by 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#p12593I 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#p12604in 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).
[quote="Gordon Watson"][quote="gill1109"]
[GW edit: [u]The opening supposition[/u].] Suppose four numbers are such that the sum of any three of them, minus the fourth, lies between -2 and +2
[/quote]
[/quote]
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:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=451&p=12608#p12593
I showed you how to derive [b]eight one-sided Bell-CHSH four-correlation inequalities[/b] from local realism, using elementary probability theory. This is that fundamental, elementary, pure probability core:
LEMMA 1. [i]Derivation of Bell type inequalities after we have assumed existence of functions A(a, lambda) and B(b, lambda) with the usual properties[/i].
[quote="gill1109"]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]
W – X
| |
Z – Y
[/code]
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.[/quote]
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
http://www.sciphysicsforums.com/spfbb1/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 [b]eight one-sided Bell-CHSH four-correlation inequalities[/b] of Lemma 1 and it’s corollary.
LEMMA 2.[i] Bringing in those absolute values.[/i]
[quote="gill1109"]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[/quote]
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. [i]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).[/i]