Bell's theorem refuted via elementary probability theory

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

Re: Bell's theorem refuted via elementary probability theory

Postby Esail » Sat Jan 02, 2021 2:45 am

Gordon Watson wrote:
Esail wrote:
Gordon Watson wrote:How then do you move, in a mathematically correct way, from Bell's 1964:(14a) to his (14b)?

Gordon:

With A(b,lambda) *A(b,lambda)=1 see Bell's eq.(1) you will easily arrive from Bell's 14a at 14b.
With A(a,lambda)*A(b,lambda) <1 you will arrive from 14b at 14c provided A(a,lambda) exists.
But A(a,lambda) exists unambiguously only with noncontextual models. Thus, with a noncontextual model Bell's equation is correct but does not apply to QM.

Eugen


Thanks Eugen, I should have been more specific. We accept the truth of Bell (14a) but differ over the truth of (14b).

So please show me the (14aa), (14ab), (14ac) — say, if you need 3 steps — that YOU use to arrive at (14b).
.


Gordon:

The derivation of equation 15 from equations 1 and 14a, b, c is simple school math. You can try yourself with the explanations I gave you. Where is the problem?
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Sat Jan 02, 2021 2:59 am

Better proof still [and correction of earlier mistakes!]: consider three random variables X, Y, Z which take values +/-1 only. In any one realisation x, y, z of the triple, there must be two which are equal to one another (if not all three). You can't have inequality simultaneously in every pair. The event {X = Y} OR {Y = Z} OR {Z = X} is certain. The three events overlap, and cover all possibilities. Hence Pr{X = Y} + Pr{Y = Z} + Pr{X = Z} cannot be less than 1. That's the elementary probability theory behind Bell's 1964 inequalities. One can convert an inequality in terms of probabilities to an inequality in terms of correlations by substituting (1 + E(XY))/2 for P(X = Y). One can obtain a total of 2 x 2 x 2 = 8 inequalities by flipping the signs of some or all of X, Y and Z. Some of those inequalities turn out to be equivalent to one another: flipping signs of all three changes nothing; flipping signs of 2 is the same as flipping the sign of the third, only. That means there are essentially four inequalities, not six as I wrote earlier. Sorry. Too late to edit that posting.

Similarly, CHSH is got by playing similar games with four variables, W, X, Y, Z instead of three. Looking at the sides of the square
Code: Select all
W – X
|   |
Z – Y

it's clear that the number of equalities among W = X (top side), X = Y (right hand side), Y = Z (bottom side), Z = W (left hand side) must be even. In particular, three equalities imply the fourth. The event {Z = W} is contained in the union of {W = X}, {X = Y}, {Y = Z}. So Pr(Z = W) cannot exceed Pr(W = X) + Pr(X = Y) + Pr(Y = Z).

Take opposite corners W and Y to be Alice's two measurement outcomes, and opposite corners X and Z to be Bob's two measurement outcomes. Convert probabilities of equality to correlations, and we have one of the 8 one-sided CHSH inequalities. Flipping some of the signs of some of the variables generates a total of 16 one-sided inequalities but there are only 8 different ones.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sat Jan 02, 2021 5:43 pm

Esail wrote:
Gordon Watson wrote:
Esail wrote:
Gordon Watson wrote:How then do you move, in a mathematically correct way, from Bell's 1964:(14a) to his (14b)?

Gordon:

With A(b,lambda) *A(b,lambda)=1 see Bell's eq.(1) you will easily arrive from Bell's 14a at 14b.
With A(a,lambda)*A(b,lambda) <1 you will arrive from 14b at 14c provided A(a,lambda) exists.
But A(a,lambda) exists unambiguously only with noncontextual models. Thus, with a noncontextual model Bell's equation is correct but does not apply to QM.

Eugen


Thanks Eugen, I should have been more specific. We accept the truth of Bell (14a) but differ over the truth of (14b).

So please show me the (14aa), (14ab), (14ac) — say, if you need 3 steps — that YOU use to arrive at (14b).
.


Gordon:

The derivation of equation 15 from equations 1 and 14a, b, c is simple school math. You can try yourself with the explanations I gave you. Where is the problem?


Eugen,

For me, a problem is generally a deviation from an expectation! And I expected that you would happily show me YOUR "simple school math".

For we might then discuss why YOUR "simple school math" is swamped by my high-school math (which delivers an irrefutable inequality that swamps Bell's).

