Definition of "realistic" in "local-realistic"

[guest1202]

I want to correct an oversight.
A post titled "Is it impossible to violate the Bell-CH inequallities", concluded:

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.

This is all fine except for the very last assertion that "p1(a' ) "can be chosen arbitrarily". I had overlooked a consistency condition which requires that p1(a' ) = 1/2. The same condition requires that all of the probabilities be 1/2 except that p12(a, b' ) = 0. (To understand why p12(a, b' ) is different, recall that p12(a, b' ) is the probability that Alice and Bob both obtain result +1, but the condition A = (not B' ) says that they never both obtain the same result.) That is,

p1(a) = p1(a' ) = p1(b) = p1(b' ) = p12(a,b) = p12(a' , b) = p12(a' , b' ) = 1/2, but p12(a, b' ) = 0.

The counterexample is still valid under these choices of the probabilities, and even easier to check:
the middle term is 1/2 > 0.

In another reply, someone asked how one can obtain these probabilities from quantum mechanics (QM). One can't; QM does not allow Popescu/Rohlich (PR) boxes. But QM does allow what I called "probabilistic PR boxes", as discussed in a previous post on PR boxes (my post discussing the Kwiat/Hardy "quantum cakes" paper). A "probabilistic PR box" has the same (classically) contradictory features as a PR box, but one of the four defining conditions,
A = B, B = A', A' = B', B' = (not A),
only occurs with some nonzero probability (for Kwiat/Hardy, that probability is about .09) instead of occurring with probability 1 as in a genuine PR box.

Also, there seems to be some misunderstanding that the CH or CHSH inequalities may have something to do with classical or quantum mechanics. They have nothing to do with classical or quantum mechanics. They are purely mathematical theorems stating a necessary condition that certain hypothetical "marginals" like p12(a,b) and p1(a) really are marginals of some probability distribution which was previously denoted as . However, these mathematical theorems are not always stated in this way; Clauser/Horne uses an equivalent statement. Arthur Fine (J. Math. Phys. reference given in another post) uses the statement that I just gave.

[guest1202]

Gordon Watson wrote:

You [guest1202] 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

I haven't attempted to follow the above in detail, but your concluding equation (6) is correct. I had overlooked a consistency condition which implies your (6) and also that all the other probabilities are 1/2 except that P(a, b' )=0. A previous post discuses this in detail.

[guest1202]

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?
Gordon Watson

I think that you are missing something. The Bell inequalities are valid for local-realistic models. I will not attempt a definition of "local-realistic" here; the definition is (or should be) given in the hypotheses for the inequalities. Some models, like quantum mechanics, violate these inequalities. That implies that such models do not satisfy the "local-realistic" hypothesis which was assumed to derive the inequalities. It does not imply "a fallacy in that inequality's derivation".

Let us consider a simpler example.

Under the hypothesis that " is a real number" (i.e., the symbol x represents a real number), we can derive the inequality

(1) .

When you read mathematical texts, you will sometimes come across a symbol " " for which it is claimed that

.

This violates inequality (1). Does that mean that there was a fallacy in the derivation of (1)? No, it only means that does not satisfy the hypotheses under which (1) was derived. That is, it means that is not a real number.

Here the hypothesis " is a real number" is like the hypothesis "the model is local-realistic", and the violation of the inequality simply implies a violation of the hypothesis, namely " is not a real number or "the model is not local-realistic".

[Joy Christian]

***
Let me stress once again that the issues raised by "guest1202" with the derivation in my latest paper are a result of his or her own personal difficulties in reading what is so clearly explained in this paper. The discussion in the paper about what is meant by the limits etc. is quite clear and transparent to anyone familiar with the EPRB type experiments, as depicted, for example, in Fig. 1 of the paper. No one (especially with "a Ph.D. in mathematics") should have any difficulty in understanding the trivial identity of limits presented in Eq. (35) of my paper, which can be verified by immediate inspection. It is quite bizarre that anyone would raise an issue with it:



My suggested to anyone taken in by the recent comments by "guest1202" is to simply read the two papers I linked in my very first post in this thread, which also quite clearly debunks the faux issue of "realistic" raised by "guest1202": viewtopic.php?f=6&t=231#p5978 (the second paper reproduces the quantum probabilities explicitly).

