin in direction ),
then Alice obtains result 1 and Bob obtains result -1. Quantum mechanics does predict this probability.

Write .
The correlation, which my previous post denoted and is called
by Christian (where measurements and correspond to
measuring spin along axes and ), is defined by:



Thus if we are given , we can compute the correlation
But given we can't compute because that would require
solving one linear equations in the three unknowns
and . (There are only three unknowns because all the have
to sum to 1.)

Because of this, it is unclear (and looks unlikely) that Christian's model
actually can reproduce the predictions of quantum mechanics. I say
"unclear" instead of "impossible" because Christian's model does give
something more than just , namely it gives etc.
However, I haven't seen in any of his papers an explanation of how to get
the various , etc., from the correlations which his
model provides. Until this issue is addressed, it would be premature to claim
that his model provides a local-realistic way of obtaining all quantum
mechanical predictions.

[guest1202]

This post will discuss questionable mathematical exposition
in Dr. Christian's latest paper "Macroscopic Observability of Spinorial Sign Changes: A Simplied Proof"
His equation (27) contains an expression of the general form

(27)

where and are functions whose exact form will be irrelevant to the
discussion. The next equation (28) appears to transform (27) into


(28)

At a minimum, this is not standard mathematical exposition.
If interpreted literally, it is wrong because s cannot approach a and b
simultaneously unless a = b (and in the context a=b does not necessarily hold).

I haven't been able to think of a reasonable interpretation of (28).
Christian claims it follows from his (correct) equation (35), using
"elementary properties of limits", but it doesn't if his (28) is interpreted
literally according to standard mathematical usage. As long as
the reader has to guess at the intended meaning of such things,
I couldn't agree with Christian's characterization of his own paper as
"quite clearly written".

Rick Lockyer raised the same point in a previous post in this thread,
and he is correct that at a minimum, the intended meaning of (28) and the
passage from (27) to (28) requires more explanation.

A reply to Lockyer's post in effect accused him of not understanding limits,
and stated in the most insulting way that Christian would "not be responding to any comments from
him [Lockyer]". This is an inappropriate attitude. I don't say that
Christian's intended meaning for (28) and how it follows from (27) cannot
be explained. But even if so, Lockyer's objection is reasonable and should be
answered with respect.

I have wondered if like Lockyer, I will now be accused of not understanding limits.
For what it's worth, I do have a Ph.D. in mathematics and a modest professional reputation.

I think that there are various other questionable mathematical manipulations
in Christian's paper, but it is too tedious to set them out in this forum.

[FrediFizzx]

This post will discuss questionable mathematical exposition
in Dr. Christian's latest paper "Macroscopic Observability of Spinorial Sign Changes: A Simplied Proof"
His equation (27) contains an expression of the general form

(27)

where and are functions whose exact form will be irrelevant to the
discussion. The next equation (28) appears to transform (27) into


(28)

At a minimum, this is not standard mathematical exposition.
If interpreted literally, it is wrong because s cannot approach a and b
simultaneously unless a = b (and in the context a=b does not necessarily hold).

I haven't been able to think of a reasonable interpretation of (28).

That is because you are not taking the physics of the situation into account. Think of two separated particles that both started out with a common s.

[Joy Christian]

***

I have just linked the following paper above, as I have done earlier in this very thread, which reproduces ALL of the probabilistic predictions of the EPRB scenario:

http://arxiv.org/abs/1405.2355

This is just one example which shows that "guest1202" has not read my papers.

My complete local-realistic model for ALL quantum mechanical correlations (no matter how complicated the underlying quantum state) is presented in my book.

My latest paper below, which is quite clearly written, presents a local-realistic derivation of the EPRB correlation in a most transparent and elegant manner:

https://www.academia.edu/19235737/Macro ... fied_Proof

I have no reason to believe that "guest1202" has "a Ph.D. in mathematics and a modest professional reputation." The impression I get is quite the opposite.

***

[guest1202]

FrediFizzx wrote:

Well, then I will make it more explicit. From the CH paper we have the following expression right before their eq. (4),

-1 ≤ p1(λ, a)p2(λ, b) - p1(λ, a)p2(λ, b') + p1(λ, a')p2(λ, b) + p1(λ, a' )p2(λ, b') - p1(λ, a') - p2(λ, b) ≤ 0

Now we will label each of the probability terms with A, A', B, and B' so we have,

-1 ≤ AB - AB' + A'B + A'B' - A' - B ≤ 0

So it is easy to see that the terms AB and AB' depend on the same A. And so forth with the other terms. IOW, the terms in the inequality are dependent on elements of each other and that is what makes the upper bound of 0 mathematically impossible to violate by anything. Now from the "Violating" Clauser-Horne... paper that I linked to above, you will see that the QM probabilities can be formulated in such a way as to have that same dependency between the terms thus satisfying Bell-CH and not violating it. I am sure an equivalent could be done for Joy's local-realistic model. So as I said before, the proof of the theorem doesn't matter. But thanks for the references anyways.
FrediFizzx
Independent Physics Researcher

I have seen such statements before in this forum, and they have always puzzled me. I can't imagine what can lead to the conclusion that "the upper bound of 0 [is] mathematically impossible to violate by anything."