HTH.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sat Jan 02, 2021 6:01 pm

gill1109 wrote:X1 = X2 and X2 = X3 => X1 = X3
X1 != X3 => X1 != X2 or X2 != X3
Pr(X1 != X3) </= Pr(X1 != X2) + Pr(X2 != X3)
2 * Pr(X1 != X3) </= 2 * Pr(X1 != X2) + 2 * Pr(X2 != X3)
1 + 1 - 2 * Pr(X1 != X3) >/= 1 - 2 * Pr(X1 != X2) + 1 - 2 * Pr(X2 != X3)
1 + E(X1 * X3) >/= E(X1 * X2) + E(X2 * X3)

[the three variables take only the values +/-1, for such r.v.s, E(XY) = Pr(X == Y) - Pr(X != Y) while Pr(X == Y) + Pr(X != Y) = 1]


Richard,

Given your revised version, I've wiped my pained reply to the above. But I leave two notes (as questions):

1. As a mathematician, can you use LaTeX?

2. Aren't we dealing with quantum observables under EPRB?

EDIT; draft footnote c/fwd:

Footnote: With his move from (14a) to (14b), Bell creates an inequality that is satisfied by , etc.

So, in the quantum context known as EPRB, Bell "delivers" an inequality which exceeds a related classical limit of and falls short of the quantum limit , etc.

Against this, via high-school math, my irrefutable inequalities hold classically and quantum-mechanically. So Bell's inequality lands in no-man's land.

(With consequences that I suspect some of his supporters miss.)

I thus seek suggestions re the best terminology to define Bell's error.
.
Last edited by Gordon Watson on Sat Jan 02, 2021 6:25 pm, edited 1 time in total.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Sat Jan 02, 2021 6:21 pm

Gordon Watson wrote:
gill1109 wrote:X1 = X2 and X2 = X3 => X1 = X3
X1 != X3 => X1 != X2 or X2 != X3
Pr(X1 != X3) </= Pr(X1 != X2) + Pr(X2 != X3)
2 * Pr(X1 != X3) </= 2 * Pr(X1 != X2) + 2 * Pr(X2 != X3)
1 + 1 - 2 * Pr(X1 != X3) >/= 1 - 2 * Pr(X1 != X2) + 1 - 2 * Pr(X2 != X3)
1 + E(X1 * X3) >/= E(X1 * X2) + E(X2 * X3)

[the three variables take only the values +/-1, for such r.v.s, E(XY) = Pr(X == Y) - Pr(X != Y) while Pr(X == Y) + Pr(X != Y) = 1]


Richard,

Given your revised version, I've wiped my pained reply to the above. But I leave two notes (as questions):

1. As a mathematician, can you use LaTeX?

2. Aren't we dealing with quantum observables under EPRB?
.

1. Yes I use LaTeX all the time but mathematicians also develop linear formula writing for quick communication without recourse to expensive type-setting. I do not send people emails as pdf documents, lovingly typeset with lots of $...$ and even $$...$$. LaTeX formulas are computer code. The bulletin board creates bitmapped images for each formula. It’s pretty gruesome technology. I have respect for it, but I don’t love it.

2. No we are not dealing with quantum observables under EPRB. We are dealing with an underlying theory of classical type which explains the quantum correlations without recourse to quantum theory mumbo jumbo. We are dealing with a deterministic mechanistic explanation “behind the scenes” which just gives the illusion of quantum magic. The conclusion is that we can’t fake quantum magic. It can’t be done. It seems to be for real.

Surprise, surprise. The mysteries of nature are explained (? Well, maybe not “explained”, but they are perfectly and economically predicted by the non-commutation of Hilbert space operators).
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Re: Bell's theorem refuted via elementary probability theory

Postby Esail » Tue Jan 05, 2021 12:22 am

gill1109 wrote:Bell defines A(a, lambda) to be the result “A” of measuring the observable “sigma_1 cdot a” when, just before measurement, the state of particles and measurement devices is lambda.


lambda cannot be the full status of the measurement devices as lambda is fixed after the particles have left the source. If the settigs of the instruments are changed (see Weihs doctoral thesis) after the particles have left the source but before they hit the detector a new context is established not described by lambda.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Tue Jan 05, 2021 12:40 am

gill1109 wrote:Better proof still [and correction of earlier mistakes!]: consider three random variables X, Y, Z which take values +/-1 only. ... ...


Richard,

Since we cannot both be right, I want to understand your "proofs" from their opening assumptions to the concluded inequalities.