***

[FrediFizzx]

***
Let me stress once again that the issues raised by "guest1202" with the derivation in my latest paper are a result of his or her own personal difficulties in reading what is so clearly explained in this paper. The discussion in the paper about what is meant by the limits etc. is quite clear and transparent to anyone familiar with the EPRB type experiments, as depicted, for example, in Fig. 1 of the paper. No one (especially with "a Ph.D. in mathematics") should have any difficulty in understanding the trivial identity of limits presented in Eq. (35) of my paper, which can be verified by immediate inspection. It is quite bizarre that anyone would raise an issue with it:



My suggested to anyone taken in by the recent comments by "guest1202" is to simply read the two papers I linked in my very first post in this thread, which also quite clearly debunks the faux issue of "realistic" raised by "guest1202": viewtopic.php?f=6&t=231#p5978 (the second paper reproduces the quantum probabilities explicitly).

***

I would just add that s_1 = s_2 per iteration. Maybe I missed it?

[FrediFizzx]

I think that you are missing something. The Bell inequalities are valid for local-realistic models. I will not attempt a definition of "local-realistic" here; the definition is (or should be) given in the hypotheses for the inequalities. Some models, like quantum mechanics, violate these inequalities. That implies that such models do not satisfy the "local-realistic" hypothesis which was assumed to derive the inequalities. It does not imply "a fallacy in that inequality's derivation".

Let us consider a simpler example.

Under the hypothesis that " is a real number" (i.e., the symbol x represents a real number), we can derive the inequality

(1) .

When you read mathematical texts, you will sometimes come across a symbol " " for which it is claimed that

.

This violates inequality (1). Does that mean that there was a fallacy in the derivation of (1)? No, it only means that does not satisfy the hypotheses under which (1) was derived. That is, it means that is not a real number.

Here the hypothesis " is a real number" is like the hypothesis "the model is local-realistic", and the violation of the inequality simply implies a violation of the hypothesis, namely " is not a real number or "the model is not local-realistic".

Ahh... we got you now. :- ) By the logic you present here, saying that QM violates the Bell inequalities is like comparing apples to oranges. It is not even valid to do it; it is not a valid violation which is an oxymoron since there is no such thing as a valid violation. If you put QM on an equal footing with LHV, it does not violate the inequalities as shown by the paper I linked above ""Violating" Clauser-Horne ...". IOW, QM needs a different set of inequalities from the Bell ones and QM never violates the set of inequalities appropriate for QM.

[Joy Christian]

I would just add that s_1 = s_2 per iteration. Maybe I missed it?

I think you did:

[FrediFizzx]

I want to correct an oversight.
A post titled "Is it impossible to violate the Bell-CH inequallities", concluded:

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.

This is all fine except for the very last assertion that "p1(a' ) "can be chosen arbitrarily". I had overlooked a consistency condition which requires that p1(a' ) = 1/2. The same condition requires that all of the probabilities be 1/2 except that p12(a, b' ) = 0. (To understand why p12(a, b' ) is different, recall that p12(a, b' ) is the probability that Alice and Bob both obtain result +1, but the condition A = (not B' ) says that they never both obtain the same result.) That is,

p1(a) = p1(a' ) = p1(b) = p1(b' ) = p12(a,b) = p12(a' , b) = p12(a' , b' ) = 1/2, but p12(a, b' ) = 0.

The counterexample is still valid under these choices of the probabilities, and even easier to check:
the middle term is 1/2 > 0.

In another reply, someone asked how one can obtain these probabilities from quantum mechanics (QM). One can't; QM does not allow Popescu/Rohlich (PR) boxes. But QM does allow what I called "probabilistic PR boxes", as discussed in a previous post on PR boxes (my post discussing the Kwiat/Hardy "quantum cakes" paper). A "probabilistic PR box" has the same (classically) contradictory features as a PR box, but one of the four defining conditions,
A = B, B = A', A' = B', B' = (not A),
only occurs with some nonzero probability (for Kwiat/Hardy, that probability is about .09) instead of occurring with probability 1 as in a genuine PR box. ...

****
You are skirting what I asked you to do. Prove that B = A' = B'. Otherwise you are just demonstrating handwavy junk that is typical of most Bell fans.

What you have done is shift to an inequality of the CH form where the upper bound is 1 not 0.

****

[Gordon Watson]

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?

I think that you are missing something. The Bell inequalities are valid for local-realistic models. I will not attempt a definition of "local-realistic" here; the definition is (or should be) given in the hypotheses for the inequalities. Some models, like quantum mechanics, violate these inequalities. That implies that such models do not satisfy the "local-realistic" hypothesis which was assumed to derive the inequalities. It does not imply "a fallacy in that inequality's derivation".

Let us consider a simpler example.

Under the hypothesis that " is a real number" (i.e., the symbol x represents a real number), we can derive the inequality

(1) .