Before presenting a counterexample that does violate CH, to avoid misunderstanding, I ought to straighten out the notation. In CH's equation (4) which you quote, there is no . To get the CH inequalities, one integrates over . After the integration, your term becomes , which represents the probability that Alice measures in direction a and obtains result +1, while Bob measures in direction b and also obtains +1.

The "B" at the end of the middle term represents the probability that Bob measures +1 irrespective of what Alice measures. The term "AB" is not a product of "A" and "B", though you might not have meant to imply that.

The CH inequality that you attribute to Clauser/Horne's equation (4) is actually:

-1 ≤ p12(a,b) - p12(a, b' ) + p12(a' , b) + p12(a' , b' ) - p1( a' ) - p2( b) ≤ 0 .

Here p12(a, b) represents the probability that Alice measures in direction a and obtains +1 and Bob measures in direction b and obtains +1, and similarly for the other double-subscripted terms. The p1(a' ) represents the probability that Alice measures in direction a' and obtains result +1, irrespective of Bob's measurement. The p2(b) represents the probability that Bob measures in direction b and obtains results +1 regardless of what Alice does. The singly-subscripted terms refer to measurements by one of the observers (subscript 1 for Alice, 2 for Bob) without reference to the other observer. The terms with the double subscript "12" represent joint measurement probabilities.

The counterexample that I am going to present is a "Popescu/Rohlich (PR) box", which I discussed in a previous post. The PR boxes generally provide the maximum violation of Bell/type inequalities, and hence the maximum non-classicality. This is easy to believe from the fact that the very definition of PR boxes seems classically impossible.

We use the notation A = B to mean that whenever Alice measures in direction a and Bob measures in direction b, they always obtain the same result (both +1 or both -1 ). If they always obtain opposite results (+1 for Alice and -1 for Bob or -1 for Alice and +1 for Bob), we write " A = (not B)". Similarly for A' = B, for which Alice measures in direction a` and Bob in direction b, etc. We will use a Popescu/Rohrlich box for which:

A = B, B = A' , A' = B', and B' = (not A ) .

Readers who are worried that this might be contradictory, should consult the previous post for reassurance. In a classical world, it would be contradictory, but quantum mechanics permits similar (though not identical) situations.
Then

p12(a, b) = p1(a) because A = B. That is, Alice gets +1 with some probability p1(a), and Bob always gets the same result as Alice. When you work out the other terms, the middle of the inequality becomes

p12(a,b) - p12(a, b' ) + p12(a' , b) + p12(a' , b' ) - p1( a' ) - p2( b) = p1(a) - 0 + p1(a' ) + p1(a' ) - p1(a' ) - p1 (a) = p1(a`) ,

which is always strictly positive unless p1(a' ) = 0, and p1(a' ) can be chosen arbitrarily.

[thray]

This post will discuss questionable mathematical exposition
in Dr. Christian's latest paper "Macroscopic Observability of Spinorial Sign Changes: A Simplied Proof"
His equation (27) contains an expression of the general form

(27)

where and are functions whose exact form will be irrelevant to the
discussion. The next equation (28) appears to transform (27) into


(28)

At a minimum, this is not standard mathematical exposition.

It's a parity argument. Do you agree that in the linear maps:

ΦA→B[(0(mod2)]
ΦA→B[1(mod2)]

The first half is to the even part of A and odd part of B; the second to the odd part of B and the even part of A?

[FrediFizzx]

We use the notation A = B to mean that whenever Alice measures in direction a and Bob measures in direction b, they always obtain the same result (both +1 or both -1 ). If they always obtain opposite results (+1 for Alice and -1 for Bob or -1 for Alice and +1 for Bob), we write " A = (not B)". Similarly for A' = B, for which Alice measures in direction a` and Bob in direction b, etc. We will use a Popescu/Rohrlich box for which:

A = B, B = A' , A' = B', and B' = (not A ) .

Readers who are worried that this might be contradictory, should consult the previous post for reassurance. In a classical world, it would be contradictory, but quantum mechanics permits similar (though not identical) situations. ...