So please complete your "better proofs" by taking them through to Bell's 1964 inequality and the CHSH-Bell inequality.
.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Tue Jan 05, 2021 9:34 pm

Gordon

Then please let’s go through the elementary probability proof which I quote below. That’s the pure math part. Preparatory to that, one motivates *what* X, Y, Z represent. That’s the physics part.

Bell’s 1964 argument is different from, and simpler than, the mature (Bertlmann’s socks) argument,
In 1964 he used perfect anticorrelation to prove realism.
The reason: 1964 is only about theories (QM vs. LR). Later, post CHSH, is about experiment (nature vs. LR).

Later he assumed realism as well as locality.

His aim is to disprove locality + realism by obtaining a contradiction. So his “assumptions” are what Joy Christian would call “straw men”.

Let’s agree on the pure math first.

gill1109 wrote:Better proof still [and correction of earlier mistakes!]: consider three random variables X, Y, Z which take values +/-1 only. In any one realisation x, y, z of the triple, there must be two which are equal to one another (if not all three). You can't have inequality simultaneously in every pair. The event {X = Y} OR {Y = Z} OR {Z = X} is certain. The three events overlap, and cover all possibilities. Hence Pr{X = Y} + Pr{Y = Z} + Pr{X = Z} cannot be less than 1. That's the elementary probability theory behind Bell's 1964 inequalities. One can convert an inequality in terms of probabilities to an inequality in terms of correlations by substituting (1 + E(XY))/2 for P(X = Y). One can obtain a total of 2 x 2 x 2 = 8 inequalities by flipping the signs of some or all of X, Y and Z. Some of those inequalities turn out to be equivalent to one another: flipping signs of all three changes nothing; flipping signs of 2 is the same as flipping the sign of the third, only. That means there are essentially four inequalities, not six as I wrote earlier. Sorry. Too late to edit that posting.

Similarly, CHSH is got by playing similar games with four variables, W, X, Y, Z instead of three. Looking at the sides of the square
Code: Select all
W – X
|   |
Z – Y

it's clear that the number of equalities among W = X (top side), X = Y (right hand side), Y = Z (bottom side), Z = W (left hand side) must be even. In particular, three equalities imply the fourth. The event {Z = W} is contained in the union of {W = X}, {X = Y}, {Y = Z}. So Pr(Z = W) cannot exceed Pr(W = X) + Pr(X = Y) + Pr(Y = Z).

Take opposite corners W and Y to be Alice's two measurement outcomes, and opposite corners X and Z to be Bob's two measurement outcomes. Convert probabilities of equality to correlations, and we have one of the 8 one-sided CHSH inequalities. Flipping some of the signs of some of the variables generates a total of 16 one-sided inequalities but there are only 8 different ones.


If you are interested in Bell ‘64 then I can recommend to you our new paper
https://arxiv.org/abs/2012.00719
And slides of a talk on it
https://www.math.leidenuniv.nl/~gill/gull-talk.pdf
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Wed Jan 06, 2021 8:23 am

Esail wrote:
gill1109 wrote:Bell defines A(a, lambda) to be the result “A” of measuring the observable “sigma_1 cdot a” when, just before measurement, the state of particles and measurement devices is lambda.


lambda cannot be the full status of the measurement devices as lambda is fixed after the particles have left the source. If the settings of the instruments are changed (see Weihs doctoral thesis) after the particles have left the source but before they hit the detector a new context is established not described by lambda.

Dear Eugen

We may imagine "lambda" as "fixed" [in a mathematical model having the properties of locality and realism] just *before* the particles leave the source: it is *by definition* the combined state of source and communication channels and detectors and everything else that will play a role regarding the possible future measurements *at that moment*.

The state of Alice's detector *after* setting "a" has been chosen by her still depends only on "a" and on "lambda".

The state of the detector as it interacts with the particle as it arrives at the detector still depends only on "a" and on "lambda".

The final outcome of the measurement there depends only on "a" and on "lambda".

Richard
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Re: Bell's theorem refuted via elementary probability theory

Postby Esail » Wed Jan 06, 2021 8:44 am

gill1109 wrote:The state of Alice's detector *after* setting "a" has been chosen by her still depends only on "a" and on "lambda".

The state of the detector as it interacts with the particle as it arrives at the detector still depends only on "a" and on "lambda".

The final outcome of the measurement there depends only on "a" and on "lambda".

Richard


Richard:

If the measurement context depends on "a" and "b" as is the case with contextual models it cannot depend on "lambda" as "lambda" does neither depend on "a" nor on "b".