When you read mathematical texts, you will sometimes come across a symbol " " for which it is claimed that

.

This violates inequality (1). Does that mean that there was a fallacy in the derivation of (1)? No, it only means that does not satisfy the hypotheses under which (1) was derived. That is, it means that is not a real number.

Here the hypothesis " is a real number" is like the hypothesis "the model is local-realistic", and the violation of the inequality simply implies a violation of the hypothesis, namely " is not a real number or "the model is not local-realistic".

You write: "The Bell inequalities are valid for local-realistic models." Please, guest1202, after I bring your "simpler example" into line with Bell's theorem and rebut it, would you mind answering the substantial questions that I've put to you directly re Bell's work?

Rebuttal of guest1202's "simpler example" (with corrections): Please note that for guest1202's (1) to relate to Bell's theorem it needs to be corrected like this:

(1) is an inequality under the hypothesis that " is a number" (i.e., the symbol represents a number)". Then is a violation of (1) because is a number (albeit it's a complex one). QED.

So Bell's supporters to this day search madly for the place where Bell says the equivalent of, "Oh, folks, back there in 1964: sorry, I obviously meant a real number!" (NB: For then would not be a violation of (1) because is a complex number which is a category error, outside the given context.)

So your "simpler example" is erroneous, misleading, and an unintentional mask for Bell's error. FOR Bell makes an hypothesis under local-realism AND that hypothesis is refuted under local-realism; just as your corrected example is an hypotheses under numbers and is refuted under numbers.

The difference here is that you assume and do not search. For you write: "I will not attempt a definition of "local-realistic" here; the definition is (or should be) given in the hypotheses for the inequalities." Fred says you're "hand-waving", I just say "please point" and get off the fence that you're on via the use of this phrase: "… is (or should be) …"! HINT: Have a look at Bell's endorsement of d'Espagnat's (1979) naive realism in Bell's Bertlmann's socks article; a blunder if ever there was one!

However: not only models but experiments violate Bell's inequalities! So I'd welcome your explanation as to what part of a Bell-test experiment is not-local and/or not-real.
.

[FrediFizzx]

However: not only models but experiments violate Bell's inequalities!

Yes, they "violate" Bell's inequalities only by failed logic. Comparing apples to oranges. It truly amazes me that mathematicians put up with this mathematical nonsense.

[Gordon Watson]

However: not only models but experiments violate Bell's inequalities!

Yes, they "violate" Bell's inequalities only by failed logic. Comparing apples to oranges. It truly amazes me that mathematicians put up with this mathematical nonsense.

Comparing "apples and oranges" is a good analogy. Here's another; one that is more mathematically relevant to Bell's theorem:

We consider any Bellian inequality in the context of the following analogy:

Any Bellian inequality is akin to

(1) is an inequality under the hypothesis that " is a number" (i.e., the symbol represents a number).

And every refutation of a Bellian inequality is akin to:

Then is a violation of (1) because is a number (albeit it's a complex one). QED.

Reasoning via this analogy: Bellian inequalities are derived under the limiting "Bellian local naive realism" (BLNR).* They are refuted under the less limiting "local realism" (LR). For LR is to BLNR as a complex number is to a real. That is, LR can explain situations that BLNR cannot: just like imaginary numbers solve problems that the reals cannot.

* See Bell's endorsement of d'Espagnat's (1979) 'naive realism' in Bell's "Bertlmann's socks" article; a blunder if ever there was one!
.

[Mikko]

Rebuttal of guest1202's "simpler example" (with corrections): Please note that for guest1202's (1) to relate to Bell's theorem it needs to be corrected like this:

(1) is an inequality under the hypothesis that " is a number" (i.e., the symbol represents a number)". Then is a violation of (1) because is a number (albeit it's a complex one). QED.

This "correction" is incorrect. From x is a number you cannot infer that x² ≧ 0 unless you have a non-standard definition of "number". There are standard definitions for natural numbers and real numbers and many other numbers but not numbers. The example was already good so there is no way and no need to correct or improve it, and using an unstated non-standard definition wouldn't be the way even if there were one.

[Joy Christian]

Rebuttal of guest1202's "simpler example" (with corrections): Please note that for guest1202's (1) to relate to Bell's theorem it needs to be corrected like this:

(1) is an inequality under the hypothesis that " is a number" (i.e., the symbol represents a number)". Then is a violation of (1) because is a number (albeit it's a complex one). QED.

This "correction" is incorrect. From x is a number you cannot infer that x² ≧ 0 unless you have a non-standard definition of "number". There are standard definitions for natural numbers and real numbers and many other numbers but not numbers. The example was already good so there is no way and no need to correct or improve it, and using an unstated non-standard definition wouldn't be the way even if there were one.