***
You are going to need to demonstrate how QM allows B = A' and A' = B' at the same time. IOW, prove that B = A' = B'.
***

I was merely demonstrating with the CH equation before eq. (4) how the dependency creeps into the inequality. It is there and you can't get rid of it. Now... you are a mathematician so you claim and what exactly is the point of an inequality that can be violated? That simply means that the inequality was invalid to start with. It is mathematical insanity to believe that an inequality of Bell's type can be violated. You are simply tricking yourself.

***

[Gordon Watson]

FrediFizzx wrote:

Well, then I will make it more explicit. From the CH paper we have the following expression right before their eq. (4),

-1 ≤ p1(λ, a)p2(λ, b) - p1(λ, a)p2(λ, b') + p1(λ, a')p2(λ, b) + p1(λ, a' )p2(λ, b') - p1(λ, a') - p2(λ, b) ≤ 0

Now we will label each of the probability terms with A, A', B, and B' so we have,

-1 ≤ AB - AB' + A'B + A'B' - A' - B ≤ 0

So it is easy to see that the terms AB and AB' depend on the same A. And so forth with the other terms. IOW, the terms in the inequality are dependent on elements of each other and that is what makes the upper bound of 0 mathematically impossible to violate by anything. Now from the "Violating" Clauser-Horne... paper that I linked to above, you will see that the QM probabilities can be formulated in such a way as to have that same dependency between the terms thus satisfying Bell-CH and not violating it. I am sure an equivalent could be done for Joy's local-realistic model. So as I said before, the proof of the theorem doesn't matter. But thanks for the references anyways.
FrediFizzx
Independent Physics Researcher

I have seen such statements before in this forum, and they have always puzzled me. I can't imagine what can lead to the conclusion that "the upper bound of 0 [is] mathematically impossible to violate by anything."

Before presenting a counterexample that does violate CH, to avoid misunderstanding, I ought to straighten out the notation. In CH's equation (4) which you quote, there is no . To get the CH inequalities, one integrates over . After the integration, your term becomes , which represents the probability that Alice measures in direction a and obtains result +1, while Bob measures in direction b and also obtains +1.

The "B" at the end of the middle term represents the probability that Bob measures +1 irrespective of what Alice measures. The term "AB" is not a product of "A" and "B", though you might not have meant to imply that.

The CH inequality that you attribute to Clauser/Horne's equation (4) is actually:

-1 ≤ p12(a,b) - p12(a, b' ) + p12(a' , b) + p12(a' , b' ) - p1( a' ) - p2( b) ≤ 0 .

Here p12(a, b) represents the probability that Alice measures in direction a and obtains +1 and Bob measures in direction b and obtains +1, and similarly for the other double-subscripted terms. The p1(a' ) represents the probability that Alice measures in direction a' and obtains result +1, irrespective of Bob's measurement. The p2(b) represents the probability that Bob measures in direction b and obtains results +1 regardless of what Alice does. The singly-subscripted terms refer to measurements by one of the observers (subscript 1 for Alice, 2 for Bob) without reference to the other observer. The terms with the double subscript "12" represent joint measurement probabilities.

The counterexample that I am going to present is a "Popescu/Rohlich (PR) box", which I discussed in a previous post. The PR boxes generally provide the maximum violation of Bell/type inequalities, and hence the maximum non-classicality. This is easy to believe from the fact that the very definition of PR boxes seems classically impossible.

We use the notation A = B to mean that whenever Alice measures in direction a and Bob measures in direction b, they always obtain the same result (both +1 or both -1 ). If they always obtain opposite results (+1 for Alice and -1 for Bob or -1 for Alice and +1 for Bob), we write " A = (not B)". Similarly for A' = B, for which Alice measures in direction a` and Bob in direction b, etc. We will use a Popescu/Rohrlich box for which:

A = B, B = A' , A' = B', and B' = (not A ) .

Readers who are worried that this might be contradictory, should consult the previous post for reassurance. In a classical world, it would be contradictory, but quantum mechanics permits similar (though not identical) situations.
Then

p12(a, b) = p1(a) because A = B. That is, Alice gets +1 with some probability p1(a), and Bob always gets the same result as Alice. When you work out the other terms, the middle of the inequality becomes

p12(a,b) - p12(a, b' ) + p12(a' , b) + p12(a' , b' ) - p1( a' ) - p2( b) = p1(a) - 0 + p1(a' ) + p1(a' ) - p1(a' ) - p1 (a) = p1(a`) ,

which is always strictly positive unless p1(a' ) = 0, and p1(a' ) can be chosen arbitrarily.

guest1202,

You write: "The CH inequality that you [Fred] attribute to Clauser/Horne's [CH] equation (4) is actually:

-1 ≤ p12(a,b) - p12(a, b' ) + p12(a' , b) + p12(a' , b' ) - p1(a') - p2(b) ≤ 0." (1) [My designation.]