Eugen
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Wed Jan 06, 2021 9:02 am

Esail wrote:
gill1109 wrote:The state of Alice's detector *after* setting "a" has been chosen by her still depends only on "a" and on "lambda".

The state of the detector as it interacts with the particle as it arrives at the detector still depends only on "a" and on "lambda".

The final outcome of the measurement there depends only on "a" and on "lambda".

Richard


Richard:

If the measurement context depends on "a" and "b" as is the case with contextual models it cannot depend on "lambda" as "lambda" does neither depend on "a" nor on "b".

Eugen

Under Bell's local realism (leading to CHSH inequality etc), the measurement context for Alice's measurement is a part of "lambda", together with the setting which was freely inserted from outside, "a". When that happens, all kinds of things can change in the measurement device. What happens will depend on what is already there, relevant parts of "lambda", and the change which was made, namely, someone from outside turning a dial to a position "a".

Alice's measurement depends on "lambda" and on "a"
Bob's on "lambda" and "b".

Seems you have a different definition of "contextual" from everybody else.

If what happens in Bob's lab also depends on the position to which Alice turns her dial, then OK this is contextual too, and the context of Bob's measurement is "a", "b" and "lambda". I would call that "non-local".
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Thu Jan 07, 2021 12:22 am

gill1109 wrote:Gordon

Then please let’s go through the elementary probability proof which I quote below. That’s the pure math part. Preparatory to that, one motivates *what* X, Y, Z represent. That’s the physics part.

Bell’s 1964 argument is different from, and simpler than, the mature (Bertlmann’s socks) argument,
In 1964 he used perfect anticorrelation to prove realism.
The reason: 1964 is only about theories (QM vs. LR). Later, post CHSH, is about experiment (nature vs. LR).

Later he assumed realism as well as locality.

His aim is to disprove locality + realism by obtaining a contradiction. So his “assumptions” are what Joy Christian would call “straw men”.

Let’s agree on the pure math first.

gill1109 wrote:Better proof still [and correction of earlier mistakes!]: consider three random variables X, Y, Z which take values +/-1 only. In any one realisation x, y, z of the triple, there must be two which are equal to one another (if not all three). You can't have inequality simultaneously in every pair. The event {X = Y} OR {Y = Z} OR {Z = X} is certain. The three events overlap, and cover all possibilities. Hence Pr{X = Y} + Pr{Y = Z} + Pr{X = Z} cannot be less than 1. That's the elementary probability theory behind Bell's 1964 inequalities. One can convert an inequality in terms of probabilities to an inequality in terms of correlations by substituting (1 + E(XY))/2 for P(X = Y). One can obtain a total of 2 x 2 x 2 = 8 inequalities by flipping the signs of some or all of X, Y and Z. Some of those inequalities turn out to be equivalent to one another: flipping signs of all three changes nothing; flipping signs of 2 is the same as flipping the sign of the third, only. That means there are essentially four inequalities, not six as I wrote earlier. Sorry. Too late to edit that posting.

Similarly, CHSH is got by playing similar games with four variables, W, X, Y, Z instead of three. Looking at the sides of the square
Code: Select all
W – X
|   |
Z – Y

it's clear that the number of equalities among W = X (top side), X = Y (right hand side), Y = Z (bottom side), Z = W (left hand side) must be even. In particular, three equalities imply the fourth. The event {Z = W} is contained in the union of {W = X}, {X = Y}, {Y = Z}. So Pr(Z = W) cannot exceed Pr(W = X) + Pr(X = Y) + Pr(Y = Z).

Take opposite corners W and Y to be Alice's two measurement outcomes, and opposite corners X and Z to be Bob's two measurement outcomes. Convert probabilities of equality to correlations, and we have one of the 8 one-sided CHSH inequalities. Flipping some of the signs of some of the variables generates a total of 16 one-sided inequalities but there are only 8 different ones.


If you are interested in Bell ‘64 then I can recommend to you our new paper
https://arxiv.org/abs/2012.00719
And slides of a talk on it
https://www.math.leidenuniv.nl/~gill/gull-talk.pdf


Let's not get sidetracked. In accord with your offer, and since they do not require many lines:

Please provide your complete "better way" derivation of the Bell (1964) and the CHSH-Bell inequalities,

That is: from your opening assumptions to their final forms (which include absolute-brackets).