The real issue here is that any Bell type inequality depends on an unphysical juxtaposition of three or four mutually exclusive incompatible experiments. Therefore to define the very notion of local realism on such an inequality is to put the cart before the horse. It is a complete lunacy. John Bell was a lot smarter intellectual than most of his mediocre followers (I had the privilege of meeting and discussing physics with Bell on several occasions). He did not define local realism on any inequality, but used Boole's inequality after he independently quantified Einstein's conception of local realism (as I have already noted), thus putting the horse before the cart:

[thray]

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.

No worries. I just interpret it as ignorance. I gave you a linear example of pairwise correlation. It fits Joy's framework, which is nonlinear.

It's become so painfully obvious that critics like you (and I can name many more) lack knowledge of elementary topology. Keep working on notation, and you'll find that by doing what you've always done, you'll get what you've always got. But it won't get you anywhere.

[Gordon Watson]

Rebuttal of guest1202's "simpler example" (with corrections): Please note that for guest1202's (1) to relate to Bell's theorem it needs to be corrected like this:

(1) is an inequality under the hypothesis that " is a number" (i.e., the symbol represents a number)". Then is a violation of (1) because is a number (albeit it's a complex one). QED.

This "correction" is incorrect. From x is a number you cannot infer that x² ≧ 0 unless you have a non-standard definition of "number". There are standard definitions for natural numbers and real numbers and many other numbers but not numbers. The example was already good so there is no way and no need to correct or improve it, and using an unstated non-standard definition wouldn't be the way even if there were one.

With respect:

(i) Your "correction" of my correction is incorrect. The inference that I use is analogous to Bell's erroneous adoption of naive realism (following d'Espagnat); ie, Bell neglects/forgets/overlooks a whole class of local realism that is not naive. (See the analogy in (iii) below.)

(ii) From you, here is classical bellian double-speak: "There are standard definitions for natural numbers and real numbers and many other numbers but not numbers." Please explain: How you can have " … many other numbers but not numbers"? Are there "dogs and cats and many other animals but not animals"?

(iii) My example was only intended to cover the main "number" systems: Primes ,
 Wholes ,
 Naturals ,
 Integers ,
 Irrationals ,
 Rationals ,
 Reals , Complex : with Bell (analogously) overlooking .
.

[Joy Christian]

***
In addition to my previous posts addressing the concerns of "guest1202", it is also worth noting that, given the following relation specified in Eq. (16) of this paper,



the following identity holds



which can be verified by recalling that all unit bivectors square to -1. This again demonstrates that Eq. (31) of the above paper immediately follows from Eq. (27).

***

[FrediFizzx]

I want to correct an oversight.
A post titled "Is it impossible to violate the Bell-CH inequallities", concluded:

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.

This is all fine except for the very last assertion that "p1(a' ) "can be chosen arbitrarily". I had overlooked a consistency condition which requires that p1(a' ) = 1/2. The same condition requires that all of the probabilities be 1/2 except that p12(a, b' ) = 0. (To understand why p12(a, b' ) is different, recall that p12(a, b' ) is the probability that Alice and Bob both obtain result +1, but the condition A = (not B' ) says that they never both obtain the same result.) That is,

p1(a) = p1(a' ) = p1(b) = p1(b' ) = p12(a,b) = p12(a' , b) = p12(a' , b' ) = 1/2, but p12(a, b' ) = 0.

The counterexample is still valid under these choices of the probabilities, and even easier to check:
the middle term is 1/2 > 0.

In another reply, someone asked how one can obtain these probabilities from quantum mechanics (QM). One can't; QM does not allow Popescu/Rohlich (PR) boxes. But QM does allow what I called "probabilistic PR boxes", as discussed in a previous post on PR boxes (my post discussing the Kwiat/Hardy "quantum cakes" paper). A "probabilistic PR box" has the same (classically) contradictory features as a PR box, but one of the four defining conditions,
A = B, B = A', A' = B', B' = (not A),
only occurs with some nonzero probability (for Kwiat/Hardy, that probability is about .09) instead of occurring with probability 1 as in a genuine PR box. ...

****
You are skirting what I asked you to do. Prove that B = A' = B'. Otherwise you are just demonstrating handwavy junk that is typical of most Bell fans.

What you have done is shift to an inequality of the CH form where the upper bound is 1 not 0.

****

From what we learned on the CH inequality thread, the actual absolute upper bound for CH for independent terms is 1/2 not 1.