Question (using P for the CH p). Given (1), under what circumstances can any of the following derivative equations ever be be false?

-1 ≤ P1(a)P2(b|a) - P1(a)P2(b'|a ) + P1(a')P2(b|a') + P1(a')P2(b'|a') - P1(a') - P2(b) ≤ 0; (2)

-1 ≤ (1/2)[P2(b|a) - P2(b'|a ) + P2(b|a') + P2(b'|a') - 1 - 1] ≤ 0; (3)

-2 ≤ P2(b|a) - P2(b'|a ) + P2(b|a') + P2(b'|a') - 2 ≤ 0; (4)

0 ≤ P2(b|a) - P2(b'|a ) + P2(b|a') + P2(b'|a') ≤ 2. (5)

NB: (3) follows from the fact that in QM (or any rational theory related to CH): the variables correlating each particle-pair are taken to be pairwise correlated but otherwise random. Thus:

P1(a) = P1(a') = P2(b) = 1/2; (6)

(6) being expected and confirmed in practice!

Thanks; Gordon

[Gordon Watson]

We use the notation A = B to mean that whenever Alice measures in direction a and Bob measures in direction b, they always obtain the same result (both +1 or both -1 ). If they always obtain opposite results (+1 for Alice and -1 for Bob or -1 for Alice and +1 for Bob), we write " A = (not B)". Similarly for A' = B, for which Alice measures in direction a` and Bob in direction b, etc. We will use a Popescu/Rohrlich box for which:

A = B, B = A' , A' = B', and B' = (not A ) .

Readers who are worried that this might be contradictory, should consult the previous post for reassurance. In a classical world, it would be contradictory, but quantum mechanics permits similar (though not identical) situations. ...

***
You are going to need to demonstrate how QM allows B = A' and A' = B' at the same time. IOW, prove that B = A' = B'.
***

I was merely demonstrating with the CH equation before eq. (4) how the dependency creeps into the inequality. It is there and you can't get rid of it. Now... you are a mathematician so you claim and what exactly is the point of an inequality that can be violated. That simply means that the inequality was invalid to start with. It is mathematical insanity to believe that an inequality of Bell's type can be violated. You are simply tricking yourself.

***

Fred, re this from you above:

"… what exactly is the point of an inequality that can be violated. That simply means that the inequality was invalid to start with. It is mathematical insanity to believe that an inequality of Bell's type can be violated."

For me, "the point of an inequality that can be violated" is this: Any valid violation of an inequality indicates a fallacy in that inequality's derivation.

Let's say that, in an earlier discussion with me, you incorrectly derived an inequality. Then, surely, no surprise in its violation?

Alas, Bell had no discussion with me [though I'm sure he would have, had I not been prematurely told that he was dead]. But his inequality is false in the context (EPRB) in which he derived it. So what 's the point …?

Any valid violation of a Bellian inequality indicates a fallacy in that inequality's derivation.

Or am I missing something?

[FrediFizzx]

Alas, Bell had no discussion with me [though I'm sure he would have, had I not been prematurely told that he was dead]. But his inequality is false in the context (EPRB) in which he derived it. So what 's the point …?

Any valid violation of a Bellian inequality indicates a fallacy in that inequality's derivation.

Or am I missing something?

You are not missing anything AFAICT. So far all I have seen from the Bell fans is some handwavy arguments that are always shown to be false under scrutiny.

[guest1202]

user "thray" wrote:

It's a parity argument. Do you agree that in the linear maps:

ΦA→B[(0(mod2)]
ΦA→B[1(mod2)]

The first half is to the even part of A and odd part of B; the second to the odd part of B and the even part of A?
thray

That's all he wrote, except that it was preceded by a quote from a previous post of mine which contains none of the symbols A, B, or . I have no idea what these symbols represent to "thray". Also,


is not standard mathematical language, and I wouldn't know how to interpret it even if I knew what the symbols represented.
I have wondered if he might have meant



which could have a meaning in certain contexts, but then I wouldn't know how to interpret the following [0(mod 2)]. Could he mean " " ?

I am answering this out of politeness because I imagine that "thray" thinks he has asked a clear and valid question. But I'm afraid that I won't have time to answer questions like this in the future. I hope no one will interpret this as discourtesy.