For I would like to be able to say:

1. Here's Richard Gill's "better way" derivation of Bell's (1964) inequality and here's my acceptance.**

2. Here's Richard Gill's "better way" derivation of the CHSH-Bell inequality and here's my acceptance.**

** If your "better way" is valid.
.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Thu Jan 07, 2021 2:15 am

Gordon Watson wrote:Let's not get sidetracked. In accord with your offer, and since they do not require many lines:

Please provide your complete "better way" derivation of the Bell (1964) and the CHSH-Bell inequalities,

That is: from your opening assumptions to their final forms (which include absolute-brackets).

For I would like to be able to say:

1. Here's Richard Gill's "better way" derivation of Bell's (1964) inequality and here's my acceptance.**

2. Here's Richard Gill's "better way" derivation of the CHSH-Bell inequality and here's my acceptance.**

** If your "better way" is valid.

Nope. Let's derive *my* four Bell 1964 inequalities, and *my* 8 Bell-CHSH inequalities (and also prove that they are not only necessary but also sufficient). That part is elementary probability theory, and it's fundamental.

The fact that the set of three Bell 1964 three-correlations inequalities are a N&SC for local realism was proven by Boole (1853), Bass (1955), Vorob'ev (1962); and for the four-correlation inequality of CHSH by Fine (1982).

This small number of simpler inequalities is absolutely fundamental to understanding the whole field. They are immediate and trivial and transparent and generalisable to more complex situations, e.g. the p x q x r case: p parties, each with q settings, and such that each party-setting combination has r outcomes.

I would say that it is an exercise to the student (you) to derive from them the less transparent, more ugly, inequalities which you are obsessing about. It's called obfuscation. Take something simple, make it more complicated, and then see difficulties which are not there at all.

We can talk separately about the physics assumptions on which Bell based his elementary probability theory.

But let's agree on some simple elementary probability theory first.

By the way, we can also do CHSH first. Then suppose that one of the four correlations is +/-1. The 8 one-sided CHSH four correlation inequalities will collapse to the 4 one-sided Bell (1964) three correlation inequalities.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Thu Jan 07, 2021 4:27 am

Suppose four numbers are such that the sum of any three of them, minus the fourth, lies between -2 and +2
Altogether: 8 one-sided inequalities, right?
Then
x + y + z - w < 2
x + y + w - z < 2
Hence
x + y + |w -z| < 2
Also
-2 < x + y + z - w
-2 < x + y +w - z
Hence
-2 < x + y - |z - w|
Hence
- (x + y) + |z - w| < 2
Hence
|x + y| + |z - w| < 2

Now suppose x = 1

Then the third inequality above becomes
|w - z| < 1 - y
This is a Bell three correlation inequality
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Fri Jan 08, 2021 5:15 am

gill1109 wrote:
[GW edit: The opening supposition.] Suppose four numbers are such that the sum of any three of them, minus the fourth, lies between -2 and +2
Altogether: 8 one-sided inequalities, right?
Then
x + y + z - w < 2
x + y + w - z < 2
Hence
x + y + |w -z| < 2
Also
-2 < x + y + z - w
-2 < x + y +w - z
Hence
-2 < x + y - |z - w|
Hence
- (x + y) + |z - w| < 2
Hence
|x + y| + |z - w| < 2 [GW edit: This is the CHSH-Bell inequality.]

Now suppose x = 1

Then the third inequality above becomes
|w - z| < 1 - y
This is a Bell three correlation inequality. [GW edit: This is Bell's 1964 inequality.]

gill1109
Mathematical Statistician


Thanks Richard,

1. We agree: the opening supposition leads to the CHSH-Bell inequality; which in turn leads to Bell's 1964 inequality.

2. However, in the context that Bell first chose (ie, EPRB), the Bell-based erroneousness*** of this supposition underlies the silliness of the above inequalities; a silliness which Bell half-expected in 1990 but never saw.

3. So, consistent with the theme of this thread — ie, using [unconstrained] elementary probability theory — let's see.

4. Let's consider four expectations like those in eqn (11) of https://vixra.org/pdf/2010.0068v5.pdf. (A draft I have yet to revise in line with some of the discussion in this thread. But such will not affect the essence of what follows.) Thus:



5. Then, using nothing but high-school math as in eqns (12) to (15) in

https://vixra.org/pdf/2010.0068v5.pdf

. (A1), say.

6. And via similar math we find:

. (A2), say.

7. So, via the LHS of your formulation of the CHSH-Bell inequality; as above:

+ . (A3) QED.

*** And thus is Bellian-silliness exposed. (Again.) For that opening supposition — with its Bell-based error — suppresses two vital RHS terms in (A3); vital in that irrefutable (A3) is valid for any experiment (classical or quantum) that satisfies #4 above.

8. Finally, let the angle .

Then (A3) reduces to my prior refutation of Bell's famous (1964) inequality:

; (A4) QED.

(A4) being eqn (15) in the draft https://vixra.org/pdf/2010.0068v5.pdf

9. In sum: consistent with the theme of this thread. (i) Elementary probability theory here refutes Bell's 1964 inequality and the CHSH-Bell inequality. (ii) However, since the refutation of a defective proof does not necessarily refute a related theorem: the above draft — via elementary probability theory alone — refutes Bell's theorem first via eqns (1)-(9). (iii) Thanks to some good discussion here (at Fred's), that section will be strengthened in the next draft.

.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Fri Jan 08, 2021 5:34 am

Gordon, you are still completely blind to the main point. Bell’s elementary probability follows from some elementary but controversial physical assumptions (nowadays called local realism). You (like Bohr) disagree with those physics assumptions. Nature violates the inequalities, hence Bell’s (correct, elementary) math and logic proves that the physics assumptions of local realism are indeed, as you say, naïeve. Not just naïeve: they must be wrong. You, me, and Bell agree on this.

Bell is not silly. But lots of people are too silly to follow his argument.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Fri Jan 08, 2021 8:11 am

Gordon Watson wrote:
gill1109 wrote:
[GW edit: The opening supposition.] Suppose four numbers are such that the sum of any three of them, minus the fourth, lies between -2 and +2


Gordon, what *you* now call my “opening supposition” was merely a conclusion which I derived earlier from the probabilistic interpretation of Bell’s opening physics supposition.

Bell’s truly opening *physics* supposition is called (by me) *local realism*. We haven’t talked about it yet. You don’t have to agree with it. (Bell doesn’t believe it either. Nor do I. But Einstein would have taken it on board. Bohr on the other hand, not).

In this posting I will just summarise (yet again) the elementary probability which Bell in effect applies to his physics supposition. *After* you have agreed with it, we can discuss the reasoning behind the classical physics set-up. (I repeat - you don’t have to agree with it. We are not a priori assuming EPR-B or QM).

In an earlier post:
viewtopic.php?f=6&t=451&p=12608#p12593
I showed you how to derive eight one-sided Bell-CHSH four-correlation inequalities from local realism, using elementary probability theory. This is that fundamental, elementary, pure probability core:

LEMMA 1. Derivation of Bell type inequalities after we have assumed existence of functions A(a, lambda) and B(b, lambda) with the usual properties.
gill1109 wrote:Consider three random variables X, Y, Z which take values +/-1 only. In any one realisation x, y, z of the triple, there must be two which are equal to one another (if not all three). You can't have inequality simultaneously in every pair. The event {X = Y} OR {Y = Z} OR {Z = X} is certain. The three events overlap, and cover all possibilities. Hence Pr{X = Y} + Pr{Y = Z} + Pr{X = Z} cannot be less than 1. That's the elementary probability theory behind Bell's 1964 inequalities. One can convert an inequality in terms of probabilities to an inequality in terms of correlations by substituting (1 + E(XY))/2 for P(X = Y). One can obtain a total of 2 x 2 x 2 = 8 inequalities by flipping the signs of some or all of X, Y and Z. Some of those inequalities turn out to be equivalent to one another: flipping signs of all three changes nothing; flipping signs of 2 is the same as flipping the sign of the third, only. That means there are essentially four inequalities.
Similarly, CHSH is got by playing similar games with four variables, W, X, Y, Z instead of three. Looking at the sides of the square
Code: Select all
W – X
|   |
Z – Y

it's clear that the number of equalities among W = X (top side), X = Y (right hand side), Y = Z (bottom side), Z = W (left hand side) must be even. In particular, three equalities imply the fourth. The event {Z = W} is contained in the union of {W = X}, {X = Y}, {Y = Z}. So Pr(Z = W) cannot exceed Pr(W = X) + Pr(X = Y) + Pr(Y = Z).
Take opposite corners W and Y to be Alice's two measurement outcomes, and opposite corners X and Z to be Bob's two measurement outcomes. Convert probabilities of equality to correlations, and we have one of the 8 one-sided CHSH inequalities. Flipping some of the signs of some of the variables generates a total of 16 one-sided inequalities but there are only 8 different ones.


COROLLARY. We can get the 8 one-sided inequalities for correlations from the 8 one-sided inequalities for probabilities of the previous lemma, by simple rewriting using E(XY) = 2 Pr(X = Y) - 1, etc.

In a later post
viewtopic.php?f=6&t=451&p=12608#p12604
in response to your repeatedly expressed wish, I derived the CHSH inequality in that horrible form which you are so attached to (with two “absolute value” bracketed expressions) from the earlier derived eight one-sided Bell-CHSH four-correlation inequalities of Lemma 1 and it’s corollary.

LEMMA 2. Bringing in those absolute values.
gill1109 wrote:Suppose four numbers are such that the sum of any three of them, minus the fourth, lies between -2 and +2.
Then
x + y + z - w < 2
x + y + w - z < 2
Hence
x + y + |w -z| < 2
Also
-2 < x + y + z - w
-2 < x + y +w - z
Hence
-2 < x + y - |z - w|
Hence
- (x + y) + |z - w| < 2
Hence
|x + y| + |z - w| < 2
Now suppose x = 1.
Then the third inequality above becomes
|w - z| < 1 - y
This is a Bell three correlation inequality


The four numbers x, y, z, w are the four correlations E(XY) = 2 Pr{X = Y} -1 etc. which we get from the earlier posting, plus the remark about conversion from probabilities of equality to correlations.

What we did not yet discuss is where the binary random variables X, Y, Z, W come from. They come from naïeve physical assumptions called “local realism”.

If you will ever be able to bring yourself to focus on *my* elementary probability, and confirm those two lemma’s, then, and only then, is there any point in discussing physics and EPR-B.

NOTE. Bell does *not* assume the usual QM modelling which we call EPR-B nowadays. He assumes a classical physical description of an experiment with binary measurements being made, with binary settings, in two separated labs. He asks the question: could classical physics generate the same correlations as those which QM predicts for EPR-B. His answer is no. You agree with him. End of story. He proves rigorously something which you find completely natural (not in need of explanation).
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sat Jan 30, 2021 1:41 pm

Richard,

Apologies for this delayed reply. Among other things, I've been trying to make sense of your math, etc. Including the Boole-based work that you repeat (the post above) after I refuted it (3 posts above).

gill1109 wrote:Gordon, you are still completely blind to the main point. Bell’s elementary probability follows from [?? GW emphasis] some elementary but controversial physical assumptions (nowadays called local realism).


Let's identify this as "Richard Gill's main point". And let's recall that I have already [3 posts above], refuted "Gill's better-way" of probabilistically deriving the Bell and CHSH-Bell inequalities.

As for Bell: I say that "elementary probability" begins best with expectations: the averages that we learn about in primary school.

And surely that's where QM excels and Bell begins: see Bell 1964:(2)! And surely you've seen that that's where I continue when refuting your "better-way"?

Thus, in short: "Richard Gill's main point" is back-to-front and quite silly. For, rather than following from some elementary but controversial physical assumptions (nowadays called local realism), Bell’s "elementary probability" begins correctly ––– via (2) and (14a) –– with two valid expectations: AND ––– independent of any physical assumptions ––– continues erroneously to invoke a third expectation in the next line, (14b).

So, with Bell's first error being his move from (14a) to (14b): please explain the "controversial physical assumptions" that Bell uses. It's not enough to say "local realism" without you define each term. Further, if you say that Bell uses "naive realism" — how is it that Bell and Bellians are still on the horns of a dilemma 25 years later? (See endnote below.) Kids have normally rejected "naive realism" by the age of 10.

gill1109 wrote:You (like Bohr) disagree with those physics assumptions.


I've said before: I disagree with Bohr on some related issues. So please specify the Bohrian assumption that you think I agree with.

gill1109 wrote:Nature violates the inequalities, hence Bell’s (correct, elementary) math and logic proves that the physics assumptions of local realism are indeed, as you say, naïeve.


When you define Bell's version of "local realism" we might agree on Bell's naivety. BUT I show that "Bell’s (correct, elementary) math and logic" is FALSE, independent of any theorising.

gill1109 wrote: Not just naïeve: they must be wrong. You, me, and Bell agree on this.


They are, indeed, wrong. So let's get it clear about Bell's naivety; see above. For, as I see it: I disagree with you [your "better-way" is false] and I disagree with Bell [his MATH is false; his 1990 dilemma rather silly (see below)].

gill1109 wrote:Bell is not silly. But lots of people are too silly to follow his argument.


Maybe.

In Bell's words, 25 years after the publication of his theorem, with my emphasis:

p.5: ‘I cannot say that action at a distance (AAD is required in physics. But I can say that you cannot get way with no AAD. You cannot separate off what happens in one place and what happens in another. Somehow they have to be described and explained jointly.’ p.6: ‘The Einstein program fails, that’s too bad for Einstein, but should we worry about that? So what? ... it might be that we have to learn to accept not so much AAD, but the inadequacy of no AAD.’ p.7: ‘And that is the dilemma. We are led by analysing this situation to admit that in somehow distant things are connected, or at least not disconnected. ... So the connections have to be very subtle, and I have told you all that I know about them.’ p.9: ‘It’s my feeling that all this AAD and no AAD business will go the same way [as the ether]. But someone will come up with the answer, with a reasonable way of looking at these things. If we are lucky it will be to some big new development like the theory of relativity. Maybe someone will just point out that we were being rather silly, and it won’t lead to a big new development. But anyway, I believe the questions will be resolved.’ p.10: ‘I think somebody will find a way of saying that [relativity and QM] are compatible. For me it’s very hard to put them together, but I think somebody will put them together, and we’ll just see that my imagination was too limited.’ p.12: ‘I don’t know any conception of locality that works with QM. So I think we’re stuck with nonlocality.’ p.13: ‘... I step back from asserting that there is AAD, and I say only that you cannot get away with locality. You cannot explain things by events in their neighbourhood. But I am careful not to assert that there is AAD,’ after Bell seminar (1990) http://www.quantumphil.org./Bell-indeterminism-and-nonlocality.pdf.


.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Sat Jan 30, 2021 7:21 pm

Gordon, you think you have refuted something; I don’t. You merely have made weaker assumptions than Bell, and derived a weaker conclusion. That doesn’t prove anything. It is completely unsurprising!

You want a definition of local realism? “Realism” is Counterfactual Definiteness. “Local” is Relativistic Local Causality. Read the Bertlmann’s socks paper by Bell, https://hal.archives-ouvertes.fr/jpa-00220688/document , and read Tsirelson’s Citizendium article on quantum entanglement, https://en.citizendium.org/wiki/Quantum_entanglement
Or read my paper (with Dilara Karakozak) on Gull’s proof of Bell’s theorem, https://arxiv.org/abs/2012.00719
Actually, there should be a new version on arXiv in about a week, it should be more easy to read (better organised).

Today I would say that local realism is the idea that the part of the world as we experience it can be modelled as a probabilistic cellular automaton. Think of Conway’s game of life but allow the rules to be stochastic. 3D space is thought of as made up of tiny adjacent cells. Each cell has a state. Time runs in tiny discrete time-steps. At each time-step, the state of each cell is updated according to a rule which fixes the probability of each possible new state, given the present states of the cell in question and its neighbours.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Mon Feb 01, 2021 1:32 pm

gill1109 wrote:Gordon, you think you have refuted something; I don’t. You merely have made weaker assumptions than Bell, and derived a weaker conclusion. That doesn’t prove anything. It is completely unsurprising!

You want a definition of local realism? “Realism” is Counterfactual Definiteness. “Local” is Relativistic Local Causality. Read the Bertlmann’s socks paper by Bell, https://hal.archives-ouvertes.fr/jpa-00220688/document , and read Tsirelson’s Citizendium article on quantum entanglement, https://en.citizendium.org/wiki/Quantum_entanglement
Or read my paper (with Dilara Karakozak) on Gull’s proof of Bell’s theorem, https://arxiv.org/abs/2012.00719
Actually, there should be a new version on arXiv in about a week, it should be more easy to read (better organised).

Today I would say that local realism is the idea that the part of the world as we experience it can be modelled as a probabilistic cellular automaton. Think of Conway’s game of life but allow the rules to be stochastic. 3D space is thought of as made up of tiny adjacent cells. Each cell has a state. Time runs in tiny discrete time-steps. At each time-step, the state of each cell is updated according to a rule which fixes the probability of each possible new state, given the present states of the cell in question and its neighbours.


Richard,

1. In my experience, weaker assumptions lead to stronger conclusions. And that's what my refutation of Bell's theorem, and his inequality, show.

2. I expect to have a revised version of https://vixra.org/pdf/2010.0068v5.pdf -- Bell's theorem refuted via elementary probability theory -- up at viXra.org this week (and will announce it here).

3. In the meantime, you might like to refute the 4 high-school-level equations, (15)-(17), that refute Bell's famous inequality. They remain unchanged.

4. It will be a positive step when we each have an improved version of our ideas as the basis for future discussion.

PS: Please, if you cannot refute those 4 high-school-level equations, (15)-(17) -- say, 5 minutes work -- let me know.

All the best,

Gordon